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authorsotech117 <michael_foiani@brown.edu>2024-05-02 19:29:49 -0400
committersotech117 <michael_foiani@brown.edu>2024-05-02 19:29:49 -0400
commitc048f9f2219897ff3cc20a50d459911db3496a0a (patch)
tree59dd4883ddc1dcb75c67703ba9af93dfe2ac7b8a
parente650ed1e1e908e51c78c1b047bec0da7c4fea366 (diff)
major update, with final project rough draft code
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-rw-r--r--t/procs/energies-modes-sum-beta-9.1.pngbin0 -> 93221 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.125.pngbin0 -> 93558 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.15.pngbin0 -> 93637 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.175.pngbin0 -> 93789 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.2.pngbin0 -> 96247 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.225.pngbin0 -> 96585 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.25.pngbin0 -> 96754 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.275.pngbin0 -> 96944 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.3.pngbin0 -> 97144 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.325.pngbin0 -> 97002 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.35.pngbin0 -> 96792 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.375.pngbin0 -> 96735 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.4.pngbin0 -> 96741 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.425.pngbin0 -> 97132 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.45.pngbin0 -> 96998 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.475.pngbin0 -> 97346 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.5.pngbin0 -> 97601 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.525.pngbin0 -> 97469 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.55.pngbin0 -> 97413 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.575.pngbin0 -> 96993 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.6.pngbin0 -> 94487 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.625.pngbin0 -> 94800 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.65.pngbin0 -> 94776 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.675.pngbin0 -> 94444 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.7.pngbin0 -> 94546 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.725.pngbin0 -> 95179 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.75.pngbin0 -> 94833 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.775.pngbin0 -> 95005 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.8.pngbin0 -> 94784 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.825.pngbin0 -> 94444 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.85.pngbin0 -> 94275 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.875.pngbin0 -> 94566 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.9.pngbin0 -> 94228 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.925.pngbin0 -> 91697 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.95.pngbin0 -> 91772 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.975.pngbin0 -> 91931 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-9.pngbin0 -> 96689 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-90.pngbin0 -> 84119 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-91.pngbin0 -> 76377 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-92.pngbin0 -> 71698 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-93.pngbin0 -> 74522 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-94.pngbin0 -> 70966 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-95.pngbin0 -> 72925 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-96.pngbin0 -> 70917 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-97.pngbin0 -> 78240 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-98.pngbin0 -> 73695 bytes
-rw-r--r--t/procs/energies-modes-sum-beta-99.pngbin0 -> 71749 bytes
-rw-r--r--t/semi-oldish/animated-fpu-chaotic.mp4bin0 -> 4409143 bytes
-rw-r--r--t/semi-oldish/animated-fpu.mp4bin0 -> 6925 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-masses-beta-0.3.mp4bin0 -> 40346 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-masses-beta-1.3.mp4bin0 -> 187631 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-masses-beta-1.4.mp4bin0 -> 221205 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-masses-beta-1.5.mp4bin0 -> 260381 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-masses-beta-2.3.mp4bin0 -> 257854 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-modes-over-beta-dec.mp4bin0 -> 1211562 bytes
-rw-r--r--t/semi-oldish/animated-phase-space-modes-over-beta.mp4bin0 -> 2867084 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-0.0-log.pngbin0 -> 46452 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-0.0.pngbin0 -> 39931 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-0.3-log.pngbin0 -> 40953 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-0.3.pngbin0 -> 27323 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-1.0-log.pngbin0 -> 43338 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-1.0.pngbin0 -> 28918 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-3.0-log.pngbin0 -> 41479 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-3.0.pngbin0 -> 24314 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-5.0-log.pngbin0 -> 40949 bytes
-rw-r--r--t/semi-oldish/difference-in-states-beta-5.0.pngbin0 -> 24153 bytes
-rw-r--r--t/semi-oldish/energies-beta15-9modes.pngbin0 -> 60210 bytes
-rw-r--r--t/semi-oldish/energies-beta15.pngbin0 -> 40026 bytes
-rw-r--r--t/semi-oldish/energies-mode1-beta15.pngbin0 -> 102029 bytes
-rw-r--r--t/semi-oldish/energies-modes-1-3-5-7-9-11-13-beta15.pngbin0 -> 49513 bytes
-rw-r--r--t/semi-oldish/energies-modes-separate-1.3-smalltimescale.pngbin0 -> 73708 bytes
-rw-r--r--t/semi-oldish/energies-modes-separate-1.3beta.pngbin0 -> 104931 bytes
-rw-r--r--t/semi-oldish/energies-modes-separate-beta-0.0.pngbin0 -> 20439 bytes
-rw-r--r--t/semi-oldish/energies-modes-separate-beta-0.3.pngbin0 -> 35990 bytes
-rw-r--r--t/semi-oldish/energies-modes-separate-beta-2.3.pngbin0 -> 53327 bytes
-rw-r--r--t/semi-oldish/energies-modes-sum-1.3beta.pngbin0 -> 53752 bytes
-rw-r--r--t/semi-oldish/energies-modes-sum-beta-0.0.pngbin0 -> 100744 bytes
-rw-r--r--t/semi-oldish/energies-modes-sum-beta-0.3.pngbin0 -> 100149 bytes
-rw-r--r--t/semi-oldish/energies-modes-sum-beta-2.3.pngbin0 -> 120117 bytes
-rw-r--r--t/semi-oldish/energies-modes-sum-beta1.3-smallt.pngbin0 -> 90688 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-0.3.pngbin0 -> 34919 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-1.0.pngbin0 -> 38946 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-1.3.pngbin0 -> 38148 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-1.4.pngbin0 -> 35136 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-1.5.pngbin0 -> 38998 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-2.3.pngbin0 -> 39145 bytes
-rw-r--r--t/semi-oldish/initial-final-beta-3.0.pngbin0 -> 39470 bytes
-rw-r--r--t/semi-oldish/initial-final-beta15.pngbin0 -> 34894 bytes
-rw-r--r--t/semi-oldish/modes-beta15.pngbin0 -> 48822 bytes
-rw-r--r--t/semi-oldish/phase-space-mass10-beta15.pngbin0 -> 73743 bytes
-rw-r--r--t/semi-oldish/phase-space-mass11-beta15.pngbin0 -> 68234 bytes
-rw-r--r--t/semi-oldish/phase-space-mass12-beta15.pngbin0 -> 60372 bytes
-rw-r--r--t/semi-oldish/phase-space-mass17-beta15-smalltimespetm13beta.pngbin0 -> 186312 bytes
-rw-r--r--t/semi-oldish/phase-space-mass17-beta15.pngbin0 -> 49482 bytes
-rw-r--r--t/semi-oldish/phase-space-mass2-beta15.pngbin0 -> 133344 bytes
-rw-r--r--t/semi-oldish/phase-space-mass3-beta15.pngbin0 -> 123030 bytes
-rw-r--r--t/semi-oldish/phase-space-mass4-beta15.pngbin0 -> 78903 bytes
-rw-r--r--t/semi-oldish/phase-space-mass5-beta15.pngbin0 -> 120409 bytes
-rw-r--r--t/semi-oldish/phase-space-mass6-beta15.pngbin0 -> 115723 bytes
-rw-r--r--t/semi-oldish/phase-space-mass7-beta15.pngbin0 -> 102313 bytes
-rw-r--r--t/semi-oldish/phase-space-mass8-beta15.pngbin0 -> 89319 bytes
-rw-r--r--t/semi-oldish/phase-space-mass9-beta15.pngbin0 -> 82229 bytes
-rw-r--r--t/semi-oldish/phase-space-mode-3-beta-2.3.pngbin0 -> 56426 bytes
-rw-r--r--t/semi-oldish/plz.gifbin0 -> 11532644 bytes
-rw-r--r--t/velocity-over-time.pngbin49230 -> 0 bytes
5108 files changed, 1197 insertions, 323 deletions
diff --git a/.DS_Store b/.DS_Store
index 2b90e36..cd84abd 100644
--- a/.DS_Store
+++ b/.DS_Store
Binary files 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
--- /dev/null
+++ b/hw7/magnetization_vs_field.png
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diff --git a/t/.DS_Store b/t/.DS_Store
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index 0000000..446c2ed
--- /dev/null
+++ b/t/.DS_Store
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diff --git a/t/animate-positions.mp4 b/t/animate-positions.mp4
deleted file mode 100644
index 3633358..0000000
--- a/t/animate-positions.mp4
+++ /dev/null
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diff --git a/t/animated-fpu.mp4 b/t/animated-fpu.mp4
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index 0000000..87c2082
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diff --git a/t/breakdown-time-over-beta-od.png b/t/breakdown-time-over-beta-od.png
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diff --git a/t/breakdown-time-over-beta-short.png b/t/breakdown-time-over-beta-short.png
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diff --git a/t/breakdown-time-over-beta.png b/t/breakdown-time-over-beta.png
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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)
-plot_starting_and_final_positions(sol.u, sol.t)
-animate_positions(sol.u, sol.t, 0, 1, my_vel)
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@@ -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
--- /dev/null
+++ b/t/old/initial-final-positions.png
Binary files differ
diff --git a/t/old/initial-final-velocities.png b/t/old/initial-final-velocities.png
new file mode 100644
index 0000000..b7cc859
--- /dev/null
+++ b/t/old/initial-final-velocities.png
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diff --git a/t/old/modes-beta15.png b/t/old/modes-beta15.png
new file mode 100644
index 0000000..dbb632d
--- /dev/null
+++ b/t/old/modes-beta15.png
Binary files differ
diff --git a/t/old/phase-space-.5vel.png b/t/old/phase-space-.5vel.png
new file mode 100644
index 0000000..100cd2e
--- /dev/null
+++ b/t/old/phase-space-.5vel.png
Binary files differ
diff --git a/t/old/phase-space-chos.png b/t/old/phase-space-chos.png
new file mode 100644
index 0000000..cbbbf09
--- /dev/null
+++ b/t/old/phase-space-chos.png
Binary files differ
diff --git a/t/old/phase-space-stable.png b/t/old/phase-space-stable.png
new file mode 100644
index 0000000..2ffa7c6
--- /dev/null
+++ b/t/old/phase-space-stable.png
Binary files differ
diff --git a/t/old/phase-space.01strings.png b/t/old/phase-space.01strings.png
new file mode 100644
index 0000000..5e78901
--- /dev/null
+++ b/t/old/phase-space.01strings.png
Binary files differ
diff --git a/t/old/phase-space.png b/t/old/phase-space.png
new file mode 100644
index 0000000..48e5368
--- /dev/null
+++ b/t/old/phase-space.png
Binary files differ
diff --git a/t/plot_data.png b/t/old/plot_data.png
index d052983..d052983 100644
--- a/t/plot_data.png
+++ b/t/old/plot_data.png
Binary files differ
diff --git a/t/old/plz.gif b/t/old/plz.gif
new file mode 100644
index 0000000..febf678
--- /dev/null
+++ b/t/old/plz.gif
Binary files differ
diff --git a/t/old/plz.mp4 b/t/old/plz.mp4
new file mode 100644
index 0000000..6a0d7b4
--- /dev/null
+++ b/t/old/plz.mp4
Binary files differ
diff --git a/t/old/pos-over-time.png b/t/old/pos-over-time.png
new file mode 100644
index 0000000..e3a732d
--- /dev/null
+++ b/t/old/pos-over-time.png
Binary files differ
diff --git a/t/r.jl b/t/old/r.jl
index 516ab09..516ab09 100644
--- a/t/r.jl
+++ b/t/old/r.jl
diff --git a/t/old/semi-old b/t/old/semi-old
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/t/old/semi-old
diff --git a/t/t.jl b/t/old/t.jl
index 6a06777..6a06777 100644
--- a/t/t.jl
+++ b/t/old/t.jl
diff --git a/t/old/velocity-over-time.png b/t/old/velocity-over-time.png
new file mode 100644
index 0000000..3cab01f
--- /dev/null
+++ b/t/old/velocity-over-time.png
Binary files 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)
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