From e650ed1e1e908e51c78c1b047bec0da7c4fea366 Mon Sep 17 00:00:00 2001 From: sotech117 Date: Sat, 27 Apr 2024 04:25:23 -0400 Subject: testing --- .DS_Store | Bin 6148 -> 10244 bytes examples/FallingBall3.jl | 16 +- hw4/.DS_Store | Bin 8196 -> 8196 bytes hw5/9-11.jl | 713 +++++++++++++++++++++++++++ hw5/9-14.jl | 846 +++++++++++++++++++++++++++++++ hw5/9-14.png | Bin 0 -> 59247 bytes hw5/9-2.jl | 848 ++++++++++++++++++++++++++++++++ hw5/9-2.png | Bin 0 -> 903130 bytes hw5/HarmonicOscillator.jl | 71 +++ hw5/MolecularDynamics2.jl | 755 ++++++++++++++++++++++++++++ hw5/demo_2-f.png | Bin 0 -> 1842809 bytes hw5/demo_2.png | Bin 0 -> 2321892 bytes hw5/fpu-2.jl | 138 ++++++ hw5/fpu-3.jl | 78 +++ hw5/fpu.gif | Bin 0 -> 1551508 bytes hw5/fpu.jl | 35 ++ hw5/fpu.png | Bin 0 -> 44884 bytes hw5/lfg-modes.png | Bin 0 -> 40240 bytes hw5/modes-beta15.png | Bin 0 -> 57359 bytes hw5/modes-beta3.png | Bin 0 -> 59535 bytes hw5/plot_175.svg | 105 ++++ hw5/plz-modes.png | Bin 0 -> 49160 bytes hw5/plz.gif | 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create mode 100644 hw8/tightbinding.jl create mode 100644 t/1d.jl create mode 100644 t/animate-positions-10.mp4 create mode 100644 t/animate-positions.mp4 create mode 100644 t/disp.jl create mode 100644 t/initial-final-positions.png create mode 100644 t/initial-final-velocities.png create mode 100644 t/plot_data.png create mode 100644 t/plz.mp4 create mode 100644 t/r.jl create mode 100644 t/t.jl create mode 100644 t/velocity-over-time.png diff --git a/.DS_Store b/.DS_Store index 0994b43..2b90e36 100644 Binary files a/.DS_Store and b/.DS_Store differ diff --git a/examples/FallingBall3.jl b/examples/FallingBall3.jl index 123a19a..823adfb 100644 --- a/examples/FallingBall3.jl +++ b/examples/FallingBall3.jl @@ -8,13 +8,15 @@ g = 9.8 # acceleration of gravity in m/s^2 t_final = 1.0 # final time of trajectory p = 0.0 # parameters (not used here) -function tendency!(dyv::Vector{Float64}, yv::Vector{Float64}, p, t::Float64) # ! notation tells us that arguments will be modified - y = yv[1] # 2D phase space; use vcat(x, v) to combine 2 vectors - v = yv[2] # dy/dt = v - a = -g # dv/dt = -g +function tendency!(dyv::Vector{Float64}, yv::Vector{Float64}, p, t) # ! notation tells us that arguments will be modified + y = yv[1] # 2D phase space; use vcat(x, v) to combine 2 vectors + v = yv[2] # dy/dt = v + a = -g # dv/dt = -g - dyv[1] = v - dyv[2] = a + dyv[1] = v + dyv[2] = a + + println("t = ", t, " y = ", y, " v = ", v) end y0 = 10.0 # initial position in meters @@ -23,7 +25,7 @@ yv0 = [y0, v0] # initial condition in phase space tspan = (0.0, t_final) # span of time to simulate prob = ODEProblem(tendency!, yv0, tspan, p) # specify ODE -sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # solve using Tsit5 algorithm to specified accuracy +sol = solve(prob, Tsit5(), reltol = 1e-8, abstol = 1e-8) # solve using Tsit5 algorithm to specified accuracy println("\n\t Results") println("final time = ", sol.t[end]) diff --git a/hw4/.DS_Store b/hw4/.DS_Store index a5beba9..b9913e5 100644 Binary files a/hw4/.DS_Store and b/hw4/.DS_Store differ diff --git a/hw5/9-11.jl b/hw5/9-11.jl new file mode 100644 index 0000000..802d9c6 --- /dev/null +++ b/hw5/9-11.jl @@ -0,0 +1,713 @@ +# molecular dynamics 2d. + +# usage: +# at the end of this script, under the header "DEMOS", +# you'll see some functions which implement demos from GN chapter 9. +# simply load the script in your development environment +# (I strongly recommend not using jupiter) +# and in the console/REPL run +# demo_0() +# etc. + +# demos 0,1,3 can optionally make an animated gif +# if you call it with the optional argument demo_3(gif=1) + +# lmk if this script is giving you grief or if you find any bugs +# kian@brown.edu + +using Statistics +using StatsPlots +using Plots.PlotMeasures + +mutable struct ParticleSystem + N::Int64 # number of particles + L::Float64 # square box side length + T₀::Float64 # initial temperature + t::Float64 # system time + dt::Float64 # time step + state::Vector{Float64} # state space array + steps::Int64 # number of steps + sampleInterval::Int64 # interval for sampling data + timeData::Vector{Float64} # array of sampled time points + energyData::Vector{Float64} # array of sampled energy values + tempData::Vector{Float64} # array of sampled temperature values + tempAccumulator::Float64 # temperature accumulator + squareTempAccumulator::Float64 # T^2 accumulator + virialAccumulator::Float64 # virial accumulator + xData::Vector{Vector{Float64}} # array of sampled position data + vData::Vector{Vector{Float64}} # array of sampled velocity data + forceType::String # force +end + +function ParticleSystem(N::Int64=64, L::Float64=10.0, T₀::Float64=1.0) + t = 0.0 + dt = 0.001 + state = zeros(4N) # state space array, [x1,y1,x2,y2,...,vx1,vy1,...] + steps = 0 + timeData = Float64[] + energyData = Float64[] + sampleInterval = 100 + tempData = Float64[] + tempAccumulator = 0.0 + squareTempAccumulator = 0.0 + virialAccumulator = 0.0 + xData = Vector{Float64}[] + vData = Vector{Float64}[] + forceType = "lennardJones" + + return ParticleSystem( + N, + L, + T₀, + t, + dt, + state, + steps, + sampleInterval, + timeData, + energyData, + tempData, + tempAccumulator, + squareTempAccumulator, + virialAccumulator, + xData, + vData, + forceType + ) +end + +# some useful "views" of the state array +# (read the performance tips chapter of the julia manual) +@views positions(state) = state[ 1:Int64(length(state)/2) ] +@views velocities(state) = state[ (Int64(length(state)/2)+1):end ] +@views xcomponent(vector) = vector[ 1:2:end ] +@views ycomponent(vector) = vector[ 2:2:end ] +@views particle(n, vector) = [ vector[2n-1], vector[2n] ] + +# INITIALIZATION +################################################################################ + +function set_random_positions!(sys::ParticleSystem) + println("\tposition configuration: random") + positions(sys.state) .= rand(2*sys.N) .* sys.L + cool!(sys) +end + +function set_square_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: square lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end +end + +function set_triangular_lattice_positions!(sys::ParticleSystem) +end + +function add_position_jitter!(sys::ParticleSystem, jitter::Float64=0.5) + println("\tadding a wee bit of random jitter to particle positions...") + + for i = 1:length(positions(sys.state)) + sys.state[i] += rand() - jitter + end +end + +function set_random_velocities!(sys::ParticleSystem) + println("\tvelocity configuration: random") + + velocities(sys.state) .= rand(2*sys.N) .- 0.5 .+ velocities(sys.state) + zero_total_momentum!(sys) + velocities(sys.state) .*= sqrt(sys.T₀/temperature(sys)) +end + +function scale_velocities!(sys, scale) + velocities(sys.state) .*= scale +end + + +function zero_total_momentum!(sys::ParticleSystem) + xcomponent(velocities(sys.state)) .-= + mean(xcomponent(velocities(sys.state))) + ycomponent(velocities(sys.state)) .-= + mean(ycomponent(velocities(sys.state))) +end + + +# FORCES / POTENTIALS +################################################################################ + +function force(sys::ParticleSystem) + if sys.forceType == "lennardJones" + force, virial = lennard_jones_force(sys) + elseif sys.forceType == "powerLaw" + force, virial = power_law_force(sys) + end + + sys.virialAccumulator += virial + + return force +end + +# the minimum image approximation +# (periodic boundary conditions) +function minimum_image(xij::Float64, L::Float64) + if xij > (L/2) + xij -= L + elseif xij < -(L/2) + xij += L + end + return xij +end + +function lennard_jones_force(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + virial = 0.0 + force = zeros(2*sys.N) + + Threads.@threads for i = 1:(sys.N-1) + for j = (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + f = 48.0 * r2inv^7 - 24.0 * r2inv^4 + fx = dx * f + fy = dy * f + + force[2*i-1] += fx + force[2*i] += fy + force[2*j-1] -= fx + force[2*j] -= fy + + virial += fx * dx + fy * dy + end + end + + return force, 0.5 * virial +end + +function lennard_jones_potential(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + U = 0.0 + + Threads.@threads for i in 1:(sys.N-1) + for j in (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + U += r2inv^6 - r2inv^3 + end + end + return 4.0 * U +end + +function power_law_force(sys::ParticleSystem) +end + +function power_law_potential(sys::ParticleSystem) +end + +# TIME EVOLUTION +################################################################################ + +function keep_particles_in_box!(sys::ParticleSystem) + for i in 1:(2*sys.N) + if positions(sys.state)[i] > sys.L + positions(sys.state)[i] -= sys.L + elseif positions(sys.state)[i] < 0.0 + positions(sys.state)[i] += sys.L + end + end + +# # another way of doing this: use the ternary operator +# for i in 1:(2 * sys.N) +# positions(sys.state)[i] < 0.0 ? +# positions(sys.state)[i] % sys.L + sys.L : +# positions(sys.state)[i] % sys.L +# end +end + +function verlet_step!(sys::ParticleSystem) + # compute acceleration at current time + acceleration = force(sys) + + # compute positions at t + dt + positions(sys.state) .+= + velocities(sys.state) .* sys.dt .+ + 0.5 .* acceleration .* (sys.dt)^2 + + # enforce boundary conditions + # (basically check if any particles left the box and put them back) + # see function implementation for deets + keep_particles_in_box!(sys) + + # compute velocities at t + dt + velocities(sys.state) .+= + 0.5 * sys.dt .* (acceleration + force(sys)) +end + +function update_r_squared!(sys, r_squared) + # take the positions of the firs ttwo particles + p1 = particle(1, sys.state) + p2 = particle(2, sys.state) + + # calculate the distance between them + # compute the image difference + dx = minimum_image(p1[1] - p2[1], sys.L) + dy = minimum_image(p1[2] - p2[2], sys.L) + r = dx^2 + dy^2 + + # println("r_squared: ", r) + # println("t: ", sys.t) + + push!(r_squared, [sys.t, r]) +end + +function evolve!(sys::ParticleSystem, runtime::Float64=10.0, r_squared=[]) + numsteps = Int64(abs(runtime/sys.dt) + 1) + + print_evolution_message(runtime, numsteps) + + @time for step in 1:numsteps + # for problem 9.11 + update_r_squared!(sys, r_squared) + + verlet_step!(sys) + zero_total_momentum!(sys) + + if (step % sys.sampleInterval == 1) + push!(sys.timeData, sys.t) + push!(sys.energyData, energy(sys)) + push!(sys.xData, positions(sys.state)) + push!(sys.vData, velocities(sys.state)) + + T = temperature(sys) + push!(sys.tempData, T) + sys.tempAccumulator += T + sys.squareTempAccumulator += T^2 + end + + sys.t += sys.dt + sys.steps += 1 + end + println("done.") +end + +function reverse_time!(sys) + sys.dt *= -1 + println("\ntime reversed! dt = $(sys.dt)") +end + +function cool!(sys::ParticleSystem, cooltime::Float64=1.0) + numsteps = Int64(cooltime/sys.dt) + for step in 1:numsteps + verlet_step!(sys) + velocities(sys.state) .*= (1.0 - sys.dt) + end + reset_statistics!(sys) +end + +# MEASUREMENTS +################################################################################ + +function kinetic_energy(sys::ParticleSystem) + return 0.5 * sum(velocities(sys.state) .* velocities(sys.state)) +end + +function potential_energy(sys::ParticleSystem) + return lennard_jones_potential(sys) +end + +function temperature(sys::ParticleSystem) + return kinetic_energy(sys) / sys.N +end + +function energy(sys::ParticleSystem) + return potential_energy(sys) + kinetic_energy(sys) +end + +# STATISTICS +################################################################################ + +function reset_statistics!(sys::ParticleSystem) + sys.steps = 0 + sys.tempAccumulator = 0.0 + sys.squareTempAccumulator = 0.0 + sys.virialAccumulator = 0.0 + sys.xData = [] + sys.vData = [] +end + +function mean_temperature(sys::ParticleSystem) + return sys.tempAccumulator / sys.steps +end + +function mean_square_temperature(sys::ParticleSystem) + return sys.squareTempAccumulator / sys.steps +end + +function mean_pressure(sys::ParticleSystem) + # factor of half because force is calculated twice each step + meanVirial = 0.5 * sys.virialAccumulator / sys.steps + return 1.0 + 0.5 * meanVirial / (sys.N * mean_temperature(sys)) +end + +function heat_capacity(sys::ParticleSystem) + meanTemperature = mean_temperature(sys) + meanSquareTemperature = mean_square_temperature(sys) + σ2 = meanSquareTemperature - meanTemperature^2 + denom = 1.0 - σ2 * sys.N / meanTemperature^2 + return sys.N / denom +end + +function mean_energy(sys::ParticleSystem) + return mean(sys.energyData) +end + +function std_energy(sys::ParticleSystem) + return std(sys.energyData) +end + +# MATH / ADDITIONAL FUNCTIONS +################################################################################ + +function dot(v1::Vector{Float64}, v2::Vector{Float64}) + return sum(v1 .* v2) +end + +# GRAPHS +################################################################################ + +function initialize_plot() + plot( + size=(800,800), + titlefontsize=12, + guidefontsize=12, + ) +end + +function plot_positions_t(sys::ParticleSystem, t::Int64) + initialize_plot() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end +end + +function animate(sys::ParticleSystem, interval::Int64=1) + println("\ngenerating gif...") + + scatter!() + animation = @animate for t in 1:length(sys.xData) + scatter() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + #markersize = 4.0, + markercolor = n, + #markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + end every interval + + gif(animation, "./animation.gif") + println("done.") +end + +function plot_positions(sys::ParticleSystem) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + title!("positions at t=$(round(sys.t, digits=4))") +end + +function plot_trajectories(sys::ParticleSystem, particles::Vector{Int64}=[ 1 ]) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + + for n in collect(particles) + xdata = [ sys.xData[i][2n-1] for i in 1:length(sys.xData) ] + ydata = [ sys.xData[i][2n] for i in 1:length(sys.xData) ] + + # plot trajectory line for nth particle + scatter!( + xdata, + ydata, + color = n, + #markerstrokewidth = 0, + markerstrokecolor = n, + markersize = 0.7, + markeralpha = 0.5, + label = false, + widen = false, + ) + + # plot initial position for nth particle + scatter!( + [ sys.xData[1][2n-1] ], + [ sys.xData[1][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 0.3, + #label = "pcl. $n @t=t₀", + widen = false, + ) + + # plot final position for nth particle + scatter!( + [ sys.xData[end][2n-1] ], + [ sys.xData[end][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 1.0, + #label = "pcl $n @t=t", + widen = false, + ) + end + title!("positions & trajectories at time t=$(round(sys.t, digits=2))") + plot!() +end + +function plot_temperature(sys::ParticleSystem) + initialize_plot() + plot!( + sys.timeData, + sys.tempData, + #widen = true, + legend = false, + ) + ylims!( + mean(sys.tempData) - std(sys.tempData) * 1.5, + mean(sys.tempData) + std(sys.tempData) * 3, + ) + xlabel!("t") + ylabel!("T(t)") + title!("temperature vs time") +end + +function plot_energy(sys::ParticleSystem, ylimit::Float64=1.0) + initialize_plot() + plot!( + sys.timeData, + sys.energyData, + #widen = true, + legend = false, + ) + ylims!( + #ylimit * (mean(sys.energyData) - 1), + #ylimit * (mean(sys.energyData) + 1) + mean(sys.energyData) - std(sys.energyData) * 10, + mean(sys.energyData) + std(sys.energyData) * 10, + ) + xlabel!("t") + ylabel!("E(t)") + title!("energy vs time") +end + +function plot_speed_distribution(sys::ParticleSystem, numSamples::Int64=5) + initialize_plot() + + numDataPoints = Int64(length(sys.vData)) + interval = Int64(floor(numDataPoints / numSamples)) + + samples = collect(1:interval:numDataPoints) + for s in samples + speed = sqrt.( + xcomponent(sys.vData[s]).^2 .* + ycomponent(sys.vData[s]).^2 + ) + density!( + sys.vData[s], + normalize = :pdf, + label = "t = $(round(sys.timeData[s], digits=2))", + ) + end + xlabel!("speed") + title!("speed distribution") +end + +# CONSOLE PRINT DATA +################################################################################ + +function print_hello() + println("\nmolecular dynamics!") + println("number of threads: ", Threads.nthreads()) +end + +function print_bonjour() + println("\nbonjour") +end + +function print_system_parameters(sys::ParticleSystem) + println("\nsystem parameters:") + println("\tN = $(sys.N) (number of particles)") + println("\tL = $(sys.L) (side length of square box)") + println("\tDT = $(sys.dt) (time step)") +end + +function print_system_data(sys::ParticleSystem) + println("\nsystem data at time t=$(round(sys.t, digits=4))") + + if sys.steps == 0 + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + else + println("\tsteps evolved: $(sys.steps)") + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + println("\tmean energy: $(mean_energy(sys))") + println("\tstd energy: $(std_energy(sys))") + println("\theat capacity: $(heat_capacity(sys))") + println("\tPV/NkT: $(mean_pressure(sys))") + end +end + +function print_evolution_message(runtime, numsteps) + println("\nevolving...") +end + +# DEMOS +################################################################################ + +# DEMO 3: MELTING TRANSITION +function problem_9_11!(sys) + println("\nProblem 9.11") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + set_square_lattice_positions!(sys) + # set_random_velocities!(sys) + print_system_data(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + r_squared = [] + evolve!(sys, 20.0, r_squared) + print_system_data(sys) + + # now, increase the temperature of the system by giving the particles + # some velocity. evolve the system and plot the trajectories. + scale_velocities!(sys, 1.5) + evolve!(sys, 10.0, r_squared) + print_system_data(sys) + + scale_velocities!(sys, 1.5) + evolve!(sys, 10.0, r_squared) + print_system_data(sys) + + scale_velocities!(sys, 1.5) + evolve!(sys, 10.0, r_squared) + print_system_data(sys) + + scale_velocities!(sys, 1.5) + evolve!(sys, 10.0, r_squared) + print_system_data(sys) + + # some more plots + p4 = plot_energy(sys, 0.0) + p5 = plot_temperature(sys) + + t = [] + r = [] + for i in 1:length(r_squared) + push!(t, r_squared[i][1]) + push!(r, r_squared[i][2]) + end + + # plot r_squared + p_r = plot(t, r, xlabel="t", ylabel="r^2", label="r^2(t)", title="Pair Separation for $(sys.N) particles, box length 4", legend=false, + top_margin=5mm, bottom_margin=5mm, left_margin=5mm) + + return p4, p5, p_r +end + +sys_9 = ParticleSystem(9, 4.0, 0.0) +p_e_9, p_t_9, p_r_9 = problem_9_11!(sys_9) + +sys_16 = ParticleSystem(16, 4.0, 0.0) +p_e_16, p_t_16, p_r_16 = problem_9_11!(sys_16) + +sys_25 = ParticleSystem(25, 4.0, 0.0) +p_e_25, p_t_25, p_r_25 = problem_9_11!(sys_25) + +sys_36 = ParticleSystem(36, 4.0, 0.0) +p_e_36, p_t_36, p_r_36 = problem_9_11!(sys_36) + +# plot( +# p_e_100, p_t_100, p_r_100, +# p_e_16, p_t_16, p_r_16, +# p_e_36, p_t_36, p_r_36, +# layout=(3,3), size=(1000,1000)) +# savefig("problem_9_11.png") + +# plot(p_r_16, p_t_16, layout=(1,2), size=(1000,500)) + +plot( + p_r_9, p_t_9, + p_r_16, p_t_16, + p_r_25, p_t_25, + p_r_36, p_t_36, + layout=(4,2), size=(1250, 1600) +) + +# only plot the 100 particle system +savefig("problem_9_11_t.png") \ No newline at end of file diff --git a/hw5/9-14.jl b/hw5/9-14.jl new file mode 100644 index 0000000..7829e40 --- /dev/null +++ b/hw5/9-14.jl @@ -0,0 +1,846 @@ +# molecular dynamics 2d. + +# usage: +# at the end of this script, under the header "DEMOS", +# you'll see some functions which implement demos from GN chapter 9. +# simply load the script in your development environment +# (I strongly recommend not using jupiter) +# and in the console/REPL run +# demo_0() +# etc. + +# demos 0,1,3 can optionally make an animated gif +# if you call it with the optional argument demo_3(gif=1) + +# lmk if this script is giving you grief or if you find any bugs +# kian@brown.edu + +using Statistics +using StatsPlots + +mutable struct ParticleSystem + N::Int64 # number of particles + L::Float64 # square box side length + T₀::Float64 # initial temperature + t::Float64 # system time + dt::Float64 # time step + state::Vector{Float64} # state space array + steps::Int64 # number of steps + sampleInterval::Int64 # interval for sampling data + timeData::Vector{Float64} # array of sampled time points + energyData::Vector{Float64} # array of sampled energy values + tempData::Vector{Float64} # array of sampled temperature values + tempAccumulator::Float64 # temperature accumulator + squareTempAccumulator::Float64 # T^2 accumulator + virialAccumulator::Float64 # virial accumulator + xData::Vector{Vector{Float64}} # array of sampled position data + vData::Vector{Vector{Float64}} # array of sampled velocity data + forceType::String # force +end + +function ParticleSystem(N::Int64=64, L::Float64=10.0, T₀::Float64=1.0) + t = 0.0 + dt = 0.001 + state = zeros(4N) # state space array, [x1,y1,x2,y2,...,vx1,vy1,...] + steps = 0 + timeData = Float64[] + energyData = Float64[] + sampleInterval = 100 + tempData = Float64[] + tempAccumulator = 0.0 + squareTempAccumulator = 0.0 + virialAccumulator = 0.0 + xData = Vector{Float64}[] + vData = Vector{Float64}[] + forceType = "lennardJones" + + return ParticleSystem( + N, + L, + T₀, + t, + dt, + state, + steps, + sampleInterval, + timeData, + energyData, + tempData, + tempAccumulator, + squareTempAccumulator, + virialAccumulator, + xData, + vData, + forceType + ) +end + +# some useful "views" of the state array +# (read the performance tips chapter of the julia manual) +@views positions(state) = state[ 1:Int64(length(state)/2) ] +@views velocities(state) = state[ (Int64(length(state)/2)+1):end ] +@views xcomponent(vector) = vector[ 1:2:end ] +@views ycomponent(vector) = vector[ 2:2:end ] +@views particle(n, vector) = [ vector[2n-1], vector[2n] ] + +# INITIALIZATION +################################################################################ + +function set_random_positions!(sys::ParticleSystem) + println("\tposition configuration: random") + positions(sys.state) .= rand(2*sys.N) .* sys.L + cool!(sys) +end + +function set_square_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: square lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end +end + +function set_triangular_lattice_positions!(sys::ParticleSystem) +end + +function add_position_jitter!(sys::ParticleSystem, jitter::Float64=0.5) + println("\tadding a wee bit of random jitter to particle positions...") + + for i = 1:length(positions(sys.state)) + sys.state[i] += rand() - jitter + end +end + +function set_random_velocities!(sys::ParticleSystem) + println("\tvelocity configuration: random") + + velocities(sys.state) .= rand(2*sys.N) .- 0.5 + zero_total_momentum!(sys) + velocities(sys.state) .*= sqrt(sys.T₀/temperature(sys)) +end + +function zero_total_momentum!(sys::ParticleSystem) + xcomponent(velocities(sys.state)) .-= + mean(xcomponent(velocities(sys.state))) + ycomponent(velocities(sys.state)) .-= + mean(ycomponent(velocities(sys.state))) +end + + +# FORCES / POTENTIALS +################################################################################ + +function force(sys::ParticleSystem) + if sys.forceType == "lennardJones" + force, virial = lennard_jones_force(sys) + elseif sys.forceType == "powerLaw" + force, virial = power_law_force(sys) + end + + sys.virialAccumulator += virial + + return force +end + +# the minimum image approximation +# (periodic boundary conditions) +function minimum_image(xij::Float64, L::Float64) + if xij > (L/2) + xij -= L + elseif xij < -(L/2) + xij += L + end + return xij +end + +function lennard_jones_force(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + virial = 0.0 + force = zeros(2*sys.N) + + Threads.@threads for i = 1:(sys.N-1) + for j = (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + f = 48.0 * r2inv^7 - 24.0 * r2inv^4 + fx = dx * f + fy = dy * f + + force[2*i-1] += fx + force[2*i] += fy + force[2*j-1] -= fx + force[2*j] -= fy + + virial += fx * dx + fy * dy + end + end + + return force, 0.5 * virial +end + +function lennard_jones_potential(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + U = 0.0 + + Threads.@threads for i in 1:(sys.N-1) + for j in (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + U += r2inv^6 - r2inv^3 + end + end + return 4.0 * U +end + +function power_law_force(sys::ParticleSystem) +end + +function power_law_potential(sys::ParticleSystem) +end + +# TIME EVOLUTION +################################################################################ + +function keep_particles_in_box!(sys::ParticleSystem) + for i in 1:(2*sys.N) + if positions(sys.state)[i] > sys.L + positions(sys.state)[i] -= sys.L + elseif positions(sys.state)[i] < 0.0 + positions(sys.state)[i] += sys.L + end + end + +# # another way of doing this: use the ternary operator +# for i in 1:(2 * sys.N) +# positions(sys.state)[i] < 0.0 ? +# positions(sys.state)[i] % sys.L + sys.L : +# positions(sys.state)[i] % sys.L +# end +end + +function verlet_step!(sys::ParticleSystem) + # compute acceleration at current time + acceleration = force(sys) + + # compute positions at t + dt + positions(sys.state) .+= + velocities(sys.state) .* sys.dt .+ + 0.5 .* acceleration .* (sys.dt)^2 + + # enforce boundary conditions + # (basically check if any particles left the box and put them back) + # see function implementation for deets + keep_particles_in_box!(sys) + + # compute velocities at t + dt + velocities(sys.state) .+= + 0.5 * sys.dt .* (acceleration + force(sys)) +end + +function update_histogram!(histogram_data, bins, t_interval, sys) + # for problem 9-14 + # if sys.t % t_interval != 0 + # return # only update histogram at t = n * t_interval + # end + + # take the first particle as the origin + origin = particle(1, sys.state) + # println("origin: ", origin) + + # for each particle, calculate the distance from the origin + # and update the histogram + for i in 2:sys.N + pcl = particle(i, sys.state) + dx = minimum_image(pcl[1] - origin[1], sys.L) + dy = minimum_image(pcl[2] - origin[2], sys.L) + r = sqrt(dx^2 + dy^2) + + # find the bin that r belongs to + for j in 1:length(bins)-1 + if r >= bins[j] && r < bins[j+1] + histogram_data[j] += 1 + break + end + end + end +end + +function evolve!(sys, runtime, histogram_data, bins, t_interval) + numsteps = Int64(abs(runtime/sys.dt) + 1) + + print_evolution_message(runtime, numsteps) + + @time for step in 1:numsteps + verlet_step!(sys) + zero_total_momentum!(sys) + + if (step % sys.sampleInterval == 1) + # for problem, 9-14 update historgram here + update_histogram!(histogram_data, bins, t_interval, sys) + + push!(sys.timeData, sys.t) + push!(sys.energyData, energy(sys)) + push!(sys.xData, positions(sys.state)) + push!(sys.vData, velocities(sys.state)) + + T = temperature(sys) + push!(sys.tempData, T) + sys.tempAccumulator += T + sys.squareTempAccumulator += T^2 + end + + sys.t += sys.dt + sys.steps += 1 + end + println("done.") +end + +function reverse_time!(sys) + sys.dt *= -1 + println("\ntime reversed! dt = $(sys.dt)") +end + +function cool!(sys::ParticleSystem, cooltime::Float64=1.0) + numsteps = Int64(cooltime/sys.dt) + for step in 1:numsteps + verlet_step!(sys) + velocities(sys.state) .*= (1.0 - sys.dt) + end + reset_statistics!(sys) +end + +# MEASUREMENTS +################################################################################ + +function kinetic_energy(sys::ParticleSystem) + return 0.5 * sum(velocities(sys.state) .* velocities(sys.state)) +end + +function potential_energy(sys::ParticleSystem) + return lennard_jones_potential(sys) +end + +function temperature(sys::ParticleSystem) + return kinetic_energy(sys) / sys.N +end + +function energy(sys::ParticleSystem) + return potential_energy(sys) + kinetic_energy(sys) +end + +# STATISTICS +################################################################################ + +function reset_statistics!(sys::ParticleSystem) + sys.steps = 0 + sys.tempAccumulator = 0.0 + sys.squareTempAccumulator = 0.0 + sys.virialAccumulator = 0.0 + sys.xData = [] + sys.vData = [] +end + +function mean_temperature(sys::ParticleSystem) + return sys.tempAccumulator / sys.steps +end + +function mean_square_temperature(sys::ParticleSystem) + return sys.squareTempAccumulator / sys.steps +end + +function mean_pressure(sys::ParticleSystem) + # factor of half because force is calculated twice each step + meanVirial = 0.5 * sys.virialAccumulator / sys.steps + return 1.0 + 0.5 * meanVirial / (sys.N * mean_temperature(sys)) +end + +function heat_capacity(sys::ParticleSystem) + meanTemperature = mean_temperature(sys) + meanSquareTemperature = mean_square_temperature(sys) + σ2 = meanSquareTemperature - meanTemperature^2 + denom = 1.0 - σ2 * sys.N / meanTemperature^2 + return sys.N / denom +end + +function mean_energy(sys::ParticleSystem) + return mean(sys.energyData) +end + +function std_energy(sys::ParticleSystem) + return std(sys.energyData) +end + +# MATH / ADDITIONAL FUNCTIONS +################################################################################ + +function dot(v1::Vector{Float64}, v2::Vector{Float64}) + return sum(v1 .* v2) +end + +# GRAPHS +################################################################################ + +function initialize_plot() + plot( + size=(800,800), + titlefontsize=12, + guidefontsize=12, + ) +end + +function plot_positions_t(sys::ParticleSystem, t::Int64) + initialize_plot() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end +end + +function animate(sys::ParticleSystem, interval::Int64=1) + println("\ngenerating gif...") + + scatter!() + animation = @animate for t in 1:length(sys.xData) + scatter() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + #markersize = 4.0, + markercolor = n, + #markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + end every interval + + gif(animation, "./animation.gif") + println("done.") +end + +function plot_positions(sys::ParticleSystem) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + title!("positions at t=$(round(sys.t, digits=4))") +end + +function plot_trajectories(sys::ParticleSystem, particles::Vector{Int64}=[ 1 ]) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + + for n in collect(particles) + xdata = [ sys.xData[i][2n-1] for i in 1:length(sys.xData) ] + ydata = [ sys.xData[i][2n] for i in 1:length(sys.xData) ] + + # plot trajectory line for nth particle + scatter!( + xdata, + ydata, + color = n, + #markerstrokewidth = 0, + markerstrokecolor = n, + markersize = 0.7, + markeralpha = 0.5, + label = false, + widen = false, + ) + + # plot initial position for nth particle + scatter!( + [ sys.xData[1][2n-1] ], + [ sys.xData[1][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 0.3, + #label = "pcl. $n @t=t₀", + widen = false, + ) + + # plot final position for nth particle + scatter!( + [ sys.xData[end][2n-1] ], + [ sys.xData[end][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 1.0, + #label = "pcl $n @t=t", + widen = false, + ) + end + title!("positions & trajectories at time t=$(round(sys.t, digits=2))") + plot!() +end + +function plot_temperature(sys::ParticleSystem) + initialize_plot() + plot!( + sys.timeData, + sys.tempData, + #widen = true, + ) + ylims!( + mean(sys.tempData) - std(sys.tempData) * 20, + mean(sys.tempData) + std(sys.tempData) * 20, + ) + xlabel!("t") + ylabel!("T(t)") + title!("temperature vs time") + +end + +function plot_energy(sys::ParticleSystem, ylimit::Float64=1.0) + initialize_plot() + plot!( + sys.timeData, + sys.energyData, + #widen = true, + ) + ylims!( + #ylimit * (mean(sys.energyData) - 1), + #ylimit * (mean(sys.energyData) + 1) + mean(sys.energyData) - std(sys.energyData) * 10, + mean(sys.energyData) + std(sys.energyData) * 10, + ) + xlabel!("t") + ylabel!("E(t)") + title!("energy vs time") +end + +function plot_speed_distribution(sys::ParticleSystem, numSamples::Int64=5) + initialize_plot() + + numDataPoints = Int64(length(sys.vData)) + interval = Int64(floor(numDataPoints / numSamples)) + + samples = collect(1:interval:numDataPoints) + for s in samples + speed = sqrt.( + xcomponent(sys.vData[s]).^2 .+ + ycomponent(sys.vData[s]).^2 + ) + density!( + sys.vData[s], + normalize = :pdf, + label = "t = $(round(sys.timeData[s], digits=2))", + ) + end + xlabel!("speed") + title!("speed distribution") +end + +# CONSOLE PRINT DATA +################################################################################ + +function print_hello() + println("\nmolecular dynamics!") + println("number of threads: ", Threads.nthreads()) +end + +function print_bonjour() + println("\nbonjour") +end + +function print_system_parameters(sys::ParticleSystem) + println("\nsystem parameters:") + println("\tN = $(sys.N) (number of particles)") + println("\tL = $(sys.L) (side length of square box)") + println("\tDT = $(sys.dt) (time step)") +end + +function print_system_data(sys::ParticleSystem) + println("\nsystem data at time t=$(round(sys.t, digits=4))") + + if sys.steps == 0 + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + else + println("\tsteps evolved: $(sys.steps)") + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + println("\tmean energy: $(mean_energy(sys))") + println("\tstd energy: $(std_energy(sys))") + println("\theat capacity: $(heat_capacity(sys))") + println("\tPV/NkT: $(mean_pressure(sys))") + end +end + +function print_evolution_message(runtime, numsteps) + println("\nevolving...") +end + +# DEMOS +################################################################################ + + +# DEMO 0: APPROACH TO EQUILIBRIUM +function demo_0(;gif=0) + println("\nDEMO 0: APPROACH TO EQUILIBRIUM") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 20.0) + print_system_data(sys) + + p2 = plot_trajectories(sys, collect(1:64)) + p3 = plot_energy(sys) + p4 = plot_temperature(sys) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, + layout = grid(2,2, heights=[0.7,0.3]), + size = (1280,720) + ) +end + +# DEMO 1: TIME REVERSAL TEST +function demo_1(;gif=0) + println("\nDEMO 1: TIME REVERSAL TEST") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 50.0) + #p2 = plot_trajectories(sys, collect(1:64)) + p2 = plot_positions(sys) + + reverse_time!(sys) + evolve!(sys, 50.0) + print_system_data(sys) + #p3 = plot_trajectories(sys, collect(1:64)) + p3 = plot_positions(sys) + + # make gif + if gif == 1 + animate(sys, 4) + end + + plot( + p1, p2, p3, + layout = (1,3), + size = (1200,400) + ) +end + +# DEMO 2: SPEED DISTRIBUTION +function demo_2() + println("\nDEMO 2: SPEED DISTRIBUTION") + println("----------------------------------------") + + sys = ParticleSystem[] + + # array for speed distribution plots + ps = Plots.Plot{Plots.GRBackend}[] + + # array for trajectory plots + pt = Plots.Plot{Plots.GRBackend}[] + + # initialize three systems with different initial conditions + # but same KE and PE, evolve, and save plots + for i = 1:3 + push!(sys, ParticleSystem(64, 120.0, 1.0)) + + println("\nSYSTEM $i") + print_system_parameters(sys[i]) + + set_square_lattice_positions!(sys[i]) + add_position_jitter!(sys[i]) + set_random_velocities!(sys[i]) + print_system_data(sys[i]) + + evolve!(sys[i], 40.0) + print_system_data(sys[i]) + push!(ps, plot_speed_distribution(sys[i], 5)) + push!(pt, plot_trajectories(sys[i], collect(1:64)) ) + end + + + # plot speed distribution and trajectory plots + plot( + ps[1], ps[2], ps[3], + pt[1], pt[2], pt[3], + layout = (2,3), + size = (1920,1080) + ) +end + +# DEMO 3: MELTING TRANSITION +function demo_3(;gif=0) + println("\nDEMO 3: MELTING TRANSITION") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + sys = ParticleSystem(100, 10.0, 5.0) + set_square_lattice_positions!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + evolve!(sys, 20.0) + print_system_data(sys) + p2 = plot_trajectories(sys, collect(1:100)) + + # now, increase the temperature of the system by giving the particles + # some velocity. evolve the system and plot the trajectories. + set_random_velocities!(sys) + evolve!(sys, 60.0) + print_system_data(sys) + p3 = plot_trajectories(sys, collect(1:100)) + + # some more plots + p4 = plot_energy(sys, 0.0) + p5 = plot_temperature(sys) + p6 = plot_speed_distribution(sys, 20) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, p5, p6, + layout = (2,3), + size = (1280,720) + ) +end + +function normalize_histogram(histogram) + println("normalizing histogram...") + println(histogram) + return histogram ./ sum(histogram) +end + +# problem 9-14 +function problem_9_14(temp, bins, t_interval) + println("\nPROBLEM 9-14: RADIAL DISTRIBUTION FUNCTION") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + sys = ParticleSystem(16, 4.0, temp) + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + histogram_data = zeros(length(bins) - 1) + evolve!(sys, 500, histogram_data, bins, t_interval) + print_system_data(sys) + + # normalize the histogram + histogram_data = normalize_histogram(histogram_data) + # get the max y value for the plot + y_max = maximum(histogram_data) + + # center the bins + centered_bins = bins .+ (bins[2] - bins[1]) / 2 + + # plot the histogram + return plot( + centered_bins[2:end], histogram_data, + marker = :circle, + xlabel = "r", + ylabel = "g(r)", + title = "Pair Correlation (T=$temp)", + label="pdf(r)" + ), y_max +end + +bins = range(.5, stop=3, length=40) +p_05, y_max_1 = problem_9_14(.5, bins, 10.0) +# ont this plot, plot vertical lines at 1.1, 1.9, 2.8 +y_max_1 += 0.025 +plot!(p_05, [1.1, 1.1], [0, y_max_1], line = :dash, label = "r = 1.1", color = :green, ylim=(0, y_max_1), lw=1.5, legend=:topright) +plot!(p_05, [1.9, 1.9], [0, y_max_1], line = :dash, label = "r = 1.9", color = :orange, lw=1.5) +plot!(p_05, [2.8, 2.8], [0, y_max_1], line = :dash, label = "r = 2.8", color = :red, lw=1.5) + +p_5, y_max = problem_9_14(5.0, bins, 10.0) +plot!(p_5, [1.1, 1.1], [0, y_max_1], line = :dash, label = "r = 1.1", color = :green, ylim=(0, y_max_1), lw=1.5, legend=:topright) +plot!(p_5, [1.9, 1.9], [0, y_max_1], line = :dash, label = "r = 1.9", lw=1.5, color = :orange) +plot!(p_5, [2.8, 2.8], [0, y_max_1], line = :dash, label = "r = 2.8", lw=1.5, color = :red) + +using Plots.PlotMeasures # for padding + +plot(p_05, p_5, layout = (1,2), size = (1280,600), bottom_margin = 10mm, left_margin = 10mm, right_margin = 10mm, top_margin = 10mm) + +savefig("9-14.png") \ No newline at end of file diff --git a/hw5/9-14.png b/hw5/9-14.png new file mode 100644 index 0000000..2521999 Binary files /dev/null and b/hw5/9-14.png differ diff --git a/hw5/9-2.jl b/hw5/9-2.jl new file mode 100644 index 0000000..ef7af10 --- /dev/null +++ b/hw5/9-2.jl @@ -0,0 +1,848 @@ +# molecular dynamics 2d. + +# usage: +# at the end of this script, under the header "DEMOS", +# you'll see some functions which implement demos from GN chapter 9. +# simply load the script in your development environment +# (I strongly recommend not using jupiter) +# and in the console/REPL run +# demo_0() +# etc. + +# demos 0,1,3 can optionally make an animated gif +# if you call it with the optional argument demo_3(gif=1) + +# lmk if this script is giving you grief or if you find any bugs +# kian@brown.edu + +using Statistics +using StatsPlots +using Plots.PlotMeasures # for margins + +mutable struct ParticleSystem + N::Int64 # number of particles + L::Float64 # square box side length + T₀::Float64 # initial temperature + t::Float64 # system time + dt::Float64 # time step + state::Vector{Float64} # state space array + steps::Int64 # number of steps + sampleInterval::Int64 # interval for sampling data + timeData::Vector{Float64} # array of sampled time points + energyData::Vector{Float64} # array of sampled energy values + tempData::Vector{Float64} # array of sampled temperature values + tempAccumulator::Float64 # temperature accumulator + squareTempAccumulator::Float64 # T^2 accumulator + virialAccumulator::Float64 # virial accumulator + xData::Vector{Vector{Float64}} # array of sampled position data + vData::Vector{Vector{Float64}} # array of sampled velocity data + forceType::String # force +end + +function ParticleSystem(N::Int64=64, L::Float64=10.0, T₀::Float64=1.0) + t = 0.0 + dt = 0.001 + state = zeros(4N) # state space array, [x1,y1,x2,y2,...,vx1,vy1,...] + steps = 0 + timeData = Float64[] + energyData = Float64[] + sampleInterval = 100 + tempData = Float64[] + tempAccumulator = 0.0 + squareTempAccumulator = 0.0 + virialAccumulator = 0.0 + xData = Vector{Float64}[] + vData = Vector{Float64}[] + forceType = "lennardJones" + + return ParticleSystem( + N, + L, + T₀, + t, + dt, + state, + steps, + sampleInterval, + timeData, + energyData, + tempData, + tempAccumulator, + squareTempAccumulator, + virialAccumulator, + xData, + vData, + forceType + ) +end + +# some useful "views" of the state array +# (read the performance tips chapter of the julia manual) +@views positions(state) = state[ 1:Int64(length(state)/2) ] +@views velocities(state) = state[ (Int64(length(state)/2)+1):end ] +@views xcomponent(vector) = vector[ 1:2:end ] +@views ycomponent(vector) = vector[ 2:2:end ] +@views particle(n, vector) = [ vector[2n-1], vector[2n] ] + +# INITIALIZATION +################################################################################ + +function set_random_positions!(sys::ParticleSystem) + println("\tposition configuration: random") + positions(sys.state) .= rand(2*sys.N) .* sys.L + cool!(sys) +end + +function set_square_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: square lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end +end + +function set_triangular_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: triangular lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] += (j % 2) * latticeSpacing / 2 + end + end +end + +function add_position_jitter!(sys::ParticleSystem, jitter::Float64=0.5) + println("\tadding a wee bit of random jitter to particle positions...") + + for i = 1:length(positions(sys.state)) + sys.state[i] += rand() - jitter + end +end + +function set_random_velocities!(sys::ParticleSystem) + println("\tvelocity configuration: random") + + velocities(sys.state) .= rand(2*sys.N) .- 0.5 + zero_total_momentum!(sys) + velocities(sys.state) .*= sqrt(sys.T₀/temperature(sys)) +end + +function zero_total_momentum!(sys::ParticleSystem) + xcomponent(velocities(sys.state)) .-= + mean(xcomponent(velocities(sys.state))) + ycomponent(velocities(sys.state)) .-= + mean(ycomponent(velocities(sys.state))) +end + + +# FORCES / POTENTIALS +################################################################################ + +function force(sys::ParticleSystem) + if sys.forceType == "lennardJones" + force, virial = lennard_jones_force(sys) + elseif sys.forceType == "powerLaw" + force, virial = power_law_force(sys) + end + + sys.virialAccumulator += virial + + return force +end + +# the minimum image approximation +# (periodic boundary conditions) +function minimum_image(xij::Float64, L::Float64) + if xij > (L/2) + xij -= L + elseif xij < -(L/2) + xij += L + end + return xij +end + +function lennard_jones_force(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + virial = 0.0 + force = zeros(2*sys.N) + + Threads.@threads for i = 1:(sys.N-1) + for j = (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + f = 48.0 * r2inv^7 - 24.0 * r2inv^4 + fx = dx * f + fy = dy * f + + force[2*i-1] += fx + force[2*i] += fy + force[2*j-1] -= fx + force[2*j] -= fy + + virial += fx * dx + fy * dy + end + end + + return force, 0.5 * virial +end + +function lennard_jones_potential(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + U = 0.0 + + Threads.@threads for i in 1:(sys.N-1) + for j in (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + U += r2inv^6 - r2inv^3 + end + end + return 4.0 * U +end + +function power_law_force(sys::ParticleSystem) +end + +function power_law_potential(sys::ParticleSystem) +end + +# TIME EVOLUTION +################################################################################ + +function keep_particles_in_box!(sys::ParticleSystem) + for i in 1:(2*sys.N) + if positions(sys.state)[i] > sys.L + positions(sys.state)[i] -= sys.L + elseif positions(sys.state)[i] < 0.0 + positions(sys.state)[i] += sys.L + end + end + +# # another way of doing this: use the ternary operator +# for i in 1:(2 * sys.N) +# positions(sys.state)[i] < 0.0 ? +# positions(sys.state)[i] % sys.L + sys.L : +# positions(sys.state)[i] % sys.L +# end +end + +function verlet_step!(sys::ParticleSystem) + # compute acceleration at current time + acceleration = force(sys) + + # compute positions at t + dt + positions(sys.state) .+= + velocities(sys.state) .* sys.dt .+ + 0.5 .* acceleration .* (sys.dt)^2 + + # enforce boundary conditions + # (basically check if any particles left the box and put them back) + # see function implementation for deets + keep_particles_in_box!(sys) + + # compute velocities at t + dt + velocities(sys.state) .+= + 0.5 * sys.dt .* (acceleration + force(sys)) +end + +function evolve!(sys::ParticleSystem, runtime::Float64=10.0) + numsteps = Int64(abs(runtime/sys.dt) + 1) + + print_evolution_message(runtime, numsteps) + + @time for step in 1:numsteps + verlet_step!(sys) + zero_total_momentum!(sys) + + if (step % sys.sampleInterval == 1) + push!(sys.timeData, sys.t) + push!(sys.energyData, energy(sys)) + push!(sys.xData, positions(sys.state)) + push!(sys.vData, velocities(sys.state)) + + T = temperature(sys) + push!(sys.tempData, T) + sys.tempAccumulator += T + sys.squareTempAccumulator += T^2 + end + + sys.t += sys.dt + sys.steps += 1 + end + println("done.") +end + +function reverse_time!(sys) + sys.dt *= -1 + println("\ntime reversed! dt = $(sys.dt)") +end + +function cool!(sys::ParticleSystem, cooltime::Float64=1.0) + numsteps = Int64(cooltime/sys.dt) + for step in 1:numsteps + verlet_step!(sys) + velocities(sys.state) .*= (1.0 - sys.dt) + end + reset_statistics!(sys) +end + +# MEASUREMENTS +################################################################################ + +function kinetic_energy(sys::ParticleSystem) + return 0.5 * sum(velocities(sys.state) .* velocities(sys.state)) +end + +function potential_energy(sys::ParticleSystem) + return lennard_jones_potential(sys) +end + +function temperature(sys::ParticleSystem) + return kinetic_energy(sys) / sys.N +end + +function energy(sys::ParticleSystem) + return potential_energy(sys) + kinetic_energy(sys) +end + +# STATISTICS +################################################################################ + +function reset_statistics!(sys::ParticleSystem) + sys.steps = 0 + sys.tempAccumulator = 0.0 + sys.squareTempAccumulator = 0.0 + sys.virialAccumulator = 0.0 + sys.xData = [] + sys.vData = [] +end + +function mean_temperature(sys::ParticleSystem) + return sys.tempAccumulator / sys.steps +end + +function mean_square_temperature(sys::ParticleSystem) + return sys.squareTempAccumulator / sys.steps +end + +function mean_pressure(sys::ParticleSystem) + # factor of half because force is calculated twice each step + meanVirial = 0.5 * sys.virialAccumulator / sys.steps + return 1.0 + 0.5 * meanVirial / (sys.N * mean_temperature(sys)) +end + +function heat_capacity(sys::ParticleSystem) + meanTemperature = mean_temperature(sys) + meanSquareTemperature = mean_square_temperature(sys) + σ2 = meanSquareTemperature - meanTemperature^2 + denom = 1.0 - σ2 * sys.N / meanTemperature^2 + return sys.N / denom +end + +function mean_energy(sys::ParticleSystem) + return mean(sys.energyData) +end + +function std_energy(sys::ParticleSystem) + return std(sys.energyData) +end + +# MATH / ADDITIONAL FUNCTIONS +################################################################################ + +function dot(v1::Vector{Float64}, v2::Vector{Float64}) + return sum(v1 .* v2) +end + +# GRAPHS +################################################################################ + +function initialize_plot() + plot( + size=(800,800), + titlefontsize=12, + guidefontsize=12, + ) +end + +function plot_positions_t(sys::ParticleSystem, t::Int64) + initialize_plot() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end +end + +function animate(sys::ParticleSystem, interval::Int64=1) + println("\ngenerating gif...") + + scatter!() + animation = @animate for t in 1:length(sys.xData) + scatter() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + #markersize = 4.0, + markercolor = n, + #markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + end every interval + + gif(animation, "./animation.gif") + println("done.") +end + +function plot_positions(sys::ParticleSystem) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + title!("positions at t=$(round(sys.t, digits=4))") +end + +function plot_trajectories(sys::ParticleSystem, particles::Vector{Int64}=[ 1 ], t="trajectories") + initialize_plot() + # for n = 1:sys.N + # scatter!( + # [ xcomponent(positions(sys.state))[n] ], + # [ ycomponent(positions(sys.state))[n] ], + # markersize = 4.0, + # markercolor = n, + # markerstrokewidth = 0.4, + # grid = true, + # framestyle = :box, + # legend = false, + # ) + # end + + for n in collect(particles) + xdata = [ sys.xData[i][2n-1] for i in 1:length(sys.xData) ] + ydata = [ sys.xData[i][2n] for i in 1:length(sys.xData) ] + + # plot trajectory line for nth particle + # scatter!( + # xdata, + # ydata, + # color = n, + # #markerstrokewidth = 0, + # markerstrokecolor = n, + # markersize = 0.7, + # markeralpha = 0.5, + # label = false, + # widen = false, + # ) + + # plot initial position for nth particle + scatter!( + [ sys.xData[1][2n-1] ], + [ sys.xData[1][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + #label = "pcl. $n @t=t₀", + widen = false, + legend = false, + top_margin=10mm, + bottom_margin=10mm, + left_margin=10mm, + right_margin=10mm + ) + + # plot final position for nth particle + # scatter!( + # [ sys.xData[end][2n-1] ], + # [ sys.xData[end][2n] ], + # markersize = 4.0, + # markercolor = n, + # markerstrokewidth = 0.4, + # markeralpha = 1.0, + # #label = "pcl $n @t=t", + # widen = false, + # ) + end + title!(t) + plot!() +end + +function plot_temperature(sys::ParticleSystem) + initialize_plot() + plot!( + sys.timeData, + sys.tempData, + #widen = true, + ) + ylims!( + mean(sys.tempData) - std(sys.tempData) * 20, + mean(sys.tempData) + std(sys.tempData) * 20, + ) + xlabel!("t") + ylabel!("T(t)") + title!("temperature vs time") + +end + +function plot_energy(sys::ParticleSystem, ylimit::Float64=1.0) + initialize_plot() + plot!( + sys.timeData, + sys.energyData, + #widen = true, + ) + ylims!( + #ylimit * (mean(sys.energyData) - 1), + #ylimit * (mean(sys.energyData) + 1) + mean(sys.energyData) - std(sys.energyData) * 10, + mean(sys.energyData) + std(sys.energyData) * 10, + ) + xlabel!("t") + ylabel!("E(t)") + title!("energy vs time") +end + +function plot_speed_distribution(sys::ParticleSystem, numSamples::Int64=5, title="speed distribution") + initialize_plot() + + numDataPoints = Int64(length(sys.vData)) + interval = Int64(floor(numDataPoints / numSamples)) + + samples = collect(1:interval:numDataPoints) + for s in samples + speed = sqrt.( + xcomponent(sys.vData[s]).^2 .+ + ycomponent(sys.vData[s]).^2 + ) + # assert that speed is not negative + speed_min = minimum(speed) + speed_max = maximum(speed) + density!( + speed, + normalize = :pdf, + xlims = (speed_min, speed_max), + label = "t = $(round(sys.timeData[s], digits=2))", + top_margin=10mm, + bottom_margin=10mm, + left_margin=10mm, + right_margin=10mm, + ) + end + + + xlabel!("speed") + ylabel!("pdf(speed)") + title!(title) +end + +# CONSOLE PRINT DATA +################################################################################ + +function print_hello() + println("\nmolecular dynamics!") + println("number of threads: ", Threads.nthreads()) +end + +function print_bonjour() + println("\nbonjour") +end + +function print_system_parameters(sys::ParticleSystem) + println("\nsystem parameters:") + println("\tN = $(sys.N) (number of particles)") + println("\tL = $(sys.L) (side length of square box)") + println("\tDT = $(sys.dt) (time step)") +end + +function print_system_data(sys::ParticleSystem) + println("\nsystem data at time t=$(round(sys.t, digits=4))") + + if sys.steps == 0 + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + else + println("\tsteps evolved: $(sys.steps)") + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + println("\tmean energy: $(mean_energy(sys))") + println("\tstd energy: $(std_energy(sys))") + println("\theat capacity: $(heat_capacity(sys))") + println("\tPV/NkT: $(mean_pressure(sys))") + end +end + +function print_evolution_message(runtime, numsteps) + println("\nevolving...") +end + +# DEMOS +################################################################################ + + +# DEMO 0: APPROACH TO EQUILIBRIUM +function demo_0(;gif=0) + println("\nDEMO 0: APPROACH TO EQUILIBRIUM") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 40.0) + print_system_data(sys) + + p2 = plot_trajectories(sys, collect(1:64)) + p3 = plot_energy(sys) + p4 = plot_temperature(sys) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, + layout = grid(2,2, heights=[0.7,0.3]), + size = (1280,720) + ) +end + +# DEMO 1: TIME REVERSAL TEST +function demo_1(;gif=0) + println("\nDEMO 1: TIME REVERSAL TEST") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 50.0) + #p2 = plot_trajectories(sys, collect(1:64)) + p2 = plot_positions(sys) + + reverse_time!(sys) + evolve!(sys, 50.0) + print_system_data(sys) + #p3 = plot_trajectories(sys, collect(1:64)) + p3 = plot_positions(sys) + + # make gif + if gif == 1 + animate(sys, 4) + end + + plot( + p1, p2, p3, + layout = (1,3), + size = (1200,400) + ) +end + +# DEMO 2: SPEED DISTRIBUTION +function demo_2() + println("\nDEMO 2: SPEED DISTRIBUTION") + println("----------------------------------------") + + sys = ParticleSystem[] + + # array for speed distribution plots + ps = Plots.Plot{Plots.GRBackend}[] + + # array for trajectory plots + pt = Plots.Plot{Plots.GRBackend}[] + + # initialize three systems with different initial conditions + # but same KE and PE, evolve, and save plots + for i = 1:3 + push!(sys, ParticleSystem(64, 120.0, 1.0)) + + println("\nSYSTEM $i") + print_system_parameters(sys[i]) + + set_square_lattice_positions!(sys[i]) + add_position_jitter!(sys[i]) + set_random_velocities!(sys[i]) + print_system_data(sys[i]) + + evolve!(sys[i], 500.0) + print_system_data(sys[i]) + push!(ps, plot_speed_distribution(sys[i], 5)) + push!(pt, plot_trajectories(sys[i], collect(1:64)) ) + end + + + # plot speed distribution and trajectory plots + plot( + ps[1], ps[2], ps[3], + pt[1], pt[2], pt[3], + layout = (2,3), + size = (1920,1080) + ) +end + +# DEMO 3: MELTING TRANSITION +function demo_3(;gif=0) + println("\nDEMO 3: MELTING TRANSITION") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + sys = ParticleSystem(100, 10.0, 5.0) + set_square_lattice_positions!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + evolve!(sys, 20.0) + print_system_data(sys) + p2 = plot_trajectories(sys, collect(1:100)) + + # now, increase the temperature of the system by giving the particles + # some velocity. evolve the system and plot the trajectories. + set_random_velocities!(sys) + evolve!(sys, 60.0) + print_system_data(sys) + p3 = plot_trajectories(sys, collect(1:100)) + + # some more plots + p4 = plot_energy(sys, 0.0) + p5 = plot_temperature(sys) + p6 = plot_speed_distribution(sys, 20) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, p5, p6, + layout = (2,3), + size = (1280,720) + ) +end + +function problem_9_2() + println("\nProblem 9.2") + println("----------------------------------------") + + sys = ParticleSystem[] + + # array for speed distribution plots + ps = Plots.Plot{Plots.GRBackend}[] + + # array for trajectory plots + pt = Plots.Plot{Plots.GRBackend}[] + + # initialize three systems with different initial conditions + # but same KE and PE, evolve, and save plots + for i = 1:3 + push!(sys, ParticleSystem(81, 1000.0, 1.0)) # large box so particles are far away + + println("\nSYSTEM $i") + print_system_parameters(sys[i]) + if i == 1 + set_square_lattice_positions!(sys[i]) + elseif i == 2 + set_triangular_lattice_positions!(sys[i]) + elseif i == 3 + set_random_positions!(sys[i]) + else + println("oops") + exit() + end + # add_position_jitter!(sys[i]) + set_random_velocities!(sys[i]) + print_system_data(sys[i]) + + evolve!(sys[i], 300.0) + print_system_data(sys[i]) + push!(ps, plot_speed_distribution(sys[i], 2, "speed distribution over time, system $i")) + push!(pt, plot_trajectories(sys[i], collect(1:64), "initial positions of particles, system $i")) + end + + + # plot speed distribution and trajectory plots + plot( + pt[1], ps[1], + pt[2], ps[2], + pt[3], ps[3], + layout = (3,2), + size = (1080,1200) + ) + + savefig("problem_9_2_long-255p.png") +end + +# demo_0() +problem_9_2() \ No newline at end of file diff --git a/hw5/9-2.png b/hw5/9-2.png new file mode 100644 index 0000000..a59182a Binary files /dev/null and b/hw5/9-2.png differ diff --git a/hw5/HarmonicOscillator.jl b/hw5/HarmonicOscillator.jl new file mode 100644 index 0000000..adfaff8 --- /dev/null +++ b/hw5/HarmonicOscillator.jl @@ -0,0 +1,71 @@ +#!/Applications/Julia-1.8.app/Contents/Resources/julia/bin/julia + +# Simulate anharmonic oscillator that may be damped and driven + +using Plots # for plotting trajectory +using DifferentialEquations # for solving ODEs + +ω0 = 1.0 # ω0^2 = k/m +β = 0.0 # β = b/m = friction +c = 10.0 # anharmonic term +f = 0.3 # forcing amplitude +ω = 2.0 # forcing frequency +param = (ω0, β, c, f, ω) # parameters of anharmonic oscillator + +function tendency!(dxp::Vector{Float64}, xp::Vector{Float64}, param, t::Float64) + + (x, p) = xp # 2d phase space + + (ω0, β, c, f, ω) = param + + a = -ω0^2 * x - β * p - c * x^3 + f * forcing(t, ω) # acceleration with m = 1 + + dxp[1] = p + dxp[2] = a + +end + +function forcing(t::Float64, ω::Float64) + + return cos(ω * t) + +end + +function energy(xp::Vector{Float64}, param) + + (x, p) = xp + + (ω0, β, c, f, ω) = param + + pe = 0.5 * ω0^2 * x^2 + (c/4.0) * x^4 + ke = 0.5 * p^2 + + return pe + ke + +end + +x0 = 0.0 # initial position in meters +p0 = 0.0 # initial velocity in m/s +xp0 = [x0, p0] # initial condition in phase space +t_final = 100.0 # final time of simulation + +tspan = (0.0, t_final) # span of time to simulate + +prob = ODEProblem(tendency!, xp0, tspan, param) # specify ODE +sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8) # solve using Tsit5 algorithm to specified accuracy + +sample_times = sol.t +println("\n\t Results") +println("final time = ", sample_times[end]) +println("Initial energy = ", energy(sol[:,1], param)) +println("Final energy = ", energy(sol[:, end], param)) + +(ω0, β, c, f, ω) = param + +# Plot of position vs. time +xt = plot(sample_times, [sol[1, :] f * forcing.(sample_times, ω)], xlabel = "t", ylabel = "x(t)", legend = false, title = "x vs. t") + +# Phase space plot +xp = plot(sol[1, :], sol[2, :], xlabel = "x", ylabel = "p", legend = false, title = "phase space") + +plot(xt, xp) \ No newline at end of file diff --git a/hw5/MolecularDynamics2.jl b/hw5/MolecularDynamics2.jl new file mode 100644 index 0000000..6580a40 --- /dev/null +++ b/hw5/MolecularDynamics2.jl @@ -0,0 +1,755 @@ +# molecular dynamics 2d. + +# usage: +# at the end of this script, under the header "DEMOS", +# you'll see some functions which implement demos from GN chapter 9. +# simply load the script in your development environment +# (I strongly recommend not using jupiter) +# and in the console/REPL run +# demo_0() +# etc. + +# demos 0,1,3 can optionally make an animated gif +# if you call it with the optional argument demo_3(gif=1) + +# lmk if this script is giving you grief or if you find any bugs +# kian@brown.edu + +using Statistics +using StatsPlots + +mutable struct ParticleSystem + N::Int64 # number of particles + L::Float64 # square box side length + T₀::Float64 # initial temperature + t::Float64 # system time + dt::Float64 # time step + state::Vector{Float64} # state space array + steps::Int64 # number of steps + sampleInterval::Int64 # interval for sampling data + timeData::Vector{Float64} # array of sampled time points + energyData::Vector{Float64} # array of sampled energy values + tempData::Vector{Float64} # array of sampled temperature values + tempAccumulator::Float64 # temperature accumulator + squareTempAccumulator::Float64 # T^2 accumulator + virialAccumulator::Float64 # virial accumulator + xData::Vector{Vector{Float64}} # array of sampled position data + vData::Vector{Vector{Float64}} # array of sampled velocity data + forceType::String # force +end + +function ParticleSystem(N::Int64=64, L::Float64=10.0, T₀::Float64=1.0) + t = 0.0 + dt = 0.001 + state = zeros(4N) # state space array, [x1,y1,x2,y2,...,vx1,vy1,...] + steps = 0 + timeData = Float64[] + energyData = Float64[] + sampleInterval = 100 + tempData = Float64[] + tempAccumulator = 0.0 + squareTempAccumulator = 0.0 + virialAccumulator = 0.0 + xData = Vector{Float64}[] + vData = Vector{Float64}[] + forceType = "lennardJones" + + return ParticleSystem( + N, + L, + T₀, + t, + dt, + state, + steps, + sampleInterval, + timeData, + energyData, + tempData, + tempAccumulator, + squareTempAccumulator, + virialAccumulator, + xData, + vData, + forceType + ) +end + +# some useful "views" of the state array +# (read the performance tips chapter of the julia manual) +@views positions(state) = state[ 1:Int64(length(state)/2) ] +@views velocities(state) = state[ (Int64(length(state)/2)+1):end ] +@views xcomponent(vector) = vector[ 1:2:end ] +@views ycomponent(vector) = vector[ 2:2:end ] +@views particle(n, vector) = [ vector[2n-1], vector[2n] ] + +# INITIALIZATION +################################################################################ + +function set_random_positions!(sys::ParticleSystem) + println("\tposition configuration: random") + positions(sys.state) .= rand(2*sys.N) .* sys.L + cool!(sys) +end + +function set_square_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: square lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end +end + +function set_triangular_lattice_positions!(sys::ParticleSystem) +end + +function add_position_jitter!(sys::ParticleSystem, jitter::Float64=0.5) + println("\tadding a wee bit of random jitter to particle positions...") + + for i = 1:length(positions(sys.state)) + sys.state[i] += rand() - jitter + end +end + +function set_random_velocities!(sys::ParticleSystem) + println("\tvelocity configuration: random") + + velocities(sys.state) .= rand(2*sys.N) .- 0.5 + zero_total_momentum!(sys) + velocities(sys.state) .*= sqrt(sys.T₀/temperature(sys)) +end + +function zero_total_momentum!(sys::ParticleSystem) + xcomponent(velocities(sys.state)) .-= + mean(xcomponent(velocities(sys.state))) + ycomponent(velocities(sys.state)) .-= + mean(ycomponent(velocities(sys.state))) +end + + +# FORCES / POTENTIALS +################################################################################ + +function force(sys::ParticleSystem) + if sys.forceType == "lennardJones" + force, virial = lennard_jones_force(sys) + elseif sys.forceType == "powerLaw" + force, virial = power_law_force(sys) + end + + sys.virialAccumulator += virial + + return force +end + +# the minimum image approximation +# (periodic boundary conditions) +function minimum_image(xij::Float64, L::Float64) + if xij > (L/2) + xij -= L + elseif xij < -(L/2) + xij += L + end + return xij +end + +function lennard_jones_force(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + virial = 0.0 + force = zeros(2*sys.N) + + Threads.@threads for i = 1:(sys.N-1) + for j = (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + f = 48.0 * r2inv^7 - 24.0 * r2inv^4 + fx = dx * f + fy = dy * f + + force[2*i-1] += fx + force[2*i] += fy + force[2*j-1] -= fx + force[2*j] -= fy + + virial += fx * dx + fy * dy + end + end + + return force, 0.5 * virial +end + +function lennard_jones_potential(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + U = 0.0 + + Threads.@threads for i in 1:(sys.N-1) + for j in (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + U += r2inv^6 - r2inv^3 + end + end + return 4.0 * U +end + +function power_law_force(sys::ParticleSystem) +end + +function power_law_potential(sys::ParticleSystem) +end + +# TIME EVOLUTION +################################################################################ + +function keep_particles_in_box!(sys::ParticleSystem) + for i in 1:(2*sys.N) + if positions(sys.state)[i] > sys.L + positions(sys.state)[i] -= sys.L + elseif positions(sys.state)[i] < 0.0 + positions(sys.state)[i] += sys.L + end + end + +# # another way of doing this: use the ternary operator +# for i in 1:(2 * sys.N) +# positions(sys.state)[i] < 0.0 ? +# positions(sys.state)[i] % sys.L + sys.L : +# positions(sys.state)[i] % sys.L +# end +end + +function verlet_step!(sys::ParticleSystem) + # compute acceleration at current time + acceleration = force(sys) + + # compute positions at t + dt + positions(sys.state) .+= + velocities(sys.state) .* sys.dt .+ + 0.5 .* acceleration .* (sys.dt)^2 + + # enforce boundary conditions + # (basically check if any particles left the box and put them back) + # see function implementation for deets + keep_particles_in_box!(sys) + + # compute velocities at t + dt + velocities(sys.state) .+= + 0.5 * sys.dt .* (acceleration + force(sys)) +end + +function evolve!(sys::ParticleSystem, runtime::Float64=10.0) + numsteps = Int64(abs(runtime/sys.dt) + 1) + + print_evolution_message(runtime, numsteps) + + @time for step in 1:numsteps + verlet_step!(sys) + zero_total_momentum!(sys) + + if (step % sys.sampleInterval == 1) + push!(sys.timeData, sys.t) + push!(sys.energyData, energy(sys)) + push!(sys.xData, positions(sys.state)) + push!(sys.vData, velocities(sys.state)) + + T = temperature(sys) + push!(sys.tempData, T) + sys.tempAccumulator += T + sys.squareTempAccumulator += T^2 + end + + sys.t += sys.dt + sys.steps += 1 + end + println("done.") +end + +function reverse_time!(sys) + sys.dt *= -1 + println("\ntime reversed! dt = $(sys.dt)") +end + +function cool!(sys::ParticleSystem, cooltime::Float64=1.0) + numsteps = Int64(cooltime/sys.dt) + for step in 1:numsteps + verlet_step!(sys) + velocities(sys.state) .*= (1.0 - sys.dt) + end + reset_statistics!(sys) +end + +# MEASUREMENTS +################################################################################ + +function kinetic_energy(sys::ParticleSystem) + return 0.5 * sum(velocities(sys.state) .* velocities(sys.state)) +end + +function potential_energy(sys::ParticleSystem) + return lennard_jones_potential(sys) +end + +function temperature(sys::ParticleSystem) + return kinetic_energy(sys) / sys.N +end + +function energy(sys::ParticleSystem) + return potential_energy(sys) + kinetic_energy(sys) +end + +# STATISTICS +################################################################################ + +function reset_statistics!(sys::ParticleSystem) + sys.steps = 0 + sys.tempAccumulator = 0.0 + sys.squareTempAccumulator = 0.0 + sys.virialAccumulator = 0.0 + sys.xData = [] + sys.vData = [] +end + +function mean_temperature(sys::ParticleSystem) + return sys.tempAccumulator / sys.steps +end + +function mean_square_temperature(sys::ParticleSystem) + return sys.squareTempAccumulator / sys.steps +end + +function mean_pressure(sys::ParticleSystem) + # factor of half because force is calculated twice each step + meanVirial = 0.5 * sys.virialAccumulator / sys.steps + return 1.0 + 0.5 * meanVirial / (sys.N * mean_temperature(sys)) +end + +function heat_capacity(sys::ParticleSystem) + meanTemperature = mean_temperature(sys) + meanSquareTemperature = mean_square_temperature(sys) + σ2 = meanSquareTemperature - meanTemperature^2 + denom = 1.0 - σ2 * sys.N / meanTemperature^2 + return sys.N / denom +end + +function mean_energy(sys::ParticleSystem) + return mean(sys.energyData) +end + +function std_energy(sys::ParticleSystem) + return std(sys.energyData) +end + +# MATH / ADDITIONAL FUNCTIONS +################################################################################ + +function dot(v1::Vector{Float64}, v2::Vector{Float64}) + return sum(v1 .* v2) +end + +# GRAPHS +################################################################################ + +function initialize_plot() + plot( + size=(800,800), + titlefontsize=12, + guidefontsize=12, + ) +end + +function plot_positions_t(sys::ParticleSystem, t::Int64) + initialize_plot() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end +end + +function animate(sys::ParticleSystem, interval::Int64=1) + println("\ngenerating gif...") + + scatter!() + animation = @animate for t in 1:length(sys.xData) + scatter() + for n = 1:sys.N + scatter!( + [ sys.xData[t][2n-1] ], + [ sys.xData[t][2n] ], + #markersize = 4.0, + markercolor = n, + #markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + end every interval + + gif(animation, "./animation.gif") + println("done.") +end + +function plot_positions(sys::ParticleSystem) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + title!("positions at t=$(round(sys.t, digits=4))") +end + +function plot_trajectories(sys::ParticleSystem, particles::Vector{Int64}=[ 1 ]) + initialize_plot() + for n = 1:sys.N + scatter!( + [ xcomponent(positions(sys.state))[n] ], + [ ycomponent(positions(sys.state))[n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + + for n in collect(particles) + xdata = [ sys.xData[i][2n-1] for i in 1:length(sys.xData) ] + ydata = [ sys.xData[i][2n] for i in 1:length(sys.xData) ] + + # plot trajectory line for nth particle + scatter!( + xdata, + ydata, + color = n, + #markerstrokewidth = 0, + markerstrokecolor = n, + markersize = 0.7, + markeralpha = 0.5, + label = false, + widen = false, + ) + + # plot initial position for nth particle + scatter!( + [ sys.xData[1][2n-1] ], + [ sys.xData[1][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 0.3, + #label = "pcl. $n @t=t₀", + widen = false, + ) + + # plot final position for nth particle + scatter!( + [ sys.xData[end][2n-1] ], + [ sys.xData[end][2n] ], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 1.0, + #label = "pcl $n @t=t", + widen = false, + ) + end + title!("positions & trajectories at time t=$(round(sys.t, digits=2))") + plot!() +end + +function plot_temperature(sys::ParticleSystem) + initialize_plot() + plot!( + sys.timeData, + sys.tempData, + #widen = true, + ) + ylims!( + mean(sys.tempData) - std(sys.tempData) * 20, + mean(sys.tempData) + std(sys.tempData) * 20, + ) + xlabel!("t") + ylabel!("T(t)") + title!("temperature vs time") + +end + +function plot_energy(sys::ParticleSystem, ylimit::Float64=1.0) + initialize_plot() + plot!( + sys.timeData, + sys.energyData, + #widen = true, + ) + ylims!( + #ylimit * (mean(sys.energyData) - 1), + #ylimit * (mean(sys.energyData) + 1) + mean(sys.energyData) - std(sys.energyData) * 10, + mean(sys.energyData) + std(sys.energyData) * 10, + ) + xlabel!("t") + ylabel!("E(t)") + title!("energy vs time") +end + +function plot_speed_distribution(sys::ParticleSystem, numSamples::Int64=5) + initialize_plot() + + numDataPoints = Int64(length(sys.vData)) + interval = Int64(floor(numDataPoints / numSamples)) + + samples = collect(1:interval:numDataPoints) + for s in samples + speed = sqrt.( + xcomponent(sys.vData[s]).^2 .* + ycomponent(sys.vData[s]).^2 + ) + density!( + sys.vData[s], + normalize = :pdf, + label = "t = $(round(sys.timeData[s], digits=2))", + ) + end + xlabel!("speed") + title!("speed distribution") +end + +# CONSOLE PRINT DATA +################################################################################ + +function print_hello() + println("\nmolecular dynamics!") + println("number of threads: ", Threads.nthreads()) +end + +function print_bonjour() + println("\nbonjour") +end + +function print_system_parameters(sys::ParticleSystem) + println("\nsystem parameters:") + println("\tN = $(sys.N) (number of particles)") + println("\tL = $(sys.L) (side length of square box)") + println("\tDT = $(sys.dt) (time step)") +end + +function print_system_data(sys::ParticleSystem) + println("\nsystem data at time t=$(round(sys.t, digits=4))") + + if sys.steps == 0 + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + else + println("\tsteps evolved: $(sys.steps)") + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + println("\tmean energy: $(mean_energy(sys))") + println("\tstd energy: $(std_energy(sys))") + println("\theat capacity: $(heat_capacity(sys))") + println("\tPV/NkT: $(mean_pressure(sys))") + end +end + +function print_evolution_message(runtime, numsteps) + println("\nevolving...") +end + +# DEMOS +################################################################################ + + +# DEMO 0: APPROACH TO EQUILIBRIUM +function demo_0(;gif=0) + println("\nDEMO 0: APPROACH TO EQUILIBRIUM") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 20.0) + print_system_data(sys) + + p2 = plot_trajectories(sys, collect(1:64)) + p3 = plot_energy(sys) + p4 = plot_temperature(sys) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, + layout = grid(2,2, heights=[0.7,0.3]), + size = (1280,720) + ) +end + +# DEMO 1: TIME REVERSAL TEST +function demo_1(;gif=0) + println("\nDEMO 1: TIME REVERSAL TEST") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 50.0) + #p2 = plot_trajectories(sys, collect(1:64)) + p2 = plot_positions(sys) + + reverse_time!(sys) + evolve!(sys, 50.0) + print_system_data(sys) + #p3 = plot_trajectories(sys, collect(1:64)) + p3 = plot_positions(sys) + + # make gif + if gif == 1 + animate(sys, 4) + end + + plot( + p1, p2, p3, + layout = (1,3), + size = (1200,400) + ) +end + +# DEMO 2: SPEED DISTRIBUTION +function demo_2() + println("\nDEMO 2: SPEED DISTRIBUTION") + println("----------------------------------------") + + sys = ParticleSystem[] + + # array for speed distribution plots + ps = Plots.Plot{Plots.GRBackend}[] + + # array for trajectory plots + pt = Plots.Plot{Plots.GRBackend}[] + + # initialize three systems with different initial conditions + # but same KE and PE, evolve, and save plots + for i = 1:3 + push!(sys, ParticleSystem(64, 120.0, 1.0)) + + println("\nSYSTEM $i") + print_system_parameters(sys[i]) + + set_square_lattice_positions!(sys[i]) + add_position_jitter!(sys[i]) + set_random_velocities!(sys[i]) + print_system_data(sys[i]) + + evolve!(sys[i], 40.0) + print_system_data(sys[i]) + push!(ps, plot_speed_distribution(sys[i], 5)) + push!(pt, plot_trajectories(sys[i], collect(1:64)) ) + end + + + # plot speed distribution and trajectory plots + plot( + ps[1], ps[2], ps[3], + pt[1], pt[2], pt[3], + layout = (2,3), + size = (1920,1080) + ) +end + +# DEMO 3: MELTING TRANSITION +function demo_3(;gif=0) + println("\nDEMO 3: MELTING TRANSITION") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + sys = ParticleSystem(100, 10.0, 5.0) + set_square_lattice_positions!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + evolve!(sys, 20.0) + print_system_data(sys) + p2 = plot_trajectories(sys, collect(1:100)) + + # now, increase the temperature of the system by giving the particles + # some velocity. evolve the system and plot the trajectories. + set_random_velocities!(sys) + evolve!(sys, 60.0) + print_system_data(sys) + p3 = plot_trajectories(sys, collect(1:100)) + + # some more plots + p4 = plot_energy(sys, 0.0) + p5 = plot_temperature(sys) + p6 = plot_speed_distribution(sys, 20) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, p5, p6, + layout = (2,3), + size = (1280,720) + ) +end + +demo_2() \ No newline at end of file diff --git a/hw5/demo_2-f.png b/hw5/demo_2-f.png new file mode 100644 index 0000000..1595c9b Binary files /dev/null and b/hw5/demo_2-f.png differ diff --git a/hw5/demo_2.png b/hw5/demo_2.png new file mode 100644 index 0000000..c84dec6 Binary files /dev/null and b/hw5/demo_2.png differ diff --git a/hw5/fpu-2.jl b/hw5/fpu-2.jl new file mode 100644 index 0000000..3190f2b --- /dev/null +++ b/hw5/fpu-2.jl @@ -0,0 +1,138 @@ +using Plots + +N = 32 # number of masses +b =.3 # cubic string spring +A = 10 # amplitude +modes = 3 # number of modes to plot +final_time = 30 # seconds +dt = .05 # seconds + + +# get the intial positions +function calculate_x_i(mass_num, node_num, num_masses, amplitutude) + return A * + sqrt(2 / (num_masses + 1)) * + sin((mass_num * node_num * π) / (num_masses + 1)) +end + +# dynamics calculations +function dynamics!( + mass_displacement, # 2d array + params) + (beta, delta_t, num_masses, num_steps) = params + + for step in 1:num_steps-1 + if step == 1 + continue + end + for mass_num in 2:num_masses-1 + + if step == 1 + prev_mass_pos = 0 + else + prev_mass_pos = mass_displacement[mass_num, step - 1] + end + + right_mass_pos = mass_displacement[mass_num + 1, step] + left_mass_pos = mass_displacement[mass_num - 1, step] + + mass_pos = mass_displacement[mass_num, step] + + xs[mass_num, step + 1] = caluclate_next_displacement( + mass_pos, prev_mass_pos, + left_mass_pos, right_mass_pos, beta, delta_t) + end + + # update the end points + mass_displacement[1, step + 1] = caluclate_next_displacement( + 0, 0, + 0, mass_displacement[2, step], beta, delta_t) + mass_displacement[num_masses, step + 1] = caluclate_next_displacement( + 0, 0, + mass_displacement[num_masses - 1, step], 0, beta, delta_t) + end + +end + +function caluclate_next_displacement( + mass_pos, prev_mass_pos, + left_mass_pos, right_mass_pos, + beta, delta_t +) + # println(mass_pos, " " , prev_mass_pos, " ", left_mass_pos, " ", right_mass_pos, '\n') + return 2 * mass_pos - prev_mass_pos + + delta_t^(2) * (right_mass_pos + left_mass_pos - 2 * mass_pos) + + beta * delta_t^(2) * ((right_mass_pos - mass_pos)^3 + (left_mass_pos - mass_pos)^3) +end + + +# energy calcuations, after the dynamics +function calculate_a_k(k, num_masses, delta_t, xs_n, beta) + sum = 0 + for i in 1:num_masses + sum += xs_n[i] * sin((k * i * π) / (num_masses + 1)) + end + return sqrt(2 / (num_masses + 1)) * sum +end + +function calculate_energy(a_k, a_k_plus1, delta_t, mode, num_masses) + kinetic = .5 * ((a_k_plus1 - a_k) / delta_t)^2 + + # sum over the three modes + w_k = 2 * sin((mode * π) / (2 * (num_masses + 1))) + potential = .5 * (w_k^2 * a_k^2) + + return kinetic + potential +end + + +# do the simulation +num_steps = Int(final_time / dt) +params = (b, dt, N, num_steps) +# build the 2d array of mass displacements to time +xs = zeros(N, num_steps) +# fill in the initial displacements +for mass_num in 2:N-1 + xs[mass_num, 1] = calculate_x_i(mass_num, 1, N, A) +end +dynamics!(xs, params) + +# println(xs) + + +# plot these displacements over time as an animation +a = @animate for i in 1:num_steps + plot(xs[:, i], legend=false, + marker = :circle, xlabel="Mass Number", ylabel="Displacement", + yaxis = (-30, 30), title="Displacement Over Time" + ) +end + +gif(a, "fpu.gif", fps=15) + +# plot the first two timespets positions +# plot(xs[:, 1], label="t=0", marker=:circle, xlabel="Mass Number", ylabel="Displacement", title="First Two Time Steps") +# plot!(xs[:, 2], label="t=1", marker=:circle) +# plot!(xs[:, 3], label="t=2", marker=:circle) +# plot!(xs[:, 4], label="t=3", marker=:circle) +# plot!(xs[:, 5], label="t=4", marker=:circle) +# plot!(xs[:, 6], label="t=5", marker=:circle) +# plot!(xs[:, 7], label="t=6", marker=:circle) + + +# # calculate the energies for each mode from the displacements +# energies = zeros(modes, num_steps) +# for mode in 1:modes +# energies[mode, :] = calculate_energy_for_mode(mode, xs, N, dt, b) +# end + +# make a range time steps +# time = range(0, final_time, length=num_steps) + +# println("e:", length(energies[1, :])) +# println("t:", length(time)) + +# plot the energies for each mode +# scatter(time, energies[1, :], label="Mode 1", marker=:circle, xlabel="Time", ylabel="Energy", title="Energy for First Three Modes") +# scatter!(time, energies[2, :], label="Mode 2", marker=:circle) +# scatter!(time, energies[3, :], label="Mode 3", marker=:circle) \ No newline at end of file diff --git a/hw5/fpu-3.jl b/hw5/fpu-3.jl new file mode 100644 index 0000000..f2ceb38 --- /dev/null +++ b/hw5/fpu-3.jl @@ -0,0 +1,78 @@ +N = 32 # number of masses +b = 10 # cubic string spring +A = 1 # amplitude +modes = 3 # number of modes to plot +final_time = 10 # seconds +dt = .05 # seconds + +function kinetic_energy(a_k, a_k_plus1, delta_t) + return .5 * ((a_k_plus1 - a_k) / delta_t)^2 +end + +function potential_energy(a_k, a_k_plus1, mode, num_masses) + w_k = 2 * sin((mode * π) / (2 * (num_masses + 1))) + return .5 * (w_k^2 * a_k^2) +end + +function calculate_energy(a_k, a_k_plus1, delta_t, mode, num_masses) + k = kinetic_energy(a_k, a_k_plus1, delta_t) + p = potential_energy(a_k, a_k_plus1, mode, num_masses) + + return k + p +end + +function update_state!(state, prev_state, dt, beta) + x_prev = prev_state + x = copy(state) + + # update the left particle (set to rest) + state[1] = 0 + + # update the right particle (set to rest) + state[N] = 0 + + # update the middle particles + for i in 2:N-1 + state[i] = 2 * x[i] - x_prev[i] + + dt * dt * ((x[i + 1] - 2 * x[i] + x[i - 1]) + + dt * dt * beta * ((x[i + 1] - x[i])^3 - (x[i - 1] - x[i])^3) + ) + end +end + +function initial_state(N, modes, beta, A) + # make the range of masses + masses = 1:N + # make the range of modes + init_state = A * sin.((modes * π * masses) / (N + 1)) + init_state[1] = 0 + init_state[N] = 0 + return init_state +end + +function run_simulation(N, modes, beta, A, dt, num_steps) + data = [] + state = initial_state(N, modes, beta, A) + prev_state = copy(state) + println("Initial state: ", state) + for i in 1:num_steps + update_state!(state, prev_state, dt, beta) + push!(data, state) + prev_state = copy(state) + end + + return data +end + +steps = Int(final_time / dt) + +p = run_simulation(N, modes, b, A, dt, steps) + +using Plots +# plot the first and final position +plot(p[1], label="Initial position") +plot!(p[end], label="Final position") + +# plot displacement for the first particle over time +t_span = 0:dt:final_time +plot(t_span, p, label="Displacement for first particle") diff --git a/hw5/fpu.gif b/hw5/fpu.gif new file mode 100644 index 0000000..e1d1612 Binary files /dev/null and b/hw5/fpu.gif differ diff --git a/hw5/fpu.jl b/hw5/fpu.jl new file mode 100644 index 0000000..327cf3a --- /dev/null +++ b/hw5/fpu.jl @@ -0,0 +1,35 @@ +N = 32 # number of masses +beta =.3 # cubic string spring +A = 10 # amplitude +modes = 3 # number of modes to plot + + +function calculate_x_i(mass_num, node_num, num_masses, amplitutude) + return A * + sqrt(2 / (num_masses + 1)) * + sin((mass_num * node_num * π) / (num_masses + 1)) +end + +function calculate_x_s_for_mode(mass_num, node_num, num_masses, amplitutude) + x = 0 + for i in 1:num_masses + x += calculate_x_i(mass_num, node_num, num_masses, amplitutude) + end + return x +end + +# calculate the x_s for the first three modes + +x_s = zeros(modes, N+2) # add two for zeros on ends +# add the first and last 0s +for i in 1:modes + for j in 1:N + x_s[i, j+1] = calculate_x_s_for_mode(j, i, N, A) + end +end + +# plot the first three modes +using Plots +plot(x_s[1, :], label="Mode 1", marker=:circle, xlabel="Mass Number", ylabel="Displacement", title="First Three Modes") +plot!(x_s[2, :], label="Mode 2", marker=:circle) +plot!(x_s[3, :], label="Mode 3", marker=:circle) \ No newline at end of file diff --git a/hw5/fpu.png b/hw5/fpu.png new file mode 100644 index 0000000..52b6839 Binary files /dev/null and b/hw5/fpu.png differ diff --git a/hw5/lfg-modes.png b/hw5/lfg-modes.png new file mode 100644 index 0000000..ceffab8 Binary files /dev/null and b/hw5/lfg-modes.png differ diff --git a/hw5/modes-beta15.png b/hw5/modes-beta15.png new file mode 100644 index 0000000..6536741 Binary files /dev/null and b/hw5/modes-beta15.png differ diff --git a/hw5/modes-beta3.png b/hw5/modes-beta3.png new file mode 100644 index 0000000..1b2833f Binary files /dev/null and b/hw5/modes-beta3.png differ diff --git a/hw5/plot_175.svg b/hw5/plot_175.svg new file mode 100644 index 0000000..4733b20 --- /dev/null +++ b/hw5/plot_175.svg @@ -0,0 +1,105 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/hw5/plz-modes.png b/hw5/plz-modes.png new file mode 100644 index 0000000..465252b Binary files /dev/null and b/hw5/plz-modes.png differ diff --git a/hw5/plz.gif b/hw5/plz.gif new file mode 100644 index 0000000..febf678 Binary files /dev/null and b/hw5/plz.gif differ diff --git a/hw5/plz.jl b/hw5/plz.jl new file mode 100644 index 0000000..d4abcbd --- /dev/null +++ b/hw5/plz.jl @@ -0,0 +1,148 @@ +function dynamics!( + state, + prev_state, + params, + states +) + # Unpack the parameters + N, modes, beta, A, dt, num_steps = params + + for i in 1:num_steps + next_state = zeros(N) + + # update the left particle (set to rest) + state[1] = 0 + + # update the right particle (set to rest) + state[N] = 0 + + # update the middle particles + for i in 2:N-1 + a = 2 * state[i] - prev_state[i] + b = dt * dt * (state[i + 1] - 2 * state[i] + state[i - 1]) + c = dt * dt * beta * ((state[i + 1] - state[i])^3 + (state[i - 1] - state[i])^3) + next_state[i] = a + b + c + end + + push!(states, next_state) + + # update the previous state + for i in 1:N + prev_state[i] = state[i] + end + # update the current state + for i in 1:N + state[i] = next_state[i] + end + end +end + +function get_initial_state( + N, + modes, + beta, + A +) + state = zeros(N) + amp = A * sqrt(2 / (N - 1)) + for i in 2:N-1 + state[i] = amp * sin((modes * π * (i - 1)) / (N - 1)) + end + return state +end + +function run_simulation( + N, + modes, + beta, + A, + dt, + num_steps +) + states = [] + state = get_initial_state(N, 1, beta, A) + prev_state = zeros(N) + for i in 1:N + prev_state[i] = state[i] + end + params = (N, modes, beta, A, dt, num_steps) + dynamics!(state, prev_state, params, states) + return states +end + +# Run the simulation +N = 32 # number of masses +beta = 1.5 # cubic string spring +A = 10 # amplitude +modes = 3 # number of modes to plot +final_time = 50 # seconds +dt = .5 # seconds +num_steps = Int(final_time / dt) +params = (N, modes, beta, A, dt, num_steps) +s = run_simulation(N, modes, beta, A, dt, num_steps) +println("Final state: ", s[end]) + +# plot the inital positions and the final positions +using Plots +# plot(get_initial_state(N, 1, beta, A), label="Initial", marker=:circle, xlabel="Mass Number", ylabel="Displacement", title="First Three Modes") +# plot!(s[end], label="Final", marker=:circle) + +# animate the s array of positions +anim = @animate for i in 1:length(s) + plot(s[i], label="t = $(i * dt)", marker=:circle, xlabel="Mass Number", ylabel="Displacement", ylim=(-3, 3)) +end +gif(anim, "plz.gif", fps = 30) + +# function caluclate_energies_for_mode(states, mode) +# total = [] +# kinetic = [] +# potential = [] +# prev_a_k = 0 +# for i in 1:length(states) - 1 +# total_energy = 0 +# kinetic_energy = 0 +# potential_energy = 0 + +# sum = 0 +# for j in 1:length(states[i]) +# sum += states[i][j] * sin((mode * j * π) / (N - 1)) +# end +# amp = A * sqrt(2 / (N - 1)) +# a_k = amp * sum + +# k = .5 * ((a_k - prev_a_k) / dt)^2 +# if i == 1 +# k = 0 +# end +# p = (2 * sin((mode * π) / (2 * (N - 1)))^2 * a_k^2) + + +# total_energy += k + p +# kinetic_energy += k +# potential_energy += p +# push!(total, total_energy) +# push!(kinetic, kinetic_energy) +# push!(potential, potential_energy) + +# prev_a_k = a_k +# end +# return total, kinetic, potential +# end + +# # calculate the energies for each mode from the displacements +# total_1, kinetic_1, potential_2 = caluclate_energies_for_mode(s, 1) +# total_2, kinetic_2, potential_2 = caluclate_energies_for_mode(s, 2) +# total_3, kinetic_3, potential_3 = caluclate_energies_for_mode(s, 3) + +# # build a timestep space +# # timesteps = [i * dt for i in 1:num_steps - 1] +# # # plot the modes energies on the same y-axis +# plot(timesteps, total_1, label="Mode 1", xlabel="Time", ylabel="Energy of Modes", title="Energy Over Time (\$\\beta = 1.5\$)") +# plot!(timesteps, total_2, label="Mode 3") +# plot!(timesteps, total_3, label="Mode 5") + +# savefig("modes-beta15.png") + + + + diff --git a/hw5/plz.png b/hw5/plz.png new file mode 100644 index 0000000..d6e7209 Binary files /dev/null and b/hw5/plz.png differ diff --git a/hw5/problem_9_-test.png b/hw5/problem_9_-test.png new file mode 100644 index 0000000..894c5be Binary files /dev/null and b/hw5/problem_9_-test.png differ diff --git a/hw5/problem_9_11.png b/hw5/problem_9_11.png new file mode 100644 index 0000000..abe5054 Binary files /dev/null and b/hw5/problem_9_11.png differ diff --git a/hw5/problem_9_11_100.png b/hw5/problem_9_11_100.png new file mode 100644 index 0000000..e1025cb Binary files /dev/null and b/hw5/problem_9_11_100.png differ diff --git a/hw5/problem_9_11_16.png b/hw5/problem_9_11_16.png new file mode 100644 index 0000000..98b4a68 Binary files /dev/null and b/hw5/problem_9_11_16.png differ diff --git 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println("derivative is too small") + return x + end + x = x - f(x, T) / df(x, T) + if abs(f(x, T)) < tol + return x + end + end + return x +end + +# initial guess +s0 = .5 + +# solve the equation +t = newton_solver(equation, derivative, s0, 1e-6, 100, 1) +println("s(T) = ", t) + +# go over Ts from 0 - 8 and plot the solutions +using Plots +Ts = .001:0.001:8 +ss = [newton_solver(equation, derivative, s0, 1e-6, 100, T) for T in Ts] +scatter( + Ts, ss, xlabel="T (units of J / k_b)", ylabel="Magnetization", label="Numerical Solution", title="Magnetization vs Temp (Mean Field Theory)", + msw=0, ms=1.5, mc=:blue, lc=:red, lw=1.5 +) + +# plot the analytical solution +function analytical_solution(T) + n1 = (3 / T) * (T / z) ^ 3 + n2 = z - T + if n1 < 0 || n2 < 0 + return 0 + end + return sqrt(n1) * sqrt(n2) +end +plot!( + Ts, [analytical_solution(T) for T in Ts], label="Analytical Solution", + lc=:green, lw=2 +) + +# make a plot of the difference between the analytical and numerical solutions +diffs = [abs(analytical_solution(T) - newton_solver(equation, derivative, s0, 1e-6, 100, T)) for T in Ts] +plot!(Ts, diffs, label="Difference in Solutions", lc=:red, lw=1.5, ls=:dot) + +# plot a vertical line where T = 3.75 +vline!([3.9], label="T = 3.9", lc=:orange, lw=1.5, ls=:dash) + +# save the plot +savefig("8-1.png") \ No newline at end of file diff --git a/hw6/8-1.png b/hw6/8-1.png new file mode 100644 index 0000000..c9d25eb Binary files /dev/null and b/hw6/8-1.png differ diff --git a/hw6/8-11.jl b/hw6/8-11.jl new file mode 100644 index 0000000..e8a0b2e --- /dev/null +++ b/hw6/8-11.jl @@ -0,0 +1,198 @@ +# FOR PROBLEM 8.11 +# author: sotech117 + +using Statistics +using Plots + +function wrap_index(i::Int, l::Int)::Int + wrap = (i - 1) % l + 1 + return (wrap <= 0) ? l + wrap : wrap +end + +mutable struct Ising2D + l::Int + n::Int + temperature::Float64 + w::Vector{Float64} # Boltzmann weights + state::Matrix + energy::Int + magnetization::Int + mc_steps::Int + accepted_moves::Int + energy_array::Vector{Int} + magnetization_array::Vector{Int} +end + +Ising2D(l::Int, temperature::Float64) = begin + n = l^2 + w = zeros(9) + w[9] = exp(-8.0 / temperature) + w[5] = exp(-4.0 / temperature) + state = ones(Int, l, l) # initially all spins up + energy = -2 * n + magnetization = n + return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, + Int[], Int[]) +end + +function reset!(ising::Ising2D) + ising.mc_steps = 0 + ising.accepted_moves = 0 + ising.energy_array = Int[] + ising.magnetization_array = Int[] +end + +function mc_step!(ising::Ising2D) + l::Int = ising.l + n::Int = ising.n + w = ising.w + + state = ising.state + accepted_moves = ising.accepted_moves + energy = ising.energy + magnetization = ising.magnetization + + random_positions = l * rand(2 * n) + random_array = rand(n) + + for k in 1:n + i = trunc(Int, random_positions[2 * k - 1]) + 1 + j = trunc(Int, random_positions[2 * k]) + 1 + + de = 2 * state[i, j] * (state[i % l + 1, j] + + state[wrap_index(i - 1, l), j] + state[i, j % l + 1] + + state[i, wrap_index(j - 1, l)]) + + if de <= 0 || w[de + 1] > random_array[k] + accepted_moves += 1 + new_spin = - state[i, j] # flip spin + state[i, j] = new_spin + energy += de + magnetization += 2 * new_spin + end + + end + + ising.state = state + ising.accepted_moves = accepted_moves + ising.energy = energy + ising.magnetization = magnetization + + append!(ising.energy_array, ising.energy) + append!(ising.magnetization_array, ising.magnetization) + ising.mc_steps = ising.mc_steps + 1 +end + +function steps!(ising::Ising2D, num::Int=100) + for i in 1:num + mc_step!(ising) + end +end + +function mean_energy(ising::Ising2D) + return mean(ising.energy_array) / ising.n +end + +function specific_heat(ising::Ising2D) + return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n +end + +function mean_magnetization(ising::Ising2D) + return mean(ising.magnetization_array) / ising.n +end + +function susceptibility(ising::Ising2D) + return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n) +end + +function observables(ising::Ising2D) + printstyled("Temperature\t\t", bold=true) + print(ising.temperature); print("\n") + + printstyled("Mean Energy\t\t", bold=true) + print(mean_energy(ising)); print("\n") + + printstyled("Mean Magnetiz.\t\t", bold=true) + print(mean_magnetization(ising)); print("\n") + + printstyled("Specific Heat\t\t", bold=true) + print(specific_heat(ising)); print("\n") + + printstyled("Susceptibility\t\t", bold=true) + print(susceptibility(ising)); print("\n") + + printstyled("MC Steps\t\t", bold=true) + print(ising.mc_steps); print("\n") + printstyled("Accepted Moves\t\t", bold=true) + print(ising.accepted_moves); print("\n") +end + + +function plot_ising(state::Matrix{Int}) + pos = Tuple.(findall(>(0), state)) + neg = Tuple.(findall(<(0), state)) + scatter(pos, markersize=5) + scatter!(neg, markersize=5) +end + + +function get_magnetization(T, n=1000) + m = Ising2D(64, T) + steps!(m, n) + + println("done with T = $T") + return mean_magnetization(m) +end + +Ts = 0:.1:5 +ms = [abs(get_magnetization(T)) for T in Ts] + +println("done with calculating magnetizations") + +function linear_regression(x, y) + n = length(x) + x̄ = sum(x) / n + ȳ = sum(y) / n + a = sum((x .- x̄) .* (y .- ȳ)) / sum((x .- x̄).^2) + b = ȳ - a * x̄ + return (a, b) +end + +# plot M^(8) over T +betas = .001:.001:1 +residuals = [] +for i in 1:length(betas) + b = betas[i] + m = ms .^ (1 / b) + # filter out the zero values + + + s = scatter(p, + Ts, m, xlabel="T (units of J / k_b)", ylabel="Magnetization", label="$b-beta", title="Magnetization vs Temp (Ising Monte Carlo)", + msw=0, ms=1.5, mc=:red, lc=:red, lw=1.5, legend=:bottomleft + ) + + + # do a linear regression + a, b = linear_regression(Ts, m) + + # plot a linear regression line + plot!(s, Ts, a*Ts .+ b, label="Linear Regression", lw=1.2, color=:red, linestyle=:dash) + + # calculate the residuals + push!(residuals, sum((m .- (a*Ts .+ b)).^2)) + + + savefig(s, "hw6/b/8-2-$i.png") +end + +# plot the residuals over beta +plot(betas, residuals, xlabel="beta", ylabel="Squared Distance", label="Residuals", title="Error from Linear Regression of M^(1/Beta)", lw=1.2, lc=:red, legend=:topright) +# find the min on the first half of the residuals +min_residuali = argmin(residuals[1:div(length(residuals), 2)]) +min_residual = betas[min_residuali] +println("Minimum Residual: ", min_residual) +vline!([min_residual], label="Minimum Point @ Beta = $min_residual", lc=:orange, lw=1.5, ls=:dash) +# plot the analityical beta of .125 +vline!([.125], label="Analytical Minimum Beta = .125", lc=:green, lw=1.5, ls=:dot) +savefig("hw6/8-2-residuals-100.png") \ No newline at end of file diff --git a/hw6/8-2-residuals-100.png b/hw6/8-2-residuals-100.png new file mode 100644 index 0000000..2731d33 Binary files /dev/null and b/hw6/8-2-residuals-100.png differ diff --git a/hw6/8-2-residuals-2.png b/hw6/8-2-residuals-2.png new file mode 100644 index 0000000..20b366a Binary files /dev/null and b/hw6/8-2-residuals-2.png differ diff --git a/hw6/8-2-residuals.png b/hw6/8-2-residuals.png new file mode 100644 index 0000000..dba70e5 Binary files /dev/null and b/hw6/8-2-residuals.png differ diff --git a/hw6/IsingModel.jl b/hw6/IsingModel.jl new file mode 100644 index 0000000..d1e04de --- /dev/null +++ b/hw6/IsingModel.jl @@ -0,0 +1,141 @@ +#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia + +using Statistics +using Plots + +function wrap_index(i::Int, l::Int)::Int + wrap = (i - 1) % l + 1 + return (wrap <= 0) ? l + wrap : wrap +end + +mutable struct Ising2D + l::Int + n::Int + temperature::Float64 + w::Vector{Float64} # Boltzmann weights + state::Matrix + energy::Int + magnetization::Int + mc_steps::Int + accepted_moves::Int + energy_array::Vector{Int} + magnetization_array::Vector{Int} +end + +Ising2D(l::Int, temperature::Float64) = begin + n = l^2 + w = zeros(9) + w[9] = exp(-8.0 / temperature) + w[5] = exp(-4.0 / temperature) + state = ones(Int, l, l) # initially all spins up + energy = -2 * n + magnetization = n + return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, + Int[], Int[]) +end + +function reset!(ising::Ising2D) + ising.mc_steps = 0 + ising.accepted_moves = 0 + ising.energy_array = Int[] + ising.magnetization_array = Int[] +end + +function mc_step!(ising::Ising2D) + l::Int = ising.l + n::Int = ising.n + w = ising.w + + state = ising.state + accepted_moves = ising.accepted_moves + energy = ising.energy + magnetization = ising.magnetization + + random_positions = l * rand(2 * n) + random_array = rand(n) + + for k in 1:n + i = trunc(Int, random_positions[2 * k - 1]) + 1 + j = trunc(Int, random_positions[2 * k]) + 1 + + de = 2 * state[i, j] * (state[i % l + 1, j] + + state[wrap_index(i - 1, l), j] + state[i, j % l + 1] + + state[i, wrap_index(j - 1, l)]) + + if de <= 0 || w[de + 1] > random_array[k] + accepted_moves += 1 + new_spin = - state[i, j] # flip spin + state[i, j] = new_spin + energy += de + magnetization += 2 * new_spin + end + + end + + ising.state = state + ising.accepted_moves = accepted_moves + ising.energy = energy + ising.magnetization = magnetization + + append!(ising.energy_array, ising.energy) + append!(ising.magnetization_array, ising.magnetization) + ising.mc_steps = ising.mc_steps + 1 +end + +function steps!(ising::Ising2D, num::Int=100) + for i in 1:num + mc_step!(ising) + end +end + +function mean_energy(ising::Ising2D) + return mean(ising.energy_array) / ising.n +end + +function specific_heat(ising::Ising2D) + return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n +end + +function mean_magnetization(ising::Ising2D) + return mean(ising.magnetization_array) / ising.n +end + +function susceptibility(ising::Ising2D) + return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n) +end + +function observables(ising::Ising2D) + printstyled("Temperature\t\t", bold=true) + print(ising.temperature); print("\n") + + printstyled("Mean Energy\t\t", bold=true) + print(mean_energy(ising)); print("\n") + + printstyled("Mean Magnetiz.\t\t", bold=true) + print(mean_magnetization(ising)); print("\n") + + printstyled("Specific Heat\t\t", bold=true) + print(specific_heat(ising)); print("\n") + + printstyled("Susceptibility\t\t", bold=true) + print(susceptibility(ising)); print("\n") + + printstyled("MC Steps\t\t", bold=true) + print(ising.mc_steps); print("\n") + printstyled("Accepted Moves\t\t", bold=true) + print(ising.accepted_moves); print("\n") +end + + +function plot_ising(state::Matrix{Int}) + pos = Tuple.(findall(>(0), state)) + neg = Tuple.(findall(<(0), state)) + scatter(pos, markersize=5) + scatter!(neg, markersize=5) +end + + +m = Ising2D(32, 2.3) +steps!(m, 1000) +observables(m) +plot_ising(m.state) diff --git a/hw6/b-old/8-2-1.png b/hw6/b-old/8-2-1.png new file mode 100644 index 0000000..321c41c Binary files /dev/null and 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a[1:end-2, 2:end-1] - + 4.0 * a[2:end-1, 2:end-1]) ./ (dx^2) + return del2 +end + +function del2_9(a::Matrix{Float64}, dx = 1.0) + #= + Returns a finite-difference approximation of the laplacian of the array a, + with lattice spacing dx, using the nine-point stencil: + + 1/6 2/3 1/6 + 2/3 -10/3 2/3 + 1/6 2/3 1/6 + =# + + del2 = zeros(size(a)) + del2[2:end-1, 2:end-1] .= (4.0 * (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] + + a[3:end, 2:end-1] + a[1:end-2, 2:end-1]) + + (a[1:end-2, 1:end-2] + a[1:end-2, 3:end] + + a[3:end, 1:end-2] + a[3:end, 3:end]) - + 20.0 * a[2:end-1, 2:end-1]) ./ (6.0 * dx^2) + return del2 +end + +function invDel2_5(b::Matrix{Float64}, dx = 1.0, N = 10000) + #= + Relaxes over N steps to a discrete approximation of the inverse laplacian + of the source term b, each step is a weighted average over the four neighboring + points and the source. This is the Jacobi algorithm. + =# + + invDel2 = zeros(size(b)) + newInvDel2 = zeros(size(b)) + + for m in 1:N + newInvDel2[2:end-1, 2:end-1] .= 0.25 * (invDel2[2:end-1, 3:end] + + invDel2[2:end-1, 1:end-2] + + invDel2[3:end, 2:end-1] + + invDel2[1:end-2, 2:end-1] - + (dx^2) * b[2:end-1, 2:end-1]) + invDel2 .= newInvDel2 + end + + diff = del2_5(invDel2, dx) - b + diffSq = diff .* diff + error = sqrt(sum(diffSq)) + + println("\nerror = ", error) + + return invDel2 +end + + +# # Define variables +# L = 10 +# dx = 1.0 + +# # Parabola of revolution has constant Laplacian +# phi = zeros((L, L)) +# for i ∈ 1:L +# x = (i + 0.5 - 0.5 * L) * dx +# for j ∈ 1:L +# y = (j + 0.5 - 0.5 * L) * dx +# phi[i, j] = x^2 + y^2 +# end +# end + +# println(size(phi)) + +# println(del2_5(phi, dx)) + +# println("\n\n") + +# println(del2_9(phi, dx)) + +# Electrostatics example: uniformly charged cylinder (rho = 1) of radius R +L = 100 +dx = 1.0 +R = 1.0 +R2 = R^2 + +# Charge distribution +rho = zeros((L, L)) +for i ∈ 1:L + y = (i - 1 + 0.5 - 0.5 * L) * dx + println("y = ", y) + for j ∈ 1:L + x = (j - 1 + 0.5 - 0.5 * L) * dx + # this is the rod here + if y < 0 && abs(x) < R + rho[i, j] = 1.0 # high voltage + elseif y > 0 && abs(x) < R + rho[i, j] = -1.0 # conducting plane + else + rho[i, j] = 0.0 + end + end +end + +rhoPlot = plot(rho[Int(L / 2), :]) + +# Exact (analytical) electric potential +# phi = zeros((L, L)) +# for i ∈ 1:L +# x = (i + 0.5 - 0.5 * L) * dx +# for j ∈ 1:L +# y = (j + 0.5 - 0.5 * L) * dx +# r2 = x^2 + y^2 +# if r2 < R2 +# phi[i, j] = -π * r2 +# else +# phi[i, j] = -π * (R2 + R2 * log(r2 / R2)) +# end +# end +# end + +# phiPlot = plot(phi[Int(L / 2), :]) + +# Charge density obtained from exact potential +# rho = -1.0 / (4.0 * π) * del2_5(phi, dx) +# rhoPlotLattice = plot(rho[Int(L / 2), :]) + +function calc_field(phi::Matrix{Float64}, dx = 1.0) + E_x = zeros(size(phi)) + E_y = zeros(size(phi)) + + E_x[2:end-1, 2:end-1] .= -(phi[3:end, 2:end-1] - phi[1:end-2, 2:end-1]) / (2 * dx) + E_y[2:end-1, 2:end-1] .= -(phi[2:end-1, 3:end] - phi[2:end-1, 1:end-2]) / (2 * dx) + + # for i ∈ 2:size(phi, 1)-1 + # for j ∈ 2:size(phi, 2)-1 + # E_x[i, j] = -(phi[i+1, j] - phi[i-1, j]) / (2 * dx) + # E_y[i, j] = -(phi[i, j+1] - phi[i, j-1]) / (2 * dx) + # end + # end + + return E_x, E_y +end + +phi = invDel2_5(-4.0 * π * rho, dx, 1000) +# phi = invDelSOR(-4.0 * π * rho, dx, 500) +phi .-= phi[Int(L / 2), Int(L / 2)] + +phiPlotInvDel = plot(phi[Int(L / 2), :]) + +# phi_contour = contourf(phi, title = "Electric Potential", color = :viridis, aspect_ratio = :equal, colorbar_title = "V") +# savefig(phi_contour, "hw7/5.6-phi.png") + +# plot the arrows of the magnatic field on the contour plot +xxs = [x for x in 1:L for y in 1:L] +xxy = [y for x in 1:L for y in 1:L] +d_x, d_y = calc_field(phi, dx) + +function df(i, j) + norm = d_x[Int64(i)] * d_x[Int64(i)] + d_y[Int64(j)] * d_y[Int64(j)] + if norm == 0 + return (0, 0) + end + return (d_x[Int64(i)] / norm, d_y[Int64(j)] / norm) +end + +quiver(xxs, xxy, quiver = df, aspect_ratio = :equal, legend = false) +savefig("hw7/5.6-test.png") + + +# d_x, d_y = calc_field(phi, dx) +# field_y = contourf(d_y, title = "Electric Field (y)", color = :viridis, aspect_ratio = :equal, colorbar_title = "V/m") + +# field_x = contourf(d_x, title = "Electric Field (x)", color = :viridis, aspect_ratio = :equal, colorbar_title = "V/m") + + +# savefig(field_y, "hw7/5.6-fy.png") +# savefig(field_x, "hw7/5.6-fx.png") + +#plot(rhoPlot, rhoPlotLattice) +#plot(phiPlot, phiPlotInvDel) diff --git a/hw7/5.6.png b/hw7/5.6.png new file mode 100644 index 0000000..f17824a Binary files /dev/null and b/hw7/5.6.png differ diff --git a/hw7/8-12.jl b/hw7/8-12.jl new file mode 100644 index 0000000..29e18ec --- /dev/null +++ b/hw7/8-12.jl @@ -0,0 +1,191 @@ +#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia + +using Statistics +using Plots + +function wrap_index(i::Int, l::Int)::Int + wrap = (i - 1) % l + 1 + return (wrap <= 0) ? l + wrap : wrap +end + +mutable struct Ising2D + l::Int + n::Int + temperature::Float64 + w::Vector{Float64} # Boltzmann weights + state::Matrix + energy::Float64 + magnetization::Int + mc_steps::Int + accepted_moves::Int + energy_array::Vector{Float64} + magnetization_array::Vector{Int} + H::Float64 +end + +Ising2D(l::Int, temperature::Float64, H=1.0) = begin + n = l^2 + w = zeros(9) + w[9] = exp(-8.0 / temperature) + w[5] = exp(-4.0 / temperature) + state = ones(Int, l, l) # initially all spins up + energy = Float64(-2 * n + 2 * H * n) + magnetization = n + return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, + Int[], Int[], H) +end + +function reset!(ising::Ising2D) + ising.mc_steps = 0 + ising.accepted_moves = 0 + ising.energy_array = Int[] + ising.magnetization_array = Int[] +end + +function mc_step!(ising::Ising2D) + l::Int = ising.l + n::Int = ising.n + w = ising.w + + state = ising.state + accepted_moves = ising.accepted_moves + energy = ising.energy + magnetization = ising.magnetization + + random_positions = l * rand(2 * n) + random_array = rand(n) + + for k in 1:n + i = trunc(Int, random_positions[2 * k - 1]) + 1 + j = trunc(Int, random_positions[2 * k]) + 1 + + changed_spins = state[i, j] * (state[i % l + 1, j] + + state[wrap_index(i - 1, l), j] + state[i, j % l + 1] + + state[i, wrap_index(j - 1, l)]) + de = 2 * changed_spins + 2 * ising.H * state[i, j] + + if de <= 0 || rand() < exp(-de / ising.temperature) + accepted_moves += 1 + new_spin = - state[i, j] # flip spin + state[i, j] = new_spin + + # add the effects of the new spin + energy += de + magnetization += 2 * new_spin + end + + end + + ising.state = state + ising.accepted_moves = accepted_moves + ising.energy = energy + ising.magnetization = magnetization + + append!(ising.energy_array, ising.energy) + append!(ising.magnetization_array, ising.magnetization) + ising.mc_steps = ising.mc_steps + 1 +end + +function steps!(ising::Ising2D, num::Int=100) + for i in 1:num + mc_step!(ising) + end +end + +function mean_energy(ising::Ising2D) + return mean(ising.energy_array) / ising.n +end + +function specific_heat(ising::Ising2D) + return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n +end + +function mean_magnetization(ising::Ising2D) + return mean(ising.magnetization_array) / ising.n +end + +function susceptibility(ising::Ising2D) + return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n) +end + +function observables(ising::Ising2D) + printstyled("Temperature\t\t", bold=true) + print(ising.temperature); print("\n") + + printstyled("Mean Energy\t\t", bold=true) + print(mean_energy(ising)); print("\n") + + printstyled("Mean Magnetiz.\t\t", bold=true) + print(mean_magnetization(ising)); print("\n") + + printstyled("Specific Heat\t\t", bold=true) + print(specific_heat(ising)); print("\n") + + printstyled("Susceptibility\t\t", bold=true) + print(susceptibility(ising)); print("\n") + + printstyled("MC Steps\t\t", bold=true) + print(ising.mc_steps); print("\n") + printstyled("Accepted Moves\t\t", bold=true) + print(ising.accepted_moves); print("\n") +end + + +function plot_ising(state::Matrix{Int}) + pos = Tuple.(findall(>(0), state)) + neg = Tuple.(findall(<(0), state)) + scatter(pos, markersize=5) + scatter!(neg, markersize=5) +end + +function find_m(H::Float64, l::Int, num::Int, T::Float64) + m = Ising2D(l, T, H) + steps!(m, num) + print("T = $T\n") + print("H = $H\n") + print("Mean Energy: $(mean_energy(m))\n") + print("Mean Magnetization: $(mean_magnetization(m))\n\n") + return mean_magnetization(m) +end + +function map_h_to_m(H_range::Vector{Float64}, l::Int, num::Int, T::Float64) + m = [] + for H in H_range + push!(m, find_m(H, l, num, T)) + end + return m +end + +function do_linear_regression(x::Vector{Float64}, y::Vector{Float64}) + n = length(x) + x̄ = mean(x) + ȳ = mean(y) + Σxy = sum((x .- x̄) .* (y .- ȳ)) + Σx² = sum((x .- x̄) .^ 2) + b = Σxy / Σx² + a = ȳ - b * x̄ + return a, b +end + +function plot_log_of_m_and_h(H_range::Vector{Float64}, l::Int, num::Int, T=2.27) + m = map_h_to_m(H_range, l, num, T) + p = scatter(H_range, m, label="M vs H", xlabel="H", ylabel="M", title="Magnetization (M) vs Field (B) for Ising Model at T_c", scale=:ln) + + # get the linear regression of the log + log_h = log.(H_range) + log_m = log.(m) + a, b = do_linear_regression(log_h, log_m) + println("a: $a, b: $b") + # plot the linear regression + plot!(p, H_range, exp.(a) .* H_range .^ b, label="linear regression = $(round(a, digits=3)) + $(round(b, digits=3))x", line=:dash, color=:red) + + return p +end + +# textbook rec +h_range = .02:.02:.2 +h_range = collect(h_range) +T = 2.27 # T_c for this system +side = 64 +steps = 5000 +plot_log_of_m_and_h(h_range, side, steps) \ No newline at end of file diff --git a/hw7/8-15-d.png b/hw7/8-15-d.png new file mode 100644 index 0000000..eb64f61 Binary files /dev/null and b/hw7/8-15-d.png differ diff --git a/hw7/8-15-scales-d.png b/hw7/8-15-scales-d.png new file mode 100644 index 0000000..5e4f2e4 Binary files /dev/null and b/hw7/8-15-scales-d.png differ diff --git a/hw7/8-15-scales.png b/hw7/8-15-scales.png new file mode 100644 index 0000000..2256167 Binary files /dev/null and b/hw7/8-15-scales.png differ diff --git a/hw7/8-15.jl b/hw7/8-15.jl new file mode 100644 index 0000000..5c833be --- /dev/null +++ b/hw7/8-15.jl @@ -0,0 +1,274 @@ +#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia + +using Statistics +using Plots + +function wrap_index(i::Int, l::Int)::Int + wrap = (i - 1) % l + 1 + return (wrap <= 0) ? l + wrap : wrap +end + +mutable struct Ising2D + l::Int + n::Int + temperature::Float64 + w::Vector{Float64} # Boltzmann weights + state::Matrix + energy::Float64 + magnetization::Int + mc_steps::Int + accepted_moves::Int + energy_array::Vector{Float64} + magnetization_array::Vector{Int} + H::Float64 +end + +Ising2D(l::Int, temperature::Float64, H = 1.0) = begin + n = l^2 + w = zeros(9) + w[9] = exp(-8.0 / temperature) + w[5] = exp(-4.0 / temperature) + state = ones(Int, l, l) # initially all spins up + energy = Float64(-2 * n + 2 * H * n) + magnetization = n + return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, + Int[], Int[], H) +end + +function reset!(ising::Ising2D) + ising.mc_steps = 0 + ising.accepted_moves = 0 + ising.energy_array = Int[] + ising.magnetization_array = Int[] +end + +function mc_step!(ising::Ising2D) + l::Int = ising.l + n::Int = ising.n + w = ising.w + + state = ising.state + accepted_moves = ising.accepted_moves + energy = ising.energy + magnetization = ising.magnetization + + random_positions = l * rand(2 * n) + random_array = rand(n) + + for k in 1:n + i = trunc(Int, random_positions[2*k-1]) + 1 + j = trunc(Int, random_positions[2*k]) + 1 + + changed_spins = state[i, j] * (state[i%l+1, j] + + state[wrap_index(i - 1, l), j] + state[i, j%l+1] + + state[i, wrap_index(j - 1, l)]) + de = 2 * changed_spins + 2 * ising.H * state[i, j] + + if de <= 0 || rand() < exp(-de / ising.temperature) + accepted_moves += 1 + new_spin = -state[i, j] # flip spin + state[i, j] = new_spin + + # add the effects of the new spin + energy += de + magnetization += 2 * new_spin + end + + end + + ising.state = state + ising.accepted_moves = accepted_moves + ising.energy = energy + ising.magnetization = magnetization + + append!(ising.energy_array, ising.energy) + append!(ising.magnetization_array, ising.magnetization) + ising.mc_steps = ising.mc_steps + 1 +end + +function steps!(ising::Ising2D, num::Int = 100) + for i in 1:num + mc_step!(ising) + end +end + +function mean_energy(ising::Ising2D) + return mean(ising.energy_array) / ising.n +end + +function specific_heat(ising::Ising2D) + return (std(ising.energy_array) / ising.temperature)^2 / ising.n +end + +function mean_magnetization(ising::Ising2D) + return mean(ising.magnetization_array) / ising.n +end + +function susceptibility(ising::Ising2D) + return (std(ising.magnetization_array))^2 / (ising.temperature * ising.n) +end + +function observables(ising::Ising2D) + printstyled("Temperature\t\t", bold = true) + print(ising.temperature) + print("\n") + + printstyled("Mean Energy\t\t", bold = true) + print(mean_energy(ising)) + print("\n") + + printstyled("Mean Magnetiz.\t\t", bold = true) + print(mean_magnetization(ising)) + print("\n") + + printstyled("Specific Heat\t\t", bold = true) + print(specific_heat(ising)) + print("\n") + + printstyled("Susceptibility\t\t", bold = true) + print(susceptibility(ising)) + print("\n") + + printstyled("MC Steps\t\t", bold = true) + print(ising.mc_steps) + print("\n") + printstyled("Accepted Moves\t\t", bold = true) + print(ising.accepted_moves) + print("\n") +end + + +function plot_ising(state::Matrix{Int}) + pos = Tuple.(findall(>(0), state)) + neg = Tuple.(findall(<(0), state)) + scatter(pos, markersize = 5) + scatter!(neg, markersize = 5) +end + +function h_to_m(H::Float64, l::Int, num::Int, T::Float64) + m = Ising2D(l, T, H) + steps!(m, num) + print("T = $T\n") + print("H = $H\n") + print("Mean Energy: $(mean_energy(m))\n") + print("Mean Magnetization: $(mean_magnetization(m))\n\n") + return mean_magnetization(m) +end + +function map_h_to_m(H_range::Vector{Float64}, l::Int, num::Int, T::Float64) + m = [] + for H in H_range + push!(m, h_to_m(H, l, num, T)) + end + return m +end + +function do_linear_regression(x::Vector{Float64}, y::Vector{Float64}) + n = length(x) + x̄ = mean(x) + ȳ = mean(y) + Σxy = sum((x .- x̄) .* (y .- ȳ)) + Σx² = sum((x .- x̄) .^ 2) + b = Σxy / Σx² + a = ȳ - b * x̄ + return a, b +end + +function plot_log_of_m_and_h(H_range::Vector{Float64}, l::Int, num::Int, T = 2.27) + m = map_h_to_m(H_range, l, num, T) + p = scatter(H_range, m, label = "M vs H", xlabel = "H", ylabel = "M", title = "Magnetization (M) vs Field (B) for Ising Model at T_c", scale = :ln) + + # get the linear regression of the log + log_h = log.(H_range) + log_m = log.(m) + a, b = do_linear_regression(log_h, log_m) + println("a: $a, b: $b") + # plot the linear regression + plot!(p, H_range, exp.(a) .* H_range .^ b, label = "linear regression = $(round(a, digits=3)) + $(round(b, digits=3))x", line = :dash, color = :red) + + return p +end + +function t_to_m(T::Float64, l::Int, num::Int, H::Float64) + m = Ising2D(l, T, H) + steps!(m, num) + print("T = $T\n") + print("H = $H\n") + print("Mean Energy: $(mean_energy(m))\n") + print("Mean Sus: $(mean_magnetization(m))\n\n") + return susceptibility(m) +end + +function plot_m_over_t(plt, T_range::Vector{Float64}, l::Int, num::Int, H = 0.0) + m = [] + for T in T_range + push!(m, t_to_m(T, l, num, H)) + end + p = scatter!(plt, T_range, m, label = "H = $H", xlabel = "T", ylabel = "X", title = "Susceptibility (X) vs Temperature (T) on Ising Model") + + return p, m +end + +function plot_m_over_t_and_h(T_range::Vector{Float64}, l::Int, num::Int, H_range::Vector{Float64}) + plt = Plots.scatter() + h = [] + for H in H_range + p, m = plot_m_over_t(plt, T_range, l, num, H) + push!(h, m) + end + + # plot the critical temp as a vertical line + plot!(plt, [2.27, 2.27], [-0.01, 30], label = "T_c = 2.27", line = :dash, color = :red) + + return plt, h +end + +function plot_scales(data, t_range, h_range) + x1 = [] + y1 = [] + x2 = [] + y2 = [] + for i in 1:length(h_range) + h = h_range[i] + for j in 1:length(t_range) + t = t_range[j] + + m = data[i][j] + + scaled_t = abs((t - 2.27) / 2.27) + scaled_m = m * (scaled_t^(7.0 / 4.0)) + scaled_h = h / (scaled_t^(15.0 / 8.0)) + if scaled_h > 30 + continue # dont add if too large + end + + if t > 2.27 + push!(x1, scaled_h) + push!(y1, scaled_m) + else + push!(x2, scaled_h) + push!(y2, scaled_m) + end + end + end + + tmp = scatter(x1, y1, label = "T > T_c", xlabel = "h / abs(t)^(15/8)", ylabel = "X * abs(t)^(7/4)", title = "Susceptibility (X) vs Field (H) for Ising Model") + scatter!(tmp, x2, y2, label = "T < T_c", xlabel = "h / abs(t)^(15/8)", ylabel = "X * abs(t)^(7/4)", title = "Susceptibility (X) vs Field (H) for Ising Model") + return tmp +end + +h_range = 0.01:0.01:0.05 +h_range = collect(h_range) +t_range = 1.5:0.1:3.0 +t_range = collect(t_range) +println("t_range: $t_range") +side = 20 +steps = 3000 +plt, data = plot_m_over_t_and_h(t_range, side, steps, h_range) + +savefig(plt, "hw7/8-15.png") + +p = plot_scales(data, t_range, h_range) +savefig(p, "hw7/8-15-scales.png") + + diff --git a/hw7/8-15.png b/hw7/8-15.png new file mode 100644 index 0000000..a5a4b28 Binary files /dev/null and b/hw7/8-15.png differ diff --git a/hw7/BoundStates.jl b/hw7/BoundStates.jl new file mode 100644 index 0000000..386a5da --- /dev/null +++ b/hw7/BoundStates.jl @@ -0,0 +1,73 @@ +#!/usr/bin/env julia + +"""Find eigenstates and eigenenergies of 1-D quantum problems""" + +using LinearAlgebra +using Plots + +N = 1000 # number of lattice points +L = 20.0 # x runs from -L/2 to L/2 +dx = L / N + +D = zeros(N, N) # discrete laplacian operator +V = zeros(N, N) # potential + +for i in 1:N + D[i, i] = -2.0 +end + +for i in 1:N-1 + D[i, i+1] = 1.0 + D[i+1, i] = 1.0 +end + +#println("\nLattice Laplacian operator") +#println(D) + +function potential(x) + """ The potential energy""" + #return 0.0 # particle in a box + #return 0.5 * x^2 # SHO with the spring constant k = 1 + #return -6.0 * x^2 + 8.0 * x^6 # potential with zero ground state energy + #return 0.1 * x^4 - 2.0 * x^2 + 0.0 * x # double-well potential + return 8*x^6 - 8*x^4 - 4*x^2 + 1 # another double-well potential +end + +for i in 1:N + x = (i + 0.5) * dx - 0.5*L # coordinates of lattice points + V[i, i] = potential(x) +end + +H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1 + +#println("\nMatrix elements of Hamiltonian = ") +#println(H) + +e, v = eigen(H) # diagonalize Hamiltonian + +println("\nGround state energy = ", e[1]) +println("\n1st excited state energy = ", e[2]) +println("\n2nd excited state energy = ", e[3]) +println("\n3rd excited state energy = ", e[4]) +println("\n4th excited state energy = ", e[5]) + + +gs(x) = exp(-0.5*x^2) # Gaussian that is exact ground state of SHO + + +plot(potential) + + +plot(v[:,1]) +#plot(v[:,2]) +#plot(gs) + +#= +eList = zeros(0) +for i in 1:20 + push!(eList, e[i]) +end + + +bar(eList, orientation = :horizontal) +=# diff --git a/hw7/Laplacians.jl b/hw7/Laplacians.jl new file mode 100644 index 0000000..10f0d48 --- /dev/null +++ b/hw7/Laplacians.jl @@ -0,0 +1,145 @@ +using Pkg +Pkg.add("Plots") +using Plots + +function del2_5(a::Matrix{Float64}, dx=1.0) + #= + Returns a finite-difference approximation of the laplacian of the array a, + with lattice spacing dx, using the five-point stencil: + + 0 1 0 + 1 -4 1 + 0 1 0 + =# + + del2 = zeros(size(a)) + del2[2:end-1, 2:end-1] .= (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] + + a[3:end, 2:end-1] + a[1:end-2, 2:end-1] - + 4.0 * a[2:end-1, 2:end-1]) ./ (dx^2) + return del2 +end + +function del2_9(a::Matrix{Float64}, dx=1.0) + #= + Returns a finite-difference approximation of the laplacian of the array a, + with lattice spacing dx, using the nine-point stencil: + + 1/6 2/3 1/6 + 2/3 -10/3 2/3 + 1/6 2/3 1/6 + =# + + del2 = zeros(size(a)) + del2[2:end-1, 2:end-1] .= (4.0 * (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] + + a[3:end, 2:end-1] + a[1:end-2, 2:end-1]) + + (a[1:end-2, 1:end-2] + a[1:end-2, 3:end] + + a[3:end, 1:end-2] + a[3:end, 3:end]) - + 20.0 * a[2:end-1, 2:end-1]) ./ (6.0 * dx^2) + return del2 +end + +function invDel2_5(b::Matrix{Float64}, dx=1.0, N=10000) + #= + Relaxes over N steps to a discrete approximation of the inverse laplacian + of the source term b, each step is a weighted average over the four neighboring + points and the source. This is the Jacobi algorithm. + =# + + invDel2 = zeros(size(b)) + newInvDel2 = zeros(size(b)) + + for m in 1:N + newInvDel2[2:end-1, 2:end-1] .= 0.25 * (invDel2[2:end-1, 3:end] + + invDel2[2:end-1, 1:end-2] + + invDel2[3:end, 2:end-1] + + invDel2[1:end-2, 2:end-1] - + (dx^2) * b[2:end-1, 2:end-1]) + invDel2 .= newInvDel2 + end + + diff = del2_5(invDel2, dx) - b + diffSq = diff .* diff + error = sqrt(sum(diffSq)) + + println("\nerror = ", error) + + return invDel2 +end + + +# Define variables +L = 10 +dx = 1.0 + +# Parabola of revolution has constant Laplacian +phi = zeros((L, L)) +for i = 1:L + x = (i + 0.5 - 0.5 * L) * dx + for j = 1:L + y = (j + 0.5 - 0.5 * L) * dx + phi[i, j] = x^2 + y^2 + end +end + +println(size(phi)) + +println(del2_5(phi, dx)) + +println("\n\n") + +println(del2_9(phi, dx)) + +# Electrostatics example: uniformly charged cylinder (rho = 1) of radius R +L = 100 +dx = 1.0 +R = 20.0 +R2 = R^2 + +# Charge distribution +rho = zeros((L, L)) +for i = 1:L + x = (i + 0.5 - 0.5 * L) * dx + for j = 1:L + y = (j + 0.5 - 0.5 * L) * dx + r2 = x^2 + y^2 + if r2 < R2 + rho[i, j] = 1.0 + else + rho[i, j] = 0.0 + end + end +end + +rhoPlot = plot(rho[Int(L/2), :]) + +# Exact (analytical) electric potential +phi = zeros((L, L)) +for i = 1:L + x = (i + 0.5 - 0.5 * L) * dx + for j = 1:L + y = (j + 0.5 - 0.5 * L) * dx + r2 = x^2 + y^2 + if r2 < R2 + phi[i, j] = -π * r2 + else + phi[i, j] = -π * (R2 + R2*log(r2/R2)) + end + end +end + +phiPlot = plot(phi[Int(L/2), :]) + +# Charge density obtained from exact potential +rho = -1.0/(4.0 * π) * del2_5(phi, dx) +rhoPlotLattice = plot(rho[Int(L/2), :]) + +phi = invDel2_5(-4.0 * π * rho, dx, 20000) +#phi = invDelSOR(-4.0 * π * rho, dx, 500) +phi .-= phi[Int(L/2), Int(L/2)] + +phiPlotInvDel = plot(phi[Int(L/2), :]) + +contourf(phi) + +#plot(rhoPlot, rhoPlotLattice) +#plot(phiPlot, phiPlotInvDel) diff --git a/hw7/RadialBoundStates.jl b/hw7/RadialBoundStates.jl new file mode 100644 index 0000000..afb91b0 --- /dev/null +++ b/hw7/RadialBoundStates.jl @@ -0,0 +1,59 @@ +#!/usr/bin/env julia + +"""Find eigenstates and eigenenergies of central potential problems""" + +using LinearAlgebra +using Plots + +N = 5000 # number of lattice points +L = 20.0 # r runs from 0 to L +dr = L / N + +D = zeros(N, N) # discrete radial 2nd derivative operator +V = zeros(N, N) # potential + +for i in 1:N + D[i, i] = -2.0 +end + +for i in 1:N-1 + D[i, i+1] = 1.0 + D[i+1, i] = 1.0 +end + +#println("\nLattice Laplacian operator") +#println(D) + +function potential(r, ℓ = 0) + """ The potential energy""" + #return 0.5 * ell * (ℓ+1.0) * pow(r, -2.0) # V=0: Free particle in spherical coordinates + + return -1.0/r + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # Hydrogen atom + + #return -r^(-1.1) + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # modified Coulomb potential +end + +for i in 1:N + r = (i + 0.5) * dr # radial coordinates of lattice points + V[i, i] = potential(r, 0) +end + +H = -0.5 * dr^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1 + +#println("\nMatrix elements of Hamiltonian = ") +#println(H) + +e, v = eigen(H) # diagonalize Hamiltonian + +println("\nGround state energy = ", e[1]) +println("\n1st excited state energy = ", e[2]) +println("\n2nd excited state energy = ", e[3]) +println("\n3rd excited state energy = ", e[4]) +println("\n4th excited state energy = ", e[5]) + + +plot(potential) + + +plot(v[:,1]) +#plot(v[:,2]) diff --git a/hw7/mvsb.svg b/hw7/mvsb.svg new file mode 100644 index 0000000..9e4afbb --- /dev/null +++ b/hw7/mvsb.svg @@ -0,0 +1,55 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/hw7/mvsb2.svg b/hw7/mvsb2.svg new file mode 100644 index 0000000..9415495 --- /dev/null +++ b/hw7/mvsb2.svg @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/hw7/tightbinding (1).jl b/hw7/tightbinding (1).jl new file mode 100644 index 0000000..0d1cc8c --- /dev/null +++ b/hw7/tightbinding (1).jl @@ -0,0 +1,24 @@ +using LinearAlgebra + +# Define the tight binding Hamiltonian +function tight_binding_hamiltonian(n_sites::Int64, t::Float64) + H = zeros(n_sites, n_sites) + for i in 1:(n_sites-1) + H[i, i+1] = t + H[i+1, i] = t + end + H +end + +# Find the eigenenergies of the Hamiltonian +function eigenenergies(H::Array{Float64, 2}) + eigvals(H) +end + +n_sites = 10 # number of sites in the chain +t = 1.0 # hopping parameter +H = tight_binding_hamiltonian(n_sites, t) +energies = eigenenergies(H) +println(energies) + + diff --git a/hw8/10-14-old.png b/hw8/10-14-old.png new file mode 100644 index 0000000..c1d5618 Binary files /dev/null and b/hw8/10-14-old.png differ diff --git a/hw8/10-14.jl b/hw8/10-14.jl new file mode 100644 index 0000000..27d1b5c --- /dev/null +++ b/hw8/10-14.jl @@ -0,0 +1,82 @@ +#!/usr/bin/env julia + +using Plots +using LinearAlgebra +using DifferentialEquations + +# useful functions: + +function H(psi, dx) # action of Hamiltonian on wavefunction + Hpsi = zeros(ComplexF64, size(psi)) + # -(1/2) * laplacian(psi) (m = hbar = 1) + Hpsi[2:end-1] = 0.5 * (2.0 * psi[2:end-1] - psi[3:end] - psi[1:end-2]) / (dx * dx) + + # periodic boundary conditions + #Hpsi[1] = 0.5 * (2.0*psi[1] - psi[2] - psi[end])/(dx*dx) + #Hpsi[end] = 0.5 * (2.0*psi[end] - psi[end-1] - psi[1])/(dx*dx) + return Hpsi +end + +derivative(psi, dx) = -1.0im * H(psi, dx) + +function initialWavefunction(x::Vector{Float64}, x0 = 10.0, Delta = 1.0, k = 4.0) + Delta2 = Delta^2 + return exp.(-(x .- x0) .^ 2 / Delta2) .* exp.(1.0im * k * x) +end + +function normalization(psi, dx) # normalization of wavefunction + n2 = dot(psi, psi) * dx + return sqrt(n2) +end + +prob(psi) = real(psi .* conj(psi)) + +# The actual simulation +N = 400 # number of lattice points +L = 20.0 # x runs from 0 to L +dx = L / N + +x = range(0.0, L, N) |> collect # lattice along x-axis +#println(x) + +# initial wavefunction has position (x0), width (Delta), and momentum (k) +psi0 = initialWavefunction(x, 10.0, 1.0, 1000.0) + +# normalize wavefunction +n = normalization(psi0, dx) +psi0 = psi0 / n +println("norm = ", normalization(psi0, dx)) + +# plot initial wavefunction and probability density +plot(prob(psi0)) + +# integrate forward in time +tf = 1.0 +dt = 0.1 +tspan = (0.0, tf) + +function timeEvolve(psi0, tf, dt) # second order Runge-Kutta algorithm + psi = psi0 + for t in range(0, stop = tf, step = dt) + psiMid = psi + 0.5 * dt * derivative(psi, dx) + psi = psi + dt * derivative(psiMid, dx) + end + return psi +end + +# tendency(psi, dx, t) = derivative(psi, dx) # use ODE solver +# problem = ODEProblem(tendency, psi0, tspan, dx) # specify ODE +# sol = solve(problem, Tsit5(), reltol = 1e-12, abstol = 1e-12) # solve using Tsit5 algorithm to specified accuracy + +# compare initial and final wavefunction probabilities +psi = timeEvolve(psi0, tf, dt) +# psi = sol[:, end] +times = sol.t + +# check that normalization hasn't deviated too far from 1.0 +println("norm = ", normalization(psi, dx)) + +plot([prob(psi0), prob(psi)]) +savefig("10-14.png") + + diff --git a/hw8/10-14.png b/hw8/10-14.png new file mode 100644 index 0000000..3e18e5b Binary files /dev/null and b/hw8/10-14.png differ diff --git a/hw8/10-14t.png b/hw8/10-14t.png new file mode 100644 index 0000000..3e18e5b Binary files /dev/null and b/hw8/10-14t.png differ diff --git a/hw8/10-17.jl b/hw8/10-17.jl new file mode 100644 index 0000000..6dfff2b --- /dev/null +++ b/hw8/10-17.jl @@ -0,0 +1,81 @@ +#!/usr/bin/env julia + +using Plots +using LinearAlgebra +using DifferentialEquations + +# useful functions: + +function H(psi, dx) # action of Hamiltonian on wavefunction + Hpsi = zeros(ComplexF64, size(psi)) + # -(1/2) * laplacian(psi) (m = hbar = 1) + Hpsi[2:end-1] = 0.5 * (2.0 * psi[2:end-1] - psi[3:end] - psi[1:end-2]) / (dx * dx) + + # periodic boundary conditions + #Hpsi[1] = 0.5 * (2.0*psi[1] - psi[2] - psi[end])/(dx*dx) + #Hpsi[end] = 0.5 * (2.0*psi[end] - psi[end-1] - psi[1])/(dx*dx) + return Hpsi +end + +derivative(psi, dx) = -1.0im * H(psi, dx) + +function initialWavefunction(x::Vector{Float64}, x0 = 10.0, Delta = 1.0, k = 4.0) + Delta2 = Delta^2 + return exp.(-(x .- x0) .^ 2 / Delta2) .* exp.(1.0im * k * x) +end + +function normalization(psi, dx) # normalization of wavefunction + n2 = dot(psi, psi) * dx + return sqrt(n2) +end + +prob(psi) = real(psi .* conj(psi)) + +# The actual simulation +N = 400 # number of lattice points +L = 20.0 # x runs from 0 to L +dx = L / N + +x = range(0.0, L, N) |> collect # lattice along x-axis +#println(x) + +# initial wavefunction has position (x0), width (Delta), and momentum (k) +psi0 = initialWavefunction(x, 10.0, 0.05, 700.0) + +# normalize wavefunction +n = normalization(psi0, dx) +psi0 = psi0 / n +println("norm = ", normalization(psi0, dx)) + +# plot initial wavefunction and probability density +plot(prob(psi0)) + +# integrate forward in time +tf = 1.0 +dt = 5e-7 +tspan = (0.0, tf) + +function timeEvolve(psi0, tf, dt) # second order Runge-Kutta algorithm + psi = psi0 + for t in range(0, stop = tf, step = dt) + psiMid = psi + 0.5 * dt * derivative(psi, dx) + psi = psi + dt * derivative(psiMid, dx) + end + return psi +end + +tendency(psi, dx, t) = derivative(psi, dx) # use ODE solver +problem = ODEProblem(tendency, psi0, tspan, dx) # specify ODE +sol = solve(problem, Tsit5(), reltol = 1e-12, abstol = 1e-12) # solve using Tsit5 algorithm to specified accuracy + +# compare initial and final wavefunction probabilities +#psi = timeEvolve(psi0, tf, dt) +psi = sol[:, end] +times = sol.t + +# check that normalization hasn't deviated too far from 1.0 +println("norm = ", normalization(psi, dx)) + +plot([prob(psi0), prob(psi)]) + + diff --git a/hw8/10-3.jl b/hw8/10-3.jl new file mode 100644 index 0000000..8616a69 --- /dev/null +++ b/hw8/10-3.jl @@ -0,0 +1,92 @@ +#!/usr/bin/env julia + +"""Find eigenstates and eigenenergies of 1-D quantum problems""" + +using LinearAlgebra +using Plots + +#println("\nLattice Laplacian operator") +#println(D) + +function potential(x, n) + """ The potential energy""" + #return 0.0 # particle in a box + #return 0.5 * x^2 # SHO with the spring constant k = 1 + #return -6.0 * x^2 + 8.0 * x^6 # potential with zero ground state energy + #return 0.1 * x^4 - 2.0 * x^2 + 0.0 * x # double-well potential + # return 8 * x^6 - 8 * x^4 - 4 * x^2 + 1 # another double-well potential + return abs(x)^n +end + +#println("\nMatrix elements of Hamiltonian = ") +#println(H) + + +function map_n_to_energies(n) + N = 1000 # number of lattice points + L = 20.0 # x runs from -L/2 to L/2 + dx = L / N + + D = zeros(N, N) # discrete laplacian operator + V = zeros(N, N) # potential + + for i in 1:N + D[i, i] = -2.0 + end + + for i in 1:N-1 + D[i, i+1] = 1.0 + D[i+1, i] = 1.0 + end + + + for i in 1:N + x = (i + 0.5) * dx - 0.5 * L # coordinates of lattice points + V[i, i] = potential(x, n) + end + + H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1 + + + e, v = eigen(H) # diagonalize Hamiltonian + + println("\n n = ", n) + println("Ground state energy = ", e[1]) + println("1st excited state energy = ", e[2]) + println("2nd excited state energy = ", e[3]) + println("3rd excited state energy = ", e[4]) + println("4th excited state energy = ", e[5], "\n") + + return e +end + +n_max = 18 +n_to_e = [map_n_to_energies(n) for n in 1:n_max] + +# plot e[0] for all N +eList = zeros(0) +for i in 1:n_max + push!(eList, n_to_e[i][1]) +end + +plot(eList) + +# gs(x) = exp(-0.5 * x^2) # Gaussian that is exact ground state of SHO + + +# plot(potential) + + +# plot(v[:, 1]) +#plot(v[:,2]) +# plot(gs) + +#= +eList = zeros(0) +for i in 1:20 + push!(eList, e[i]) +end + + +bar(eList, orientation = :horizontal) +=# diff --git a/hw8/BoundStates.jl b/hw8/BoundStates.jl new file mode 100644 index 0000000..22e608c --- /dev/null +++ b/hw8/BoundStates.jl @@ -0,0 +1,73 @@ +#!/usr/bin/env julia + +"""Find eigenstates and eigenenergies of 1-D quantum problems""" + +using LinearAlgebra +using Plots + +N = 1000 # number of lattice points +L = 20.0 # x runs from -L/2 to L/2 +dx = L / N + +D = zeros(N, N) # discrete laplacian operator +V = zeros(N, N) # potential + +for i in 1:N + D[i, i] = -2.0 +end + +for i in 1:N-1 + D[i, i+1] = 1.0 + D[i+1, i] = 1.0 +end + +#println("\nLattice Laplacian operator") +#println(D) + +function potential(x) + """ The potential energy""" + #return 0.0 # particle in a box + #return 0.5 * x^2 # SHO with the spring constant k = 1 + #return -6.0 * x^2 + 8.0 * x^6 # potential with zero ground state energy + #return 0.1 * x^4 - 2.0 * x^2 + 0.0 * x # double-well potential + return 8 * x^6 - 8 * x^4 - 4 * x^2 + 1 # another double-well potential +end + +for i in 1:N + x = (i + 0.5) * dx - 0.5 * L # coordinates of lattice points + V[i, i] = potential(x) +end + +H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1 + +#println("\nMatrix elements of Hamiltonian = ") +#println(H) + +e, v = eigen(H) # diagonalize Hamiltonian + +println("\nGround state energy = ", e[1]) +println("\n1st excited state energy = ", e[2]) +println("\n2nd excited state energy = ", e[3]) +println("\n3rd excited state energy = ", e[4]) +println("\n4th excited state energy = ", e[5]) + + +gs(x) = exp(-0.5 * x^2) # Gaussian that is exact ground state of SHO + + +plot(potential) + + +plot(v[:, 1]) +#plot(v[:,2]) +#plot(gs) + +#= +eList = zeros(0) +for i in 1:20 + push!(eList, e[i]) +end + + +bar(eList, orientation = :horizontal) +=# diff --git a/hw8/RadialBoundStates.jl b/hw8/RadialBoundStates.jl new file mode 100644 index 0000000..ed7bcc9 --- /dev/null +++ b/hw8/RadialBoundStates.jl @@ -0,0 +1,59 @@ +#!/usr/bin/env julia + +"""Find eigenstates and eigenenergies of central potential problems""" + +using LinearAlgebra +using Plots + +N = 5000 # number of lattice points +L = 20.0 # r runs from 0 to L +dr = L / N + +D = zeros(N, N) # discrete radial 2nd derivative operator +V = zeros(N, N) # potential + +for i in 1:N + D[i, i] = -2.0 +end + +for i in 1:N-1 + D[i, i+1] = 1.0 + D[i+1, i] = 1.0 +end + +#println("\nLattice Laplacian operator") +#println(D) + +function potential(r, ℓ = 0) + """ The potential energy""" + #return 0.5 * ell * (ℓ+1.0) * pow(r, -2.0) # V=0: Free particle in spherical coordinates + + return -1.0 / r + 0.5 * ℓ * (ℓ + 1.0) * r^(-2.0) # Hydrogen atom + + #return -r^(-1.1) + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # modified Coulomb potential +end + +for i in 1:N + r = (i + 0.5) * dr # radial coordinates of lattice points + V[i, i] = potential(r, 0) +end + +H = -0.5 * dr^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1 + +#println("\nMatrix elements of Hamiltonian = ") +#println(H) + +e, v = eigen(H) # diagonalize Hamiltonian + +println("\nGround state energy = ", e[1]) +println("\n1st excited state energy = ", e[2]) +println("\n2nd excited state energy = ", e[3]) +println("\n3rd excited state energy = ", e[4]) +println("\n4th excited state energy = ", e[5]) + + +plot(potential) + + +plot(v[:, 1]) +#plot(v[:,2]) diff --git a/hw8/TimeDependentSchrodingerEquation.jl b/hw8/TimeDependentSchrodingerEquation.jl new file mode 100644 index 0000000..dacbced --- /dev/null +++ b/hw8/TimeDependentSchrodingerEquation.jl @@ -0,0 +1,81 @@ +#!/usr/bin/env julia + +using Plots +using LinearAlgebra +using DifferentialEquations + +# useful functions: + +function H(psi, dx) # action of Hamiltonian on wavefunction + Hpsi = zeros(ComplexF64, size(psi)) + # -(1/2) * laplacian(psi) (m = hbar = 1) + Hpsi[2:end-1] = 0.5 * (2.0*psi[2:end-1] - psi[3:end] - psi[1:end-2])/(dx*dx) + + # periodic boundary conditions + #Hpsi[1] = 0.5 * (2.0*psi[1] - psi[2] - psi[end])/(dx*dx) + #Hpsi[end] = 0.5 * (2.0*psi[end] - psi[end-1] - psi[1])/(dx*dx) + return Hpsi +end + +derivative(psi, dx) = -1.0im * H(psi, dx) + +function initialWavefunction(x::Vector{Float64}, x0 = 10.0, Delta = 1.0, k = 4.0) + Delta2 = Delta^2 + return exp.(- (x .- x0).^2 / Delta2 ) .* exp.(1.0im * k * x) +end + +function normalization(psi, dx) # normalization of wavefunction + n2 = dot(psi, psi) * dx + return sqrt(n2) +end + +prob(psi) = real(psi .* conj(psi)) + +# The actual simulation +N = 400 # number of lattice points +L = 20.0 # x runs from 0 to L +dx = L / N + +x = range(0.0, L, N) |> collect # lattice along x-axis +#println(x) + +# initial wavefunction has position (x0), width (Delta), and momentum (k) +psi0 = initialWavefunction(x, 10.0, 1.0, 0.0) + +# normalize wavefunction +n = normalization(psi0, dx) +psi0 = psi0 / n +println("norm = ", normalization(psi0, dx)) + +# plot initial wavefunction and probability density +plot(prob(psi0)) + +# integrate forward in time +tf = 1.0 +dt = 0.0001 +tspan = (0.0, tf) + +function timeEvolve(psi0, tf, dt) # second order Runge-Kutta algorithm + psi = psi0 + for t in range(0, stop=tf, step=dt) + psiMid = psi + 0.5 * dt * derivative(psi, dx) + psi = psi + dt * derivative(psiMid, dx) + end + return psi +end + +tendency(psi, dx, t) = derivative(psi, dx) # use ODE solver +problem = ODEProblem(tendency, psi0, tspan, dx) # specify ODE +sol = solve(problem, Tsit5(), reltol=1e-12, abstol=1e-12) # solve using Tsit5 algorithm to specified accuracy + +# compare initial and final wavefunction probabilities +#psi = timeEvolve(psi0, tf, dt) +psi = sol[:, end] +times = sol.t + +# check that normalization hasn't deviated too far from 1.0 +println("norm = ", normalization(psi, dx)) + +plot([prob(psi0), prob(psi)]) + + diff --git a/hw8/tightbinding.jl b/hw8/tightbinding.jl new file mode 100644 index 0000000..0d1cc8c --- /dev/null +++ b/hw8/tightbinding.jl @@ -0,0 +1,24 @@ +using LinearAlgebra + +# Define the tight binding Hamiltonian +function tight_binding_hamiltonian(n_sites::Int64, t::Float64) + H = zeros(n_sites, n_sites) + for i in 1:(n_sites-1) + H[i, i+1] = t + H[i+1, i] = t + end + H +end + +# Find the eigenenergies of the Hamiltonian +function eigenenergies(H::Array{Float64, 2}) + eigvals(H) +end + +n_sites = 10 # number of sites in the chain +t = 1.0 # hopping parameter +H = tight_binding_hamiltonian(n_sites, t) +energies = eigenenergies(H) +println(energies) + + diff --git a/t/1d.jl b/t/1d.jl new file mode 100644 index 0000000..203466c --- /dev/null +++ b/t/1d.jl @@ -0,0 +1,109 @@ +using Plots + +N = 10 # number of masses in the 1D lattice +K = 100 # elastic force constant +m = 1 # mass of particles +M = 1000 # big mass (one only) +t_final = 2 # seconds +dt = 0.001 # timestep + +initial_distance_between_masses = 10 # meters +L = initial_distance_between_masses * N # length of the 1D lattice + +# 2D array of current positions and velocities (Hamiltonian state) +q = zeros(N) +p = zeros(N) + +# initialize the positions and velocities +for i in 1:N + q[i] = initial_distance_between_masses * (i - 1) + p[i] = 0 +end + + +# plot positions +function plot_positions(q, t) + # only 1D, set y to 0 + y = zeros(N) + + # plot the masses + plot( + q, y, + seriestype = :scatter, label = "Masses @ t = $t", + # xlims = (-1, N + 1), + ylims = (-1, 1), + markerstrokewidth = 0, + ) + + # plot the middle one as red + plot!( + [q[Int(N / 2)]], [0], + seriestype = :scatter, label = "Large mass @ t = $t", + color = :red, + markerstrokewidth = 0, markersize = 10, + ) + + return plot!() +end + +# update the state +function update_state!(q, p, dt) + new_q = copy(q) + new_p = copy(p) + + # update the small masses state + for i in 2:N-1 + dx_right = q[i+1] - q[i] + dx_left = q[i] - q[i-1] + + new_q[i] += dt * p[i] / m + new_p[i] += dt * (K * dx_right - K * dx_left) + end + + # handle the ends, since our 1D system is cyclic + # case where i = 1, first particle + dx_right = q[2] - q[1] + distance_from_L = L - q[N] + dy_left = q[1] - distance_from_L + new_q[1] += dt * p[1] / m + new_p[1] += dt * (K * dx_right - K * dy_left) + + # case where i = N, last particle + distance_from_0 = q[1] + dx_right = L + q[1] - q[N] + dx_left = q[N] - q[N-1] + new_q[N] += dt * p[N] / m + new_p[N] += dt * (K * dx_right - K * dx_left) + + + # update the large mass in middle (different mass, difference case) + middle_index = Int(N / 2) + dx_right = q[middle_index+1] - q[middle_index] + dx_left = q[middle_index] - q[middle_index-1] + new_q[middle_index] += dt * p[middle_index] / M + new_p[Int(N / 2)] += dt * (K * dx_right - K * dx_left) + + # update the state + for i in 1:N + q[i] = new_q[i] + p[i] = new_p[i] + end +end + +display(plot_positions(q, 0)) + +function progress_system(q, p, dt, t_final) + t = 0 + while t < t_final + update_state!(q, p, dt) + t += dt + end +end + +progress_system(q, p, dt, t_final) + +println("Final state:") +println(q) +println(p) + +display(plot_positions(q, t_final)) diff --git a/t/animate-positions-10.mp4 b/t/animate-positions-10.mp4 new file mode 100644 index 0000000..aa00547 Binary files /dev/null and b/t/animate-positions-10.mp4 differ diff --git a/t/animate-positions.mp4 b/t/animate-positions.mp4 new file mode 100644 index 0000000..3633358 Binary files /dev/null and b/t/animate-positions.mp4 differ diff --git a/t/disp.jl b/t/disp.jl new file mode 100644 index 0000000..41ee2bb --- /dev/null +++ b/t/disp.jl @@ -0,0 +1,173 @@ +function calculate_force( + left_pos, + middle_pos, + right_pos, + K, + alpha = 0, + beta = 0, +) + linear_force = K * (middle_pos - left_pos + middle_pos - right_pos) + quadratic_force = alpha * (middle_pos - left_pos)^2 + alpha * (middle_pos - right_pos)^2 + cubic_force = beta * (middle_pos - left_pos)^3 + beta * (middle_pos - right_pos)^3 + return linear_force + quadratic_force + cubic_force +end + +function tendency!(du, u, p, t) + # unpack the params + N, K, m = p + + # get the positions and momenta + qs = u[1:2:end] + ps = u[2:2:end] + + # go over the points in the lattice and update the state + for i in 2:N-1 + mass = m + if i == 2 * Int(N / 2) - 1 || i == 2 * Int(N / 2) + mass = 10000 + end + + du[i*2-1] = ps[i] / mass + force = + du[i*2] = force / mass + end + + force_end = K * (qs[2] - 2 * qs[1] + qs[N-1]) + du[1] = ps[1] / m + du[2] = force_end / m + du[end-1] = ps[end] / m + du[end] = force_end / m +end + +function get_initial_state( + N, + initial_displacement = 2, + initial_velocity = 0, +) + state = zeros(2 * N) + + middle_index = 2 * Int(N / 2) - 1 # middle mass + state[middle_index] = initial_displacement + state[middle_index+1] = initial_velocity + return state +end + +using DifferentialEquations +function run_simulation( + N, + K, + m, + final_time, + initial_displacement = 2, + initial_velocity = 0, +) + println("Running simulation with N = $N, K = $K, m = $m, final_time = $final_time, initial_displacement = $initial_displacement, initial_velocity = $initial_velocity\n") + s_0 = get_initial_state(N, initial_displacement, initial_velocity) + + # pack the params + p = N, K, m + t_span = (0.0, final_time) + prob = ODEProblem(tendency!, s_0, t_span, p) + sol = solve(prob, Tsit5(), reltol = 1e-10, abstol = 1e-10) # control simulation + + println("Done Running Sim!\n\n") + return sol +end + +using Plots +function animate_positions( + states, + time_steps, + time_min = 0, + time_max = 30, + red_threshold = 2, +) + println("Animating positions") + anim = @animate for i in 1:length(time_steps) + t = time_steps[i] + if t < time_min + continue + end + if t > time_max + break + end + positions = states[i][1:2:end] + v_middle = states[i][Int(length(states[1]) / 2)] + # plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3)) + if v_middle >= red_threshold + plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3), + color = :red, legend = :topright, + ) + else + plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3), + color = :blue, legend = :topright, + ) + end + end + mp4(anim, "t/animate-positions.mp4", fps = 30) + println("Done animating positions") +end + +function plot_starting_and_final_positions( + states, + time_steps, +) + p1 = plot(states[1][1:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "First Three Modes") + plot!(p1, states[end][1:2:end], label = "Final", marker = :circle) + + # plot the vels + p2 = plot(states[1][2:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Velocity", title = "First Three Modes") + plot!(p2, states[end][2:2:end], label = "Final", marker = :circle) + + # save the plots + savefig(p1, "t/initial-final-positions.png") + savefig(p2, "t/initial-final-velocities.png") +end + +function analyize_vels( + states, + time_steps, + threshold = 1.975, +) + println("Analyzing velocities:\n") + output = [] + for i in 1:length(states) + if states[i][Int(length(states[i]) / 2)] >= threshold + push!(output, i) + println("Time: ", time_steps[i], " Position: ", states[i][Int(length(states[i]) / 2)]) + end + end + + # plot the first 10 seconds of Velocity + data = [] + for i in 1:length(states) + if time_steps[i] > 10 + break + end + push!(data, states[i][Int(length(states[i]) / 2)]) + end + p = plot(data, label = "Velocity Over Time", xlabel = "Time", ylabel = "Velocity") + savefig(p, "t/velocity-over-time.png") + + println("\nDone!\n\n") + return output +end + +# Run the simulation +N = 10 # number of masses +beta = 0 # cubic string spring +K = 100 # spring constant +A = 10 # amplitude +final_time = 10000 # seconds +m = 1 # mass of particles +plot_data = [] + +my_vel = 10 + +sol = run_simulation(N, K, m, final_time, 0, my_vel) + +println("final time: ", sol.t[end]) +# s = sol.u[1:2:end] +analyize_vels(sol.u, sol.t, my_vel) +plot_starting_and_final_positions(sol.u, sol.t) +animate_positions(sol.u, sol.t, 0, 1, my_vel) diff --git a/t/initial-final-positions.png b/t/initial-final-positions.png new file mode 100644 index 0000000..7fad408 Binary files /dev/null and b/t/initial-final-positions.png differ diff --git a/t/initial-final-velocities.png b/t/initial-final-velocities.png new file mode 100644 index 0000000..3763f66 Binary files /dev/null and b/t/initial-final-velocities.png differ diff --git a/t/plot_data.png b/t/plot_data.png new file mode 100644 index 0000000..d052983 Binary files /dev/null and b/t/plot_data.png differ diff --git a/t/plz.mp4 b/t/plz.mp4 new file mode 100644 index 0000000..a7dbdf7 Binary files /dev/null and b/t/plz.mp4 differ diff --git a/t/r.jl b/t/r.jl new file mode 100644 index 0000000..516ab09 --- /dev/null +++ b/t/r.jl @@ -0,0 +1,134 @@ +function dynamics!( + state, + prev_state, + params, + states, + plot_data, +) + # Unpack the parameters + N, K, beta, A, dt, num_steps = params + + for j in 1:num_steps + next_state = zeros(N) + + # update the left particle (set to rest) + state[1] = 0 + + # update the right particle (set to rest) + state[N] = 0 + + # update the middle particles + for i in 2:N-1 + a = 2 * state[i] - prev_state[i] + b = dt * dt * K * (state[i+1] - 2 * state[i] + state[i-1]) + c = dt * dt * beta * ((state[i+1] - state[i])^3 + (state[i-1] - state[i])^3) + + this_mass = 1 + if i == Int(N / 2) + # time = i * dt + # push!(plot_data, (state[i])) + this_mass = 1 + end + next_state[i] = (a + b + c) / this_mass + end + + # push!(states, next_state) + + # update the previous state + for i in 1:N + prev_state[i] = state[i] + end + # update the current state + for i in 1:N + state[i] = next_state[i] + end + + # if the middle mass is close to 2, print the time and position + if state[Int(N / 2)] > 1.995 + println("Time: ", j * dt, " Positions: ", state[Int(N / 2)]) + println("Other Positions: ", state, "\n") + push!(states, (j * dt, state)) + end + + if (j * dt % 100000) == 0 + println("TIME UPDATE: ", j * dt) + end + end +end + +function get_initial_state( + N, + modes, + beta, + A, +) + state = zeros(N) + # amp = A * sqrt(2 / (N - 1)) + # for i in 2:N-1 + # state[i] = amp * sin((modes * π * (i - 1)) / (N - 1)) + # end + # set the middle to be the largest + state[Int(N / 2)] = 2 + return state +end + +function run_simulation( + N, + modes, + beta, + A, + dt, + num_steps, + plot_data, +) + states = [] + state = get_initial_state(N, 1, beta, A) + prev_state = zeros(N) + for i in 1:N + prev_state[i] = state[i] + end + params = (N, modes, beta, A, dt, num_steps) + dynamics!(state, prev_state, params, states, plot_data) + return states +end + +# Run the simulation +N = 10 # number of masses +beta = 0 # cubic string spring +K = 100 # spring constant +A = 10 # amplitude +modes = 3 # number of modes to plot +final_time = 10000000 # seconds +dt = 0.001 # seconds +num_steps = Int(final_time / dt) +params = (N, K, beta, A, dt, num_steps) +plot_data = [] +s = run_simulation(N, K, beta, A, dt, num_steps, plot_data) +println("\nStates of interest:", s) + +# plot the inital positions and the final positions +# using Plots +# plot(get_initial_state(N, 1, beta, A), label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "First Three Modes") +# plot!(s[end], label = "Final", marker = :circle) + +# plot the middle of the lattice over time +# time_steps = [i * dt for i in 1:num_steps] +# plot(time_steps, plot_data, label = "Middle Mass Over Time", xlabel = "Time", ylabel = "Displacement") +# savefig("t/plot_data.png") + +# print the points close to the origional displacement +# output = [] +# for i in 1:length(plot_data) +# if plot_data[i] > 1.975 +# push!(output, i) +# println("Time: ", i * dt, " Position: ", plot_data[i]) +# end +# end + + +# animate the s array of positions +# anim = @animate for i in 1:length(s) +# plot(s[i], label = "t = $(round(i * dt, digits = 3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3)) +# end +# mp4(anim, "t/plz.mp4", fps = 30) +# println("Output: ", output) diff --git a/t/t.jl b/t/t.jl new file mode 100644 index 0000000..6a06777 --- /dev/null +++ b/t/t.jl @@ -0,0 +1,755 @@ +# molecular dynamics 2d. + +# usage: +# at the end of this script, under the header "DEMOS", +# you'll see some functions which implement demos from GN chapter 9. +# simply load the script in your development environment +# (I strongly recommend not using jupiter) +# and in the console/REPL run +# demo_0() +# etc. + +# demos 0,1,3 can optionally make an animated gif +# if you call it with the optional argument demo_3(gif=1) + +# lmk if this script is giving you grief or if you find any bugs +# kian@brown.edu + +using Statistics +using StatsPlots + +mutable struct ParticleSystem + N::Int64# number of particles + L::Float64# square box side length + T₀::Float64# initial temperature + t::Float64# system time + dt::Float64# time step + state::Vector{Float64}# state space array + steps::Int64# number of steps + sampleInterval::Int64# interval for sampling data + timeData::Vector{Float64}# array of sampled time points + energyData::Vector{Float64}# array of sampled energy values + tempData::Vector{Float64}# array of sampled temperature values + tempAccumulator::Float64# temperature accumulator + squareTempAccumulator::Float64# T^2 accumulator + virialAccumulator::Float64# virial accumulator + xData::Vector{Vector{Float64}} # array of sampled position data + vData::Vector{Vector{Float64}} # array of sampled velocity data + forceType::String# force +end + +function ParticleSystem(N::Int64 = 64, L::Float64 = 10.0, T₀::Float64 = 1.0) + t = 0.0 + dt = 0.001 + state = zeros(4N) # state space array, [x1,y1,x2,y2,...,vx1,vy1,...] + steps = 0 + timeData = Float64[] + energyData = Float64[] + sampleInterval = 100 + tempData = Float64[] + tempAccumulator = 0.0 + squareTempAccumulator = 0.0 + virialAccumulator = 0.0 + xData = Vector{Float64}[] + vData = Vector{Float64}[] + forceType = "lennardJones" + + return ParticleSystem( + N, + L, + T₀, + t, + dt, + state, + steps, + sampleInterval, + timeData, + energyData, + tempData, + tempAccumulator, + squareTempAccumulator, + virialAccumulator, + xData, + vData, + forceType, + ) +end + +# some useful "views" of the state array +# (read the performance tips chapter of the julia manual) +@views positions(state) = state[1:Int64(length(state) / 2)] +@views velocities(state) = state[(Int64(length(state) / 2)+1):end] +@views xcomponent(vector) = vector[1:2:end] +@views ycomponent(vector) = vector[2:2:end] +@views particle(n, vector) = [vector[2n-1], vector[2n]] + +# INITIALIZATION +################################################################################ + +function set_random_positions!(sys::ParticleSystem) + println("\tposition configuration: random") + positions(sys.state) .= rand(2 * sys.N) .* sys.L + cool!(sys) +end + +function set_square_lattice_positions!(sys::ParticleSystem) + println("\tposition configuration: square lattice") + + n = Int64(floor(sqrt(sys.N))) # num lattice points per axis + latticeSpacing = sys.L / n + + if sys.N != n^2 + println("\t\toops... your chosen N=$(sys.N) is not a square number") + println("\t\t-> resetting N to $(n^2).") + sys.N = n^2 + sys.state = zeros(4 * sys.N) + end + + for i in 0:(n-1) + for j in 0:(n-1) + sys.state[2*(i*n+j)+1] = (i + 0.5) * latticeSpacing + sys.state[2*(i*n+j)+2] = (j + 0.5) * latticeSpacing + end + end +end + +function set_triangular_lattice_positions!(sys::ParticleSystem) +end + +function add_position_jitter!(sys::ParticleSystem, jitter::Float64 = 0.5) + println("\tadding a wee bit of random jitter to particle positions...") + + for i ∈ 1:length(positions(sys.state)) + sys.state[i] += rand() - jitter + end +end + +function set_random_velocities!(sys::ParticleSystem) + println("\tvelocity configuration: random") + + velocities(sys.state) .= rand(2 * sys.N) .- 0.5 + zero_total_momentum!(sys) + velocities(sys.state) .*= sqrt(sys.T₀ / temperature(sys)) +end + +function zero_total_momentum!(sys::ParticleSystem) + xcomponent(velocities(sys.state)) .-= + mean(xcomponent(velocities(sys.state))) + ycomponent(velocities(sys.state)) .-= + mean(ycomponent(velocities(sys.state))) +end + + +# FORCES / POTENTIALS +################################################################################ + +function force(sys::ParticleSystem) + if sys.forceType == "lennardJones" + force, virial = lennard_jones_force(sys) + elseif sys.forceType == "powerLaw" + force, virial = power_law_force(sys) + end + + sys.virialAccumulator += virial + + return force +end + +# the minimum image approximation +# (periodic boundary conditions) +function minimum_image(xij::Float64, L::Float64) + if xij > (L / 2) + xij -= L + elseif xij < -(L / 2) + xij += L + end + return xij +end + +function lennard_jones_force(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + virial = 0.0 + force = zeros(2 * sys.N) + + Threads.@threads for i ∈ 1:(sys.N-1) + for j ∈ (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + f = 48.0 * r2inv^7 - 24.0 * r2inv^4 + fx = dx * f + fy = dy * f + + force[2*i-1] += fx + force[2*i] += fy + force[2*j-1] -= fx + force[2*j] -= fy + + virial += fx * dx + fy * dy + end + end + + return force, 0.5 * virial +end + +function lennard_jones_potential(sys::ParticleSystem) + x = xcomponent(positions(sys.state)) + y = ycomponent(positions(sys.state)) + U = 0.0 + + Threads.@threads for i in 1:(sys.N-1) + for j in (i+1):sys.N + dx = minimum_image(x[i] - x[j], sys.L) + dy = minimum_image(y[i] - y[j], sys.L) + + r2inv = 1.0 / (dx^2 + dy^2) + U += r2inv^6 - r2inv^3 + end + end + return 4.0 * U +end + +function power_law_force(sys::ParticleSystem) +end + +function power_law_potential(sys::ParticleSystem) +end + +# TIME EVOLUTION +################################################################################ + +function keep_particles_in_box!(sys::ParticleSystem) + for i in 1:(2*sys.N) + if positions(sys.state)[i] > sys.L + positions(sys.state)[i] -= sys.L + elseif positions(sys.state)[i] < 0.0 + positions(sys.state)[i] += sys.L + end + end + + # # another way of doing this: use the ternary operator + # for i in 1:(2 * sys.N) + # positions(sys.state)[i] < 0.0 ? + # positions(sys.state)[i] % sys.L + sys.L : + # positions(sys.state)[i] % sys.L + # end +end + +function verlet_step!(sys::ParticleSystem) + # compute acceleration at current time + acceleration = force(sys) + + # compute positions at t + dt + positions(sys.state) .+= + velocities(sys.state) .* sys.dt .+ + 0.5 .* acceleration .* (sys.dt)^2 + + # enforce boundary conditions + # (basically check if any particles left the box and put them back) + # see function implementation for deets + keep_particles_in_box!(sys) + + # compute velocities at t + dt + velocities(sys.state) .+= + 0.5 * sys.dt .* (acceleration + force(sys)) +end + +function evolve!(sys::ParticleSystem, runtime::Float64 = 10.0) + numsteps = Int64(abs(runtime / sys.dt) + 1) + + print_evolution_message(runtime, numsteps) + + @time for step in 1:numsteps + verlet_step!(sys) + zero_total_momentum!(sys) + + if (step % sys.sampleInterval == 1) + push!(sys.timeData, sys.t) + push!(sys.energyData, energy(sys)) + push!(sys.xData, positions(sys.state)) + push!(sys.vData, velocities(sys.state)) + + T = temperature(sys) + push!(sys.tempData, T) + sys.tempAccumulator += T + sys.squareTempAccumulator += T^2 + end + + sys.t += sys.dt + sys.steps += 1 + end + println("done.") +end + +function reverse_time!(sys) + sys.dt *= -1 + println("\ntime reversed! dt = $(sys.dt)") +end + +function cool!(sys::ParticleSystem, cooltime::Float64 = 1.0) + numsteps = Int64(cooltime / sys.dt) + for step in 1:numsteps + verlet_step!(sys) + velocities(sys.state) .*= (1.0 - sys.dt) + end + reset_statistics!(sys) +end + +# MEASUREMENTS +################################################################################ + +function kinetic_energy(sys::ParticleSystem) + return 0.5 * sum(velocities(sys.state) .* velocities(sys.state)) +end + +function potential_energy(sys::ParticleSystem) + return lennard_jones_potential(sys) +end + +function temperature(sys::ParticleSystem) + return kinetic_energy(sys) / sys.N +end + +function energy(sys::ParticleSystem) + return potential_energy(sys) + kinetic_energy(sys) +end + +# STATISTICS +################################################################################ + +function reset_statistics!(sys::ParticleSystem) + sys.steps = 0 + sys.tempAccumulator = 0.0 + sys.squareTempAccumulator = 0.0 + sys.virialAccumulator = 0.0 + sys.xData = [] + sys.vData = [] +end + +function mean_temperature(sys::ParticleSystem) + return sys.tempAccumulator / sys.steps +end + +function mean_square_temperature(sys::ParticleSystem) + return sys.squareTempAccumulator / sys.steps +end + +function mean_pressure(sys::ParticleSystem) + # factor of half because force is calculated twice each step + meanVirial = 0.5 * sys.virialAccumulator / sys.steps + return 1.0 + 0.5 * meanVirial / (sys.N * mean_temperature(sys)) +end + +function heat_capacity(sys::ParticleSystem) + meanTemperature = mean_temperature(sys) + meanSquareTemperature = mean_square_temperature(sys) + σ2 = meanSquareTemperature - meanTemperature^2 + denom = 1.0 - σ2 * sys.N / meanTemperature^2 + return sys.N / denom +end + +function mean_energy(sys::ParticleSystem) + return mean(sys.energyData) +end + +function std_energy(sys::ParticleSystem) + return std(sys.energyData) +end + +# MATH / ADDITIONAL FUNCTIONS +################################################################################ + +function dot(v1::Vector{Float64}, v2::Vector{Float64}) + return sum(v1 .* v2) +end + +# GRAPHS +################################################################################ + +function initialize_plot() + plot( + size = (800, 800), + titlefontsize = 12, + guidefontsize = 12, + ) +end + +function plot_positions_t(sys::ParticleSystem, t::Int64) + initialize_plot() + for n ∈ 1:sys.N + scatter!( + [sys.xData[t][2n-1]], + [sys.xData[t][2n]], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end +end + +function animate(sys::ParticleSystem, interval::Int64 = 1) + println("\ngenerating gif...") + + scatter!() + animation = @animate for t in 1:length(sys.xData) + scatter() + for n ∈ 1:sys.N + scatter!( + [sys.xData[t][2n-1]], + [sys.xData[t][2n]], + #markersize = 4.0, + markercolor = n, + #markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + end every interval + + gif(animation, "./animation.gif") + println("done.") +end + +function plot_positions(sys::ParticleSystem) + initialize_plot() + for n ∈ 1:sys.N + scatter!( + [xcomponent(positions(sys.state))[n]], + [ycomponent(positions(sys.state))[n]], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + xlims!(0, sys.L) + ylims!(0, sys.L) + xlabel!("x") + ylabel!("y") + title!("positions at t=$(round(sys.t, digits=4))") +end + +function plot_trajectories(sys::ParticleSystem, particles::Vector{Int64} = [1]) + initialize_plot() + for n ∈ 1:sys.N + scatter!( + [xcomponent(positions(sys.state))[n]], + [ycomponent(positions(sys.state))[n]], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + grid = true, + framestyle = :box, + legend = false, + ) + end + + for n in collect(particles) + xdata = [sys.xData[i][2n-1] for i in 1:length(sys.xData)] + ydata = [sys.xData[i][2n] for i in 1:length(sys.xData)] + + # plot trajectory line for nth particle + scatter!( + xdata, + ydata, + color = n, + #markerstrokewidth = 0, + markerstrokecolor = n, + markersize = 0.7, + markeralpha = 0.5, + label = false, + widen = false, + ) + + # plot initial position for nth particle + scatter!( + [sys.xData[1][2n-1]], + [sys.xData[1][2n]], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 0.3, + #label = "pcl. $n @t=t₀", + widen = false, + ) + + # plot final position for nth particle + scatter!( + [sys.xData[end][2n-1]], + [sys.xData[end][2n]], + markersize = 4.0, + markercolor = n, + markerstrokewidth = 0.4, + markeralpha = 1.0, + #label = "pcl $n @t=t", + widen = false, + ) + end + title!("positions & trajectories at time t=$(round(sys.t, digits=2))") + plot!() +end + +function plot_temperature(sys::ParticleSystem) + initialize_plot() + plot!( + sys.timeData, + sys.tempData, + #widen = true, + ) + ylims!( + mean(sys.tempData) - std(sys.tempData) * 20, + mean(sys.tempData) + std(sys.tempData) * 20, + ) + xlabel!("t") + ylabel!("T(t)") + title!("temperature vs time") + +end + +function plot_energy(sys::ParticleSystem, ylimit::Float64 = 1.0) + initialize_plot() + plot!( + sys.timeData, + sys.energyData, + #widen = true, + ) + ylims!( + #ylimit * (mean(sys.energyData) - 1), + #ylimit * (mean(sys.energyData) + 1) + mean(sys.energyData) - std(sys.energyData) * 10, + mean(sys.energyData) + std(sys.energyData) * 10, + ) + xlabel!("t") + ylabel!("E(t)") + title!("energy vs time") +end + +function plot_speed_distribution(sys::ParticleSystem, numSamples::Int64 = 5) + initialize_plot() + + numDataPoints = Int64(length(sys.vData)) + interval = Int64(floor(numDataPoints / numSamples)) + + samples = collect(1:interval:numDataPoints) + for s in samples + speed = sqrt.( + xcomponent(sys.vData[s]) .^ 2 .* + ycomponent(sys.vData[s]) .^ 2 + ) + density!( + sys.vData[s], + normalize = :pdf, + label = "t = $(round(sys.timeData[s], digits=2))", + ) + end + xlabel!("speed") + title!("speed distribution") +end + +# CONSOLE PRINT DATA +################################################################################ + +function print_hello() + println("\nmolecular dynamics!") + println("number of threads: ", Threads.nthreads()) +end + +function print_bonjour() + println("\nbonjour") +end + +function print_system_parameters(sys::ParticleSystem) + println("\nsystem parameters:") + println("\tN = $(sys.N) (number of particles)") + println("\tL = $(sys.L) (side length of square box)") + println("\tDT = $(sys.dt) (time step)") +end + +function print_system_data(sys::ParticleSystem) + println("\nsystem data at time t=$(round(sys.t, digits=4))") + + if sys.steps == 0 + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + else + println("\tsteps evolved: $(sys.steps)") + println("\ttemperature: $(temperature(sys))") + println("\tenergy: $(energy(sys))") + println("\tmean energy: $(mean_energy(sys))") + println("\tstd energy: $(std_energy(sys))") + println("\theat capacity: $(heat_capacity(sys))") + println("\tPV/NkT: $(mean_pressure(sys))") + end +end + +function print_evolution_message(runtime, numsteps) + println("\nevolving...") +end + +# DEMOS +################################################################################ + + +# DEMO 0: APPROACH TO EQUILIBRIUM +function demo_0(; gif = 0) + println("\nDEMO 0: APPROACH TO EQUILIBRIUM") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 20.0) + print_system_data(sys) + + p2 = plot_trajectories(sys, collect(1:64)) + p3 = plot_energy(sys) + p4 = plot_temperature(sys) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, + layout = grid(2, 2, heights = [0.7, 0.3]), + size = (1280, 720), + ) +end + +# DEMO 1: TIME REVERSAL TEST +function demo_1(; gif = 0) + println("\nDEMO 1: TIME REVERSAL TEST") + println("----------------------------------------") + + sys = ParticleSystem(64, 120.0, 1.0) + print_system_parameters(sys) + + set_square_lattice_positions!(sys) + set_random_velocities!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + evolve!(sys, 50.0) + #p2 = plot_trajectories(sys, collect(1:64)) + p2 = plot_positions(sys) + + reverse_time!(sys) + evolve!(sys, 50.0) + print_system_data(sys) + #p3 = plot_trajectories(sys, collect(1:64)) + p3 = plot_positions(sys) + + # make gif + if gif == 1 + animate(sys, 4) + end + + plot( + p1, p2, p3, + layout = (1, 3), + size = (1200, 400), + ) +end + +# DEMO 2: SPEED DISTRIBUTION +function demo_2() + println("\nDEMO 2: SPEED DISTRIBUTION") + println("----------------------------------------") + + sys = ParticleSystem[] + + # array for speed distribution plots + ps = Plots.Plot{Plots.GRBackend}[] + + # array for trajectory plots + pt = Plots.Plot{Plots.GRBackend}[] + + # initialize three systems with different initial conditions + # but same KE and PE, evolve, and save plots + for i ∈ 1:3 + push!(sys, ParticleSystem(64, 120.0, 1.0)) + + println("\nSYSTEM $i") + print_system_parameters(sys[i]) + + set_square_lattice_positions!(sys[i]) + add_position_jitter!(sys[i]) + set_random_velocities!(sys[i]) + print_system_data(sys[i]) + + evolve!(sys[i], 40.0) + print_system_data(sys[i]) + push!(ps, plot_speed_distribution(sys[i], 5)) + push!(pt, plot_trajectories(sys[i], collect(1:64))) + end + + + # plot speed distribution and trajectory plots + plot( + ps[1], ps[2], ps[3], + pt[1], pt[2], pt[3], + layout = (2, 3), + size = (1920, 1080), + ) +end + +# DEMO 3: MELTING TRANSITION +function demo_3(; gif = 0) + println("\nDEMO 3: MELTING TRANSITION") + println("----------------------------------------") + + # initialize system of particles on square lattice with zero velocity + sys = ParticleSystem(100, 10.0, 5.0) + set_square_lattice_positions!(sys) + print_system_data(sys) + p1 = plot_positions(sys) + + # evolve the system and watch them "crystallize" + # into a triangular lattice formation + evolve!(sys, 20.0) + print_system_data(sys) + p2 = plot_trajectories(sys, collect(1:100)) + + # now, increase the temperature of the system by giving the particles + # some velocity. evolve the system and plot the trajectories. + set_random_velocities!(sys) + evolve!(sys, 60.0) + print_system_data(sys) + p3 = plot_trajectories(sys, collect(1:100)) + + # some more plots + p4 = plot_energy(sys, 0.0) + p5 = plot_temperature(sys) + p6 = plot_speed_distribution(sys, 20) + + # make gif + if gif == 1 + animate(sys, 1) + end + + plot( + p1, p2, p3, p4, p5, p6, + layout = (2, 3), + size = (1280, 720), + ) +end + +demo_0() diff --git a/t/velocity-over-time.png b/t/velocity-over-time.png new file mode 100644 index 0000000..5eeba5b Binary files /dev/null and b/t/velocity-over-time.png differ -- cgit v1.2.3-70-g09d2