From c048f9f2219897ff3cc20a50d459911db3496a0a Mon Sep 17 00:00:00 2001 From: sotech117 Date: Thu, 2 May 2024 19:29:49 -0400 Subject: major update, with final project rough draft code --- .DS_Store | Bin 10244 -> 14340 bytes hw5/plz.jl | 124 ++-- hw7/8-12.jl | 184 ++--- hw7/magnetization_vs_field.png | Bin 0 -> 29711 bytes t/.DS_Store | Bin 0 -> 12292 bytes t/1d.jl | 109 --- t/animate-positions-10.mp4 | Bin 1016778 -> 0 bytes t/animate-positions.mp4 | Bin 312915 -> 0 bytes t/animated-fpu.mp4 | Bin 0 -> 11702 bytes t/breakdown-time-over-beta-od.png | Bin 0 -> 22753 bytes t/breakdown-time-over-beta-short.png | Bin 0 -> 22678 bytes t/breakdown-time-over-beta.png | Bin 0 -> 30522 bytes t/disp.jl | 173 ----- t/energies-modes-separate-beta-0.0.png | Bin 0 -> 24742 bytes t/energies-modes-separate-beta-0.05.png | Bin 0 -> 26815 bytes t/energies-modes-separate-beta-0.06.png | Bin 0 -> 27374 bytes t/energies-modes-separate-beta-0.07.png | Bin 0 -> 27925 bytes 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100644 Binary files a/.DS_Store and b/.DS_Store differ diff --git a/hw5/plz.jl b/hw5/plz.jl index d4abcbd..7e50e4f 100644 --- a/hw5/plz.jl +++ b/hw5/plz.jl @@ -1,73 +1,73 @@ function dynamics!( - state, - prev_state, - params, - states + 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 + # 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 + 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 + 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 + 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 + 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 @@ -76,7 +76,7 @@ beta = 1.5 # cubic string spring A = 10 # amplitude modes = 3 # number of modes to plot final_time = 50 # seconds -dt = .5 # seconds +dt = 0.5 # 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) @@ -89,7 +89,7 @@ using Plots # 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)) + plot(s[i], label = "t = $(i * dt)", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3)) end gif(anim, "plz.gif", fps = 30) diff --git a/hw7/8-12.jl b/hw7/8-12.jl index 29e18ec..b10d941 100644 --- a/hw7/8-12.jl +++ b/hw7/8-12.jl @@ -1,6 +1,6 @@ #!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia -using Statistics +using Statistics using Plots function wrap_index(i::Int, l::Int)::Int @@ -9,95 +9,95 @@ function wrap_index(i::Int, l::Int)::Int end mutable struct Ising2D - l::Int - n::Int + l::Int + n::Int temperature::Float64 w::Vector{Float64} # Boltzmann weights - state::Matrix + state::Matrix energy::Float64 magnetization::Int - mc_steps::Int - accepted_moves::Int + mc_steps::Int + accepted_moves::Int energy_array::Vector{Float64} magnetization_array::Vector{Int} - H::Float64 -end + H::Float64 +end -Ising2D(l::Int, temperature::Float64, H=1.0) = begin - n = l^2 +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) + 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[] + 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 +function mc_step!(ising::Ising2D) + l::Int = ising.l + n::Int = ising.n + w = ising.w - state = ising.state + state = ising.state accepted_moves = ising.accepted_moves - energy = ising.energy + energy = ising.energy magnetization = ising.magnetization random_positions = l * rand(2 * n) - random_array = rand(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 + 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)]) + 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 + accepted_moves += 1 + new_spin = -state[i, j] # flip spin state[i, j] = new_spin - # add the effects of the new spin + # add the effects of the new spin energy += de magnetization += 2 * new_spin end end - ising.state = state + ising.state = state ising.accepted_moves = accepted_moves - ising.energy = energy - ising.magnetization = magnetization + ising.energy = energy + ising.magnetization = magnetization - append!(ising.energy_array, ising.energy) + 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) +function steps!(ising::Ising2D, num::Int = 100) for i in 1:num mc_step!(ising) - end + end end function mean_energy(ising::Ising2D) return mean(ising.energy_array) / ising.n -end +end function specific_heat(ising::Ising2D) - return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n + return (std(ising.energy_array) / ising.temperature)^2 / ising.n end function mean_magnetization(ising::Ising2D) @@ -105,85 +105,93 @@ function mean_magnetization(ising::Ising2D) end function susceptibility(ising::Ising2D) - return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n) + 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") + 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) + 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) + 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 + 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 + 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) +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 + # 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) + 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 + savefig(p, "hw7/magnetization_vs_field.png") + return p end # textbook rec -h_range = .02:.02:.2 +h_range = 0.02:0.02:0.2 h_range = collect(h_range) T = 2.27 # T_c for this system side = 64 diff --git a/hw7/magnetization_vs_field.png b/hw7/magnetization_vs_field.png new file mode 100644 index 0000000..0ef28b5 Binary files /dev/null and b/hw7/magnetization_vs_field.png differ diff --git a/t/.DS_Store b/t/.DS_Store new file mode 100644 index 0000000..446c2ed Binary files /dev/null and b/t/.DS_Store differ diff --git a/t/1d.jl b/t/1d.jl deleted file mode 100644 index 203466c..0000000 --- a/t/1d.jl +++ /dev/null @@ -1,109 +0,0 @@ -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 deleted file mode 100644 index aa00547..0000000 Binary files a/t/animate-positions-10.mp4 and /dev/null differ diff --git a/t/animate-positions.mp4 b/t/animate-positions.mp4 deleted file mode 100644 index 3633358..0000000 Binary files a/t/animate-positions.mp4 and /dev/null differ diff --git a/t/animated-fpu.mp4 b/t/animated-fpu.mp4 new file mode 100644 index 0000000..87c2082 Binary files /dev/null and b/t/animated-fpu.mp4 differ diff --git a/t/breakdown-time-over-beta-od.png b/t/breakdown-time-over-beta-od.png new file mode 100644 index 0000000..c117be8 Binary files /dev/null and b/t/breakdown-time-over-beta-od.png differ diff --git a/t/breakdown-time-over-beta-short.png b/t/breakdown-time-over-beta-short.png new file mode 100644 index 0000000..7e8935e Binary files /dev/null and b/t/breakdown-time-over-beta-short.png differ diff --git a/t/breakdown-time-over-beta.png b/t/breakdown-time-over-beta.png new file mode 100644 index 0000000..efe8473 Binary files /dev/null and b/t/breakdown-time-over-beta.png differ diff --git a/t/disp.jl b/t/disp.jl deleted file mode 100644 index 41ee2bb..0000000 --- a/t/disp.jl +++ /dev/null @@ -1,173 +0,0 @@ -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) 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a/t/new/positions-at-t-5000-beta-0.png b/t/new/positions-at-t-5000-beta-0.png new file mode 100644 index 0000000..e593f1a Binary files /dev/null and b/t/new/positions-at-t-5000-beta-0.png differ diff --git a/t/new/positions-at-t-5000-beta-10.png b/t/new/positions-at-t-5000-beta-10.png new file mode 100644 index 0000000..b6ea36c Binary files /dev/null and b/t/new/positions-at-t-5000-beta-10.png differ diff --git a/t/old/1d.jl b/t/old/1d.jl new file mode 100644 index 0000000..203466c --- /dev/null +++ b/t/old/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/old/animate-positions-.1stringa.mp4 b/t/old/animate-positions-.1stringa.mp4 new file mode 100644 index 0000000..0964313 Binary files /dev/null and b/t/old/animate-positions-.1stringa.mp4 differ diff --git a/t/old/animate-positions-.5vel.mp4 b/t/old/animate-positions-.5vel.mp4 new file mode 100644 index 0000000..b56e372 Binary files /dev/null and b/t/old/animate-positions-.5vel.mp4 differ diff --git a/t/old/animate-positions-10.mp4 b/t/old/animate-positions-10.mp4 new file mode 100644 index 0000000..aa00547 Binary files /dev/null and b/t/old/animate-positions-10.mp4 differ diff --git a/t/old/animate-positions-80k.mp4 b/t/old/animate-positions-80k.mp4 new file mode 100644 index 0000000..3551bf4 Binary files /dev/null and b/t/old/animate-positions-80k.mp4 differ diff --git a/t/old/animate-positions-chaos.mp4 b/t/old/animate-positions-chaos.mp4 new file mode 100644 index 0000000..7edbc75 Binary files /dev/null and b/t/old/animate-positions-chaos.mp4 differ diff --git a/t/old/animate-positions-stable.mp4 b/t/old/animate-positions-stable.mp4 new file mode 100644 index 0000000..0b858a2 Binary files /dev/null and b/t/old/animate-positions-stable.mp4 differ diff --git a/t/old/animate-positions-t.mp4 b/t/old/animate-positions-t.mp4 new file mode 100644 index 0000000..a4a34e0 Binary files /dev/null and b/t/old/animate-positions-t.mp4 differ diff --git a/t/old/disp.jl b/t/old/disp.jl new file mode 100644 index 0000000..18e0a1b --- /dev/null +++ b/t/old/disp.jl @@ -0,0 +1,272 @@ +M = 10 +m = 1 # mass of particles + +function calculate_force( + left_pos, + middle_pos, + right_pos, + K, + alpha = 0.0, + beta = 1000.0, +) + linear_force = K * (left_pos + right_pos - 2 * middle_pos) + quadratic_force = alpha * (left_pos - middle_pos)^2 + alpha * (right_pos - middle_pos)^2 + cubic_force = beta * (left_pos - middle_pos)^3 + beta * (right_pos - middle_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 1:N-1 + mass = m + if i == Int(N / 2) + mass = M + end + + left_index = max(1, i - 1) + right_index = min(N, i + 1) + + du[i*2-1] = ps[i] / mass + force = calculate_force(qs[left_index], qs[i], qs[right_index], K) + du[i*2] = force / mass + end + + # make last point same as first + du[N*2-1] = du[1] = 0 # set to 0 + du[N*2] = du[2] = 0 + + + if t % 100000 == 0 + println("TIME UPDATE: ", t) + end + + # if ps[Int(N / 2)] / M >= 1 + # println("(in sim!) Time: ", t, " Vel: ", ps[Int(N / 2)] / M) + # # println("Other Positions: ", qs) + # println("Other Velocities: ", ps, "\n") + # end +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 * M + 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) + + calculate_energy(s_0) + + # 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-5, abstol = 1e-5, maxiters = 1e10) # control simulation + + calculate_energy(sol.u[end]) + + 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, + shift = true, +) + 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)] / M + p_middle = states[i][Int(length(states[1]) / 2)-1] + y_lims = shift ? (-3 + p_middle, 3 + p_middle) : (-3, 3) + # 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 = $(round(t, digits = 7)), v_middle=$(round(v_middle, digits=7))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = y_lims, + color = :red, legend = :topright, + ) + else + plot(positions, label = "t = $(round(t, digits = 7)), v_middle=$(round(v_middle, digits=7))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = y_lims, + 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, +) + # plot the positions + middle_index = Int(length(states[1]) / 2) - 1 + pos_init = [x - states[1][middle_index] for x in states[1][1:2:end]] + pos_final = [x - states[end][middle_index] for x in states[end][1:2:end]] + p1 = plot(pos_init, label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "Positions Over Time") + plot!(p1, pos_final, label = "Final", marker = :circle) + + # plot the vels + vels_init = [x / M for x in states[1][2:2:end]] + vels_final = [x / M for x in states[end][2:2:end]] + p2 = plot(states[1][2:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Velocity", title = "Velocities Over Time") + plot!(p2, states[end][2:2:end], label = "Final t = $(time_steps[end])", 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) + v = states[i][Int(length(states[i]) / 2)] / M + if v >= threshold - 10e-6 + push!(output, i) + println("Time: ", time_steps[i], " Vel: ", v) + end + end + + data = [] + for i in 1:length(states) + 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 + +function analyize_pos( + states, + time_steps, + threshold = 1.975, +) + println("Analyzing positions:\n") + output = [] + for i in 1:length(states) + if states[i][Int(length(states[i]) / 2)-1] >= 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)] - 1) + end + p = plot(data, label = "Position Over Time", xlabel = "Time", ylabel = "Position") + savefig(p, "t/pos-over-time.png") + + println("\nDone!\n\n") + return output +end + +function calculate_energy(state) + # calculate the kinetic energy + kinetic_energy = 0 + vels = state[2:2:end] + for i in 1:N + mass = i == Int(N / 2) - 1 ? M : m + # calculate the kinetic energy + kinetic_energy += 0.5 * vels[i] * vels[i] / mass + end + + # calcaute the potential energy + potential_energy = 0 + pos = state[1:2:end] + for i in 1:N-1 + left_index = max(1, i - 1) + right_index = min(N, i + 1) + potential_energy += 0.5 * K * (pos[left_index] - pos[i])^2 + potential_energy += 0.5 * K * (pos[right_index] - pos[i])^2 + end + + # print the energy + println("Kinetic Energy: ", kinetic_energy) + println("Potential Energy: ", potential_energy) + println("Total Energy: ", kinetic_energy + potential_energy, "\n") +end + +function plot_middle_mass_phase_space(states) + # get the index of the middle mass + middle_index = Int(length(states[1]) / 2) - 1 + # build an array of the pos and vel over time + pos = [] + vel = [] + for i in 1:length(states) + push!(pos, states[i][middle_index]) + push!(vel, states[i][middle_index+1]) + end + + # plot the phase space + p = plot(pos, vel, xlabel = "Position", ylabel = "Momentum", title = "Phase Space of Middle Mass in FPU", label = "Beta = 10, K = 1, N = 64, M = $M") + + # save the plot + savefig(p, "t/phase-space.png") +end + + +# Run the simulation +N = 64 # number of masses +K = 1 # spring constant +final_time = 1000 # seconds +plot_data = [] + +my_vel = 100 + +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) +# analyize_pos(sol.u, sol.t, 1.4) +plot_starting_and_final_positions(sol.u, sol.t) +# animate_positions(sol.u, sol.t, 0, 100, my_vel) # expect 80913.35854226245 for k=10?? rip +plot_middle_mass_phase_space(sol.u) diff --git a/t/old/initial-final-positions.png b/t/old/initial-final-positions.png new file mode 100644 index 0000000..e76793d Binary files /dev/null and b/t/old/initial-final-positions.png differ diff --git a/t/old/initial-final-velocities.png b/t/old/initial-final-velocities.png new file mode 100644 index 0000000..b7cc859 Binary files /dev/null and b/t/old/initial-final-velocities.png differ diff --git a/t/old/modes-beta15.png b/t/old/modes-beta15.png new file mode 100644 index 0000000..dbb632d Binary files /dev/null and b/t/old/modes-beta15.png differ diff --git a/t/old/phase-space-.5vel.png b/t/old/phase-space-.5vel.png new file mode 100644 index 0000000..100cd2e Binary files /dev/null and b/t/old/phase-space-.5vel.png differ diff --git a/t/old/phase-space-chos.png b/t/old/phase-space-chos.png new file mode 100644 index 0000000..cbbbf09 Binary files /dev/null and b/t/old/phase-space-chos.png differ diff --git a/t/old/phase-space-stable.png b/t/old/phase-space-stable.png new file mode 100644 index 0000000..2ffa7c6 Binary files /dev/null and b/t/old/phase-space-stable.png differ diff --git a/t/old/phase-space.01strings.png b/t/old/phase-space.01strings.png new file mode 100644 index 0000000..5e78901 Binary files /dev/null and b/t/old/phase-space.01strings.png differ diff --git a/t/old/phase-space.png b/t/old/phase-space.png new file mode 100644 index 0000000..48e5368 Binary files /dev/null and b/t/old/phase-space.png differ diff --git a/t/old/plot_data.png b/t/old/plot_data.png new file mode 100644 index 0000000..d052983 Binary files /dev/null and b/t/old/plot_data.png differ diff --git a/t/old/plz.gif b/t/old/plz.gif new file mode 100644 index 0000000..febf678 Binary files /dev/null and b/t/old/plz.gif differ diff --git a/t/old/plz.mp4 b/t/old/plz.mp4 new file mode 100644 index 0000000..6a0d7b4 Binary files /dev/null and b/t/old/plz.mp4 differ diff --git a/t/old/pos-over-time.png b/t/old/pos-over-time.png new file mode 100644 index 0000000..e3a732d Binary files /dev/null and b/t/old/pos-over-time.png differ diff --git a/t/old/r.jl b/t/old/r.jl new file mode 100644 index 0000000..516ab09 --- /dev/null +++ b/t/old/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/old/semi-old b/t/old/semi-old new file mode 100644 index 0000000..e69de29 diff --git a/t/old/t.jl b/t/old/t.jl new file mode 100644 index 0000000..6a06777 --- /dev/null +++ b/t/old/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/old/velocity-over-time.png b/t/old/velocity-over-time.png new file mode 100644 index 0000000..3cab01f Binary files /dev/null and b/t/old/velocity-over-time.png differ diff --git a/t/plot_data.png b/t/plot_data.png deleted file mode 100644 index d052983..0000000 Binary files a/t/plot_data.png and /dev/null differ diff --git a/t/plz.jl b/t/plz.jl new file mode 100644 index 0000000..962398a --- /dev/null +++ b/t/plz.jl @@ -0,0 +1,767 @@ +function calculate_force( + left_pos, + middle_pos, + right_pos, + K, + alpha = 0.0, + beta = 0.0, +) + linear_force = K * (left_pos + right_pos - 2 * middle_pos) + quadratic_force = alpha * (left_pos - middle_pos)^2 + alpha * (right_pos - middle_pos)^2 + cubic_force = beta * (left_pos - middle_pos)^3 + beta * (right_pos - middle_pos)^3 + + return linear_force + quadratic_force + cubic_force +end + +function tendency!(du, u, params, t) + # unpack the params + N, modes, beta, A, dt, num_steps = params + + # get the positions and momenta + qs = u[1:2:end] + ps = u[2:2:end] + + num_masses = length(qs) + + # go over the points in the lattice and update the state + for i in 2:num_masses-1 + # left_index = max(1, i - 1) + # right_index = min(N, i + 1) + + du[i*2-1] = ps[i] + force = calculate_force(qs[i-1], qs[i], qs[i+1], 1, 0, beta) + du[i*2] = force + end + + # make last point same as first at rest + du[num_masses*2-1] = du[1] = 0 # set to 0 + du[num_masses*2] = du[2] = 0 + + # if ps[Int(N / 2)] / M >= 1 + # println("(in sim!) Time: ", t, " Vel: ", ps[Int(N / 2)] / M) + # # println("Other Positions: ", qs) + # println("Other Velocities: ", ps, "\n") + # end +end + +function get_initial_state( + N, + modes, + beta, + A, +) + state = zeros(N + 2) + 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 + +using DifferentialEquations +function run_simulation( + N, + modes, + beta, + A, + dt, + num_steps, +) + println("Running simulation with params: N=$N, modes=$modes, beta=$beta, A=$A, dt=$dt, num_steps=$num_steps") + # states = [] + state = get_initial_state(N, 1, beta, A) + # prev_state = zeros(N) + # for i in 1:N + # prev_state[i] = state[i] + # end + params = (N, modes, beta, A, dt, num_steps) + # dynamics!(state, prev_state, params, states) + + # state above is position, need to add in momentum + s_0 = zeros(2 * (N + 2)) + for i in 1:length(state) + s_0[i*2-1] = state[i] + end + final_time = num_steps * dt + t_span = (0.0, final_time) + prob = ODEProblem(tendency!, s_0, t_span, params) + sol = solve(prob, Tsit5(), reltol = 1e-8, abstol = 1e-8, maxiters = 1e10) # control simulation + + return sol +end + +function get_pos_from_state(state) + return state[1:2:end] +end + +# Run the simulation +# N = 32 # number of masses +# beta = 2.3 # cubic string spring +# A = 10 # amplitude +# modes = 3 # number of modes to plot +# final_time = 100000 # seconds +# dt = 0.05 # seconds +# num_steps = Int(final_time / dt) +# params = (N, modes, beta, A, dt, num_steps) +# println("Running simulation with params: ", params) +# sol = run_simulation(N, modes, beta, A, dt, num_steps) +# states = sol.u +# timesteps = sol.t +# println("Final state: ", states[end]) + +# plot the inital positions and the final positions +using Plots +# init_pos = get_initial_state(N, 1, beta, A) +# final_pos = get_pos_from_state(states[end]) +# println("Initial: ", init_pos) +# println("Final: ", final_pos) +# plot(init_pos, label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "First Three Modes") +# plot!(final_pos, label = "Final", marker = :circle) +# println("Saving initial-final-beta15.png") +# savefig("t/initial-final-beta-$beta.png") + +function animate_states(states, timesteps, from = 1, to = 1000) + println("\nAnimating states") + # animate the s array of positions + # find the timestep that maps to the state index + i_start = 1 + i_end = length(timesteps) + for i in 1:length(timesteps) + if timesteps[i] <= from + i_start = i + continue + end + if timesteps[i] >= to + i_end = i + break + end + end + + anim = @animate for i in i_start:i_end + s = get_pos_from_state(states[i]) + t = timesteps[i] + plot(s, label = "t = $t", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3)) + end + + mp4(anim, "t/animated-fpu.mp4", fps = 30) + println("Done animating, wrote to animated-fpu.mp4\n") +end +# animate_states(states, timesteps, 149500, 150000) + +function caluclate_energies_for_mode(states, mode) + N = length(states[1]) / 2 - 2 + + total = [] + kinetic = [] + potential = [] + + for i in 1:length(states) + total_energy = 0 + kinetic_energy = 0 + potential_energy = 0 + + # find a_k for this mode + poses = get_pos_from_state(states[i]) + a_k = 0 + for j in 1:length(poses) + a_k += poses[j] * sin((mode * (j - 1) * π) / (N + 1)) + end + amp = sqrt(2 / (N + 1)) + a_k *= amp + + # get the potenital energy + omega_k = 2 * sin((mode * π) / (2 * (N + 1))) + p = 0.5 * omega_k^2 * a_k^2 + + # find the kinetic energy + v_k = 0 + vels = states[i][2:2:end] + for j in 1:length(vels) + v_k += vels[j] * sin((mode * (j - 1) * π) / (N + 1)) + end + v_k *= amp + k = 0.5 * v_k^2 + + + total_energy += k + p + kinetic_energy += k + potential_energy += p + push!(total, total_energy) + push!(kinetic, kinetic_energy) + push!(potential, potential_energy) + end + return total, kinetic, potential +end + +function calculate_total_energy_by_state(states, timesteps) + total = [] + kinetic = [] + potential = [] + for i in 1:length(states) + vels = states[i][2:2:end] + k = 0.5 * sum(vels .^ 2) + + poses = get_pos_from_state(states[i]) + p = 0 + for j in 2:length(poses) + p += beta * (poses[j] - poses[j-1])^2 + end + + push!(total, k + p) + push!(kinetic, k) + push!(potential, p) + end + + # plot all three over time on the same plot + plot(timesteps, total, label = "Total Energy", xlabel = "Time", ylabel = "Energy", title = "Total Energy Over Time") + plot!(timesteps, kinetic, label = "Kinetic Energy") + plot!(timesteps, potential, label = "Potential Energy") + + return total, kinetic, potential +end + +function print_energies_for_modes(states, max_mode = 3, index = 1) + # get the sum of total energy over all mode + total = 0 + kinetic = 0 + potential = 0 + for i in 1:max_mode + t, k, p = caluclate_energies_for_mode(states, i) + total += t[index] + kinetic += k[index] + potential += p[index] + end + println("Total Energy: ", total) + println("Kinetic Energy: ", kinetic) + println("Potential Energy: ", potential) +end + +function plot_phase_space(states, mass_number, intersecting_mass = 4) + # get the positions and momenta + qs = [] + ps = [] + + for i in 2:length(states) + # only store if the other mass is crossing 0 + if (states[i-1][intersecting_mass*2-1] < 0 && states[i][intersecting_mass*2-1] > 0) || (states[i-1][intersecting_mass*2-1] > 0 && states[i][intersecting_mass*2-1] < 0) + s = states[i] + q = s[mass_number*2-1] + p = s[mass_number*2] + push!(qs, q) + push!(ps, p) + end + end + + s = scatter( + qs, + ps, + label = "Mass $mass_number", + xlabel = "Displacement", + ylabel = "Momentum", + title = "Phase Space for Masses when Mass $intersecting_mass crosses 0", + legend = :topleft, + marker = :circle, + markersize = 2, + xlim = (-1.5, 1.5), + ylim = (-1.5, 1.5), + ) + + # savefig(s, "t/phase-space-mass$mass_number-beta15.png") + + return s +end + +function analyze_energies_of_n_modes(modes::Array{Int}, states, timesteps, beta) + println("Analyzing energies of modes: ", modes) + + total_by_mode = Dict() + kinetic_by_mode = Dict() + potential_by_mode = Dict() + for j in modes + t, k, p = caluclate_energies_for_mode(states, j) + + total_by_mode[j] = t + kinetic_by_mode[j] = k + potential_by_mode[j] = p + end + + total_energy = [] + kinetic_energy = [] + potential_energy = [] + for i in 1:length(states) + total = 0 + kinetic = 0 + potential = 0 + + for j in modes + total += total_by_mode[j][i] + kinetic += kinetic_by_mode[j][i] + potential += potential_by_mode[j][i] + end + + push!(total_energy, total) + push!(kinetic_energy, kinetic) + push!(potential_energy, potential) + end + + # plot all three over time on the same plot + plot(timesteps, total_energy, label = "Total Energy", xlabel = "Time", ylabel = "Energy", title = "Total Energy Over Time") + plot!(timesteps, kinetic_energy, label = "Kinetic Energy") + plot!(timesteps, potential_energy, label = "Potential Energy") + savefig("t/procs/energies-modes-sum-beta-$beta.png") + println("Saved energies-modes-sum-beta-$beta.png. Summary Below:\n") + + # plot the energies for each mode over time + plt = plot(title = "Energy Over Time for Modes $(join(modes, ", "))", xlabel = "Time", ylabel = "Energy") + for mode in modes + plot!(plt, timesteps, total_by_mode[mode], label = "Mode $mode") + # plot!(plt, timesteps, kinetic_by_mode[mode], label = "Mode $mode Kinetic") + # plot!(plt, timesteps, potential_by_mode[mode], label = "Mode $mode Potential") + end + savefig(plt, "t/procs/energies-modes-separate-beta-$beta.png") + + # print the energies + println("Total Energy Initial: ", total_energy[1]) + println("Total Energy Final: ", total_energy[end]) + println("Kinetic Energy Initial: ", kinetic_energy[1]) + println("Kinetic Energy Final: ", kinetic_energy[end]) + println("Potential Energy Initial: ", potential_energy[1]) + println("Potential Energy Final: ", potential_energy[end], "\n") + + println("Done analyzing energies of modes: ", modes, "\n") + + return total_energy, kinetic_energy, potential_energy, total_by_mode, kinetic_by_mode, potential_by_mode +end + +function animate_masses_phase_space(states, mass_numbers, intersecting_mass = 4) + println("\nAnimating phase space for mass $mass_numbers when mass $intersecting_mass crosses 0") + # animate the s array of positions + # find the timestep that maps to the state index + anim = @animate for i in mass_numbers + s = plot_phase_space(states, i, intersecting_mass) + s + end + + mp4(anim, "t/animated-phase-space-masses-beta-$beta.mp4", fps = 24) +end + +function plot_mode_phase_space(states, mode_num, intersecting_mode, beta) + println("\nPlotting phase space for mode $mode_num") + + # convert the state to a mode + a_modes = [] + vel_modes = [] + a_inter_modes = [] + vel_inter_modes = [] + + N = length(states[1]) / 2 - 2 + + for i in 1:length(states) + a_mode = 0 + vel_mode = 0 + a_inter_mode = 0 + vel_inter_mode = 0 + + positions = states[i][1:2:end] + velocities = states[i][2:2:end] + for j in 2:length(positions)-1 + a_mode += positions[j] * sin((mode_num * (j - 1) * π) / (N + 1)) + vel_mode += velocities[j] * sin((mode_num * (j - 1) * π) / (N + 1)) + a_inter_mode += positions[j] * sin((intersecting_mode * (j - 1) * π) / (N + 1)) + vel_inter_mode += velocities[j] * sin((intersecting_mode * (j - 1) * π) / (N + 1)) + end + + amp = sqrt(2 / (N + 1)) + a_mode *= amp + vel_mode *= amp + a_inter_mode *= amp + vel_inter_mode *= amp + + if i == 1 + push!(a_inter_modes, a_inter_mode) + push!(vel_inter_modes, vel_inter_mode) + continue + end + + if ((a_inter_modes[end] < 0 && a_inter_mode > 0) || (a_inter_modes[end] > 0 && a_inter_mode < 0)) + push!(a_modes, a_mode) + push!(vel_modes, vel_mode) + end + push!(a_inter_modes, a_inter_mode) + push!(vel_inter_modes, vel_inter_mode) + end + + + # plot the phase space + scatter(a_modes, vel_modes, label = "Beta = $beta", xlabel = "Displacement", ylabel = "Momentum", + title = "Phase Space for Mode $mode_num when a_$intersecting_mode crosses 0", + xlim = (-10, 10), + ylim = (-1, 1), + color = :red, + legend = :topright, + ) + # savefig("t/frames/phase-space-mode-$mode_num-beta-$beta.png") + println("Saved phase-space-mode-$mode_num-beta-$beta.png") +end + +function animate_phase_space_plots_over_beta(betas, t_max, desired_mode, intersecting_mode) + N = 32 # number of masses + A = 10 # amplitude + modes = 3 # number of modes to plot + final_time = t_max # seconds + dt = 0.05 # seconds + num_steps = Int(final_time / dt) + + results = [] + for b in betas + s = run_simulation(N, modes, b, A, dt, num_steps) + states = s.u + timesteps = s.t + + push!(results, states) + + println("Simualted beta: ", b, " to t=", timesteps[end]) + end + + anim = @animate for i in 1:length(betas) + plot_mode_phase_space(results[i], desired_mode, intersecting_mode, betas[i]) + end + + mp4(anim, "t/animated-phase-space-modes-over-beta.mp4", fps = 15) +end + +# animate_phase_space_plots_over_beta(collect(0:0.025:4), 25000, 1, 3) + +# analyze_energies_of_n_modes([1, 3, 5, 7, 9, 11], timesteps) + +# animate_masses_phase_space(states, collect(2:N+1), 17) + +# plot_mode_phase_space(states, 1, 3) + +function estimate_lynapov_exponent(t_max, beta, jitter = 0.001) + println("Estimating Lynapov Exponent for beta: ", beta) + + N = 32 # number of masses + A = 10 # amplitude + modes = 3 # number of modes to plot + println("t_msx: ", t_max) + final_time = t_max # seconds + dt = 0.05 # seconds + num_steps = Int(final_time / dt) + + t_span = (0.0, final_time) + params = (N, modes, beta, A, dt, num_steps) + + # run a simulation with normal parameters + is = get_initial_state(N, 1, beta, A) + # plot iniital state + plot(is, label = "Initial State", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "Initial Positions of Masses", markersize = 4) + is_0 = zeros(2 * (N + 2)) + for i in 1:length(is) + is_0[i*2-1] = is[i] + end + prob_s = ODEProblem(tendency!, is_0, t_span, params) + # solve so that the timestep is exactly set to dt + s = solve(prob_s, SSPRK33(), dt = dt) + states_control = s.u + timesteps = s.t + + + # run a simulation with a small perturbation + # dynamics!(state, prev_state, params, states) + # state above is position, need to add in momentum + is2 = get_initial_state(N, 1, beta, A) .* (1 - jitter) # perturb the initial state + # is2 = get_initial_state(N, 1, beta, A) .- jitter # perturb the initial state + plot!(is2, label = "Perturbed Initial State", marker = :cross, xlabel = "Mass Number", ylabel = "Displacement", title = "Perturbed Initial Positions of Masses", msw = 2) + savefig("t/new/inits-jitter.png") + s_0 = zeros(2 * (N + 2)) + for i in 1:length(is2) + s_0[i*2-1] = is2[i] + end + + # add random jitter to the initial state + # middle_index = Int(N / 2) * 2 + # s_0[middle_index] -= jitter # subtract a little from momenetum of middle particle + + final_time = num_steps * dt + t_span = (0.0, final_time) + prob = ODEProblem(tendency!, s_0, t_span, params) + sol = solve(prob, SSPRK33(), dt = dt) # perturbed simulation + state_perturbed = sol.u + time_perturbed = sol.t + + # println(time_perturbed .- timesteps, "\n") + + println("Finished simulations") + println("Control len: ", length(states_control), " Perturbed len: ", length(state_perturbed), " Time len: ", length(time_perturbed)) + + + # # print the difference in the two initial states + # println("Initial State Difference (pertrubed - control): ", state_perturbed[1] .- states_control[1]) + # println("Final State Difference (pertrubed - control): ", state_perturbed[end] .- states_control[end]) + + # plot the difference in the two states over time + differences = [] + for i in 1:length(state_perturbed) + # calculate mode 1 difference + a_k = 0 + amp = sqrt(2 / (N + 1)) + for j in 1:N + diff = abs(state_perturbed[i][j*2-1] - states_control[i][j*2-1]) + a_k += diff * sin((1 * (j - 1) * π) / (N + 1)) + end + a_k *= amp + push!(differences, a_k) + end + plot(time_perturbed, differences, label = "Difference in States", xlabel = "Time", ylabel = "Difference", title = "Absolute Difference in States Over Time", lw = 1.5, color = :blue) + savefig("t/frames/difference-in-states-beta-$beta.png") + # perform a linear regression on the log of the differences + # cut the data set into everythign after t=500 + # find the index where t = 500 + # index = 1 + # for i in 1:length(time_perturbed) + # if time_perturbed[i] >= 500 + # index = i + # break + # end + # end + # differences = differences[index:end] + # time_perturbed = time_perturbed[index:end] + ln_differeces = log.(differences) + (a, b) = linear_regression(time_perturbed, ln_differeces) + a = round(a, digits = 4) + b = round(b, digits = 4) + plot(time_perturbed, differences, label = "ln(difference in mode1)", xlabel = "Time", ylabel = "Difference", title = "Absolute Difference in States Over Time", msw = 0.0, yscale = :ln, color = :blue, lw = 1.5) + plot!(time_perturbed, exp.(a * time_perturbed .+ b), label = "exp($a*t + $b)", lw = 2, color = :red, linestyle = :dash) + savefig("t/frames/difference-in-states-beta-$beta-log.png") + + println("Saved difference-in-states-beta-$beta.png and difference-in-states-beta-$beta-log.png\n") + println("Lynapov Exponent: ", a) + + # check that solve doesn't explode + # animate_states(state_perturbed, time_perturbed, 0, 1) + return a +end + +# linear regression of the log of the difference +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 + +function analyze_equiparition(t_max, beta) + # run a simulation with normal parameters + N = 32 # number of masses + A = 10 # amplitude + modes = 3 # number of modes to plot + final_time = t_max # seconds + dt = 0.05 # seconds + num_steps = Int(final_time / dt) + + t_span = (0.0, final_time) + params = (N, modes, beta, A, dt, num_steps) + s = run_simulation(N, modes, beta, A, dt, num_steps) + + states = s.u + timesteps = s.t + + analyze_energies_of_n_modes([1, 3, 5, 7], states, timesteps, beta) +end + +# my_beta = 0.3 +# estimate_lynapov_exponent(10000, my_beta, 0.001) +# analyze_equiparition(10000, my_beta) + +# function plot_betas_versus_lynapov_exponent(betas, t_max, jitter = 0.001) +# lynapovs = [] +# for b in betas +# lynapov = estimate_lynapov_exponent(t_max, b, jitter) +# push!(lynapovs, lynapov) +# end + +# plot(betas, lynapovs, label = "Lynapov Exponent", xlabel = "Beta", ylabel = "Lynapov Exponent", title = "Lynapov Exponent Over Beta") +# savefig("t/lynapov-over-beta.png") +# end + +function plot_probability_distrubtion_of_mode(energy_of_mode, total_energy, timesteps, beta) + # get the probability distribution of the mode + prob = [] + for i in 1:length(timesteps) + push!(prob, energy_of_mode[i] / total_energy[1]) # energy is conserved + end + + plot(timesteps, prob, label = "Probability of Mode 1", xlabel = "Time", ylabel = "Probability", title = "Probability of Mode 1 Over Time with beta=$beta", ylim = (0, 1)) + savefig("t/probability-mode1-beta-$beta.png") +end + +function analyze_probability_distribution_of_mode(t_max, beta) + # run a simulation with normal parameters + N = 32 # number of masses + A = 10 # amplitude + modes = 3 # number of modes to plot + final_time = t_max # seconds + dt = 0.05 # seconds + num_steps = Int(final_time / dt) + + t_span = (0.0, final_time) + params = (N, modes, beta, A, dt, num_steps) + s = run_simulation(N, modes, beta, A, dt, num_steps) + + states = s.u + timesteps = s.t + + # get the energies of the mode + total_energy, kinetic_energy, potential_energy, total_by_mode, kinetic_by_mode, potential_by_mode = analyze_energies_of_n_modes([1, 3, 5, 7, 9, 11, 13], states, timesteps, beta) + + # get the probability distribution of the mode + # plot_probability_distrubtion_of_mode(total_by_mode[1], total_energy, timesteps, beta) + + # print the time when the probability gets lower than .5 + for i in 1:length(timesteps) + if (total_by_mode[1][i] / total_energy[1]) < 0.6 + println("TIME when probability of mode 1 is less than 0.6 for beta=$beta: ", timesteps[i]) + return timesteps[i] + end + end + + return -1 +end + +# estimate_lynapov_exponent(1000, 0.3, 0.001) + +function plot_pos_at_time_t(t, beta) + # run a simulation with normal parameters + N = 32 # number of masses + A = 10 # amplitude + modes = 3 # number of modes to plot + final_time = t # seconds + dt = 0.05 # seconds + num_steps = Int(final_time / dt) + + t_span = (0.0, final_time) + params = (N, modes, beta, A, dt, num_steps) + s = run_simulation(N, modes, beta, A, dt, num_steps) + + states = s.u + timesteps = s.t + positions = s.u[end][1:2:end] + inital_positions = s.u[1][1:2:end] + + plot(positions, label = "Positions at t=$t", xlabel = "Mass Number", ylabel = "Displacement", title = "Positions of Masses at t=$t, beta=$beta", lw = 2, marker = :circle) + plot!(inital_positions, label = "Initial Positions", lw = 2, marker = :circle) + savefig("t/new/positions-at-t-$t-beta-$beta.png") +end + +function get_breakdown_time(beta, t_find = 3600) + t = analyze_probability_distribution_of_mode(t_find, beta) + if t == -1 + println("Could not find time for beta: ", beta) + return -1 + end + return t +end + + +# plot_pos_at_time_t(1000, 10) +# plot_pos_at_time_t(1000, 0) + + + +# plot_betas_versus_lynapov_exponent(collect(0:1:50), 1000, 0.001) + +# analyze_probability_distribution_of_mode(100000, 2.0) + +# break_down_times = [] +# bs = collect(1:0.5:50) +# for b_s in bs +# t_find = 3600 +# t = analyze_probability_distribution_of_mode(t_find, b_s) +# if t == -1 +# println("Could not find time for beta: ", b_s) +# continue +# end +# push!(break_down_times, t) +# end +# # plot beta vs breakdown time +# plot(bs, break_down_times, label = "Breakdown Time", xlabel = "Beta", ylabel = "Time", title = "Breakdown Time Over Beta") +# savefig("t/breakdown-time-over-beta.png") + +# # plot the log of this +# one_over_breakdown_time = break_down_times .^ (-1) +# (a, b) = linear_regression(bs, one_over_breakdown_time) +# a = round(a, digits = 4) +# b = round(b, digits = 4) +# plot(bs, one_over_breakdown_time, label = "1 / (breakdown time)", xlabel = "Beta", ylabel = "1 / (breakdown time)", title = "Breakdown Time ^ -1 Over Beta", msw = 0.0, color = :blue, lw = 1.5) +# plot!(bs, a * bs .+ b, label = "1 / t = $a * beta + $b", lw = 2, color = :red, linestyle = :dash) +# savefig("t/ln-breakdown-time-over-beta.png") + +# println("Done!") +# # for a range of betas, find the breakdown time and the lynapov exponent +function tmp(betas) + break_down_times = [] + lynapovs = [] + t_find = 350000 + for b in betas + # if b < 0.5 + # t_find = 10000 + # end + t = get_breakdown_time(b, t_find) + if t == -1 + continue + end + t_find = round(Int, t) + 100 + + l = estimate_lynapov_exponent(t_find, b, 0.001) + if l == 0 + println("Lynapov for beta was zero: ", b) + continue + end + push!(break_down_times, t) + push!(lynapovs, l) + end + + # plot lynapov vs breakdown time + scatter(break_down_times, lynapovs, label = "Lynapov Exponent", xlabel = "Breakdown Time", ylabel = "Lynapov Exponent", title = "Lynapov Exponent Over Breakdown Time") + savefig("t/new/new-lynapov-over-breakdown-time.png") + + # plot this over log of lynapovs + ln_lynapovs = log.(lynapovs) + ln_break_down_times = log.(break_down_times) + (a, b) = linear_regression(ln_break_down_times, ln_lynapovs) + a = round(a, digits = 4) + b = round(b, digits = 4) + scatter(ln_break_down_times, ln_lynapovs, label = "ln(Lynapov Exponent)", xlabel = "ln(Breakdown Time)", ylabel = "ln(Lynapov Exponent)", title = "ln(Lynapov Exponent) Over ln(Breakdown Time)", msw = 0.0, color = :blue, lw = 1.5) + plot!(ln_break_down_times, a * ln_break_down_times .+ b, label = "ln(beta) = $a * ln(t) + $b", lw = 2, color = :red, linestyle = :dash) + savefig("t/new/new-ln-lynapov-over-breakdown-time.png") + + # only one log + (a, b) = linear_regression(lynapovs, ln_break_down_times) + a = round(a, digits = 4) + b = round(b, digits = 4) + scatter(lynapovs, ln_break_down_times, label = "ln(Breakdown Time)", xlabel = "Lynapov Exponent", ylabel = "ln(Breakdown Time)", title = "ln(Breakdown Time) Over Lynapov Exponent", msw = 0.0, color = :blue, lw = 1.5) + plot!(lynapovs, a * lynapovs .+ b, label = "ln(t) = $a * beta + $b", lw = 2, color = :red, linestyle = :dash) + savefig("t/new/new-new-ln-breakdown-time-over-lynapov.png") + + + + # plot the lynapoovs to the ^-1 + one_over_breakdown_time = break_down_times .^ (-1) + (a, b) = linear_regression(lynapovs, one_over_breakdown_time) + a = round(a, digits = 4) + b = round(b, digits = 4) + scatter(lynapovs, one_over_breakdown_time, label = "1 / (breakdown time)", xlabel = "Lynapov Exponent", ylabel = "1 / (breakdown time)", title = "1 / (breakdown time) Over Lynapov Exponent", msw = 0.0, color = :blue, lw = 1.5) + plot!(lynapovs, a * lynapovs .+ b, label = "1 / t = $a * beta + $b", lw = 2, color = :red, linestyle = :dash) + savefig("t/new/new-one-over-lynapov-over-breakdown-time.png") + + # plot the lynapov over betas + scatter(betas, lynapovs, label = "Lynapov Exponent", xlabel = "Beta", ylabel = "Lynapov Exponent", title = "Lynapov Exponent Over Beta") + savefig("t/new/new-lynapov-over-beta.png") +end + +tmp(collect(0.3:0.025:10)) + +# estimate_lynapov_exponent(500000, 0.3, 0.001) diff --git a/t/plz.mp4 b/t/plz.mp4 deleted file mode 100644 index a7dbdf7..0000000 Binary files a/t/plz.mp4 and /dev/null differ diff --git a/t/probability-mode1-beta-0.0.png b/t/probability-mode1-beta-0.0.png new file mode 100644 index 0000000..89030b4 Binary files /dev/null and b/t/probability-mode1-beta-0.0.png differ diff --git a/t/probability-mode1-beta-0.3-long.png b/t/probability-mode1-beta-0.3-long.png new file mode 100644 index 0000000..1c750fc Binary files /dev/null and b/t/probability-mode1-beta-0.3-long.png differ diff --git a/t/probability-mode1-beta-0.3.png b/t/probability-mode1-beta-0.3.png new file mode 100644 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- # 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/semi-oldish/animated-fpu-chaotic.mp4 b/t/semi-oldish/animated-fpu-chaotic.mp4 new file mode 100644 index 0000000..e8c41b1 Binary files /dev/null and b/t/semi-oldish/animated-fpu-chaotic.mp4 differ diff --git a/t/semi-oldish/animated-fpu.mp4 b/t/semi-oldish/animated-fpu.mp4 new file mode 100644 index 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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 deleted file mode 100644 index 5eeba5b..0000000 Binary files a/t/velocity-over-time.png and /dev/null differ -- cgit v1.2.3-70-g09d2