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# FOR PROBLEM 8.11
# author: sotech117
using Statistics
using Plots
function wrap_index(i::Int, l::Int)::Int
wrap = (i - 1) % l + 1
return (wrap <= 0) ? l + wrap : wrap
end
mutable struct Ising2D
l::Int
n::Int
temperature::Float64
w::Vector{Float64} # Boltzmann weights
state::Matrix
energy::Int
magnetization::Int
mc_steps::Int
accepted_moves::Int
energy_array::Vector{Int}
magnetization_array::Vector{Int}
end
Ising2D(l::Int, temperature::Float64) = begin
n = l^2
w = zeros(9)
w[9] = exp(-8.0 / temperature)
w[5] = exp(-4.0 / temperature)
state = ones(Int, l, l) # initially all spins up
energy = -2 * n
magnetization = n
return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0,
Int[], Int[])
end
function reset!(ising::Ising2D)
ising.mc_steps = 0
ising.accepted_moves = 0
ising.energy_array = Int[]
ising.magnetization_array = Int[]
end
function mc_step!(ising::Ising2D)
l::Int = ising.l
n::Int = ising.n
w = ising.w
state = ising.state
accepted_moves = ising.accepted_moves
energy = ising.energy
magnetization = ising.magnetization
random_positions = l * rand(2 * n)
random_array = rand(n)
for k in 1:n
i = trunc(Int, random_positions[2 * k - 1]) + 1
j = trunc(Int, random_positions[2 * k]) + 1
de = 2 * state[i, j] * (state[i % l + 1, j] +
state[wrap_index(i - 1, l), j] + state[i, j % l + 1] +
state[i, wrap_index(j - 1, l)])
if de <= 0 || w[de + 1] > random_array[k]
accepted_moves += 1
new_spin = - state[i, j] # flip spin
state[i, j] = new_spin
energy += de
magnetization += 2 * new_spin
end
end
ising.state = state
ising.accepted_moves = accepted_moves
ising.energy = energy
ising.magnetization = magnetization
append!(ising.energy_array, ising.energy)
append!(ising.magnetization_array, ising.magnetization)
ising.mc_steps = ising.mc_steps + 1
end
function steps!(ising::Ising2D, num::Int=100)
for i in 1:num
mc_step!(ising)
end
end
function mean_energy(ising::Ising2D)
return mean(ising.energy_array) / ising.n
end
function specific_heat(ising::Ising2D)
return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n
end
function mean_magnetization(ising::Ising2D)
return mean(ising.magnetization_array) / ising.n
end
function susceptibility(ising::Ising2D)
return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n)
end
function observables(ising::Ising2D)
printstyled("Temperature\t\t", bold=true)
print(ising.temperature); print("\n")
printstyled("Mean Energy\t\t", bold=true)
print(mean_energy(ising)); print("\n")
printstyled("Mean Magnetiz.\t\t", bold=true)
print(mean_magnetization(ising)); print("\n")
printstyled("Specific Heat\t\t", bold=true)
print(specific_heat(ising)); print("\n")
printstyled("Susceptibility\t\t", bold=true)
print(susceptibility(ising)); print("\n")
printstyled("MC Steps\t\t", bold=true)
print(ising.mc_steps); print("\n")
printstyled("Accepted Moves\t\t", bold=true)
print(ising.accepted_moves); print("\n")
end
function plot_ising(state::Matrix{Int})
pos = Tuple.(findall(>(0), state))
neg = Tuple.(findall(<(0), state))
scatter(pos, markersize=5)
scatter!(neg, markersize=5)
end
function get_magnetization(T, n=1000)
m = Ising2D(64, T)
steps!(m, n)
println("done with T = $T")
return mean_magnetization(m)
end
Ts = 0:.1:5
ms = [abs(get_magnetization(T)) for T in Ts]
println("done with calculating magnetizations")
function linear_regression(x, y)
n = length(x)
x̄ = sum(x) / n
ȳ = sum(y) / n
a = sum((x .- x̄) .* (y .- ȳ)) / sum((x .- x̄).^2)
b = ȳ - a * x̄
return (a, b)
end
# plot M^(8) over T
betas = .001:.001:1
residuals = []
for i in 1:length(betas)
b = betas[i]
m = ms .^ (1 / b)
# filter out the zero values
s = scatter(p,
Ts, m, xlabel="T (units of J / k_b)", ylabel="Magnetization", label="$b-beta", title="Magnetization vs Temp (Ising Monte Carlo)",
msw=0, ms=1.5, mc=:red, lc=:red, lw=1.5, legend=:bottomleft
)
# do a linear regression
a, b = linear_regression(Ts, m)
# plot a linear regression line
plot!(s, Ts, a*Ts .+ b, label="Linear Regression", lw=1.2, color=:red, linestyle=:dash)
# calculate the residuals
push!(residuals, sum((m .- (a*Ts .+ b)).^2))
savefig(s, "hw6/b/8-2-$i.png")
end
# plot the residuals over beta
plot(betas, residuals, xlabel="beta", ylabel="Squared Distance", label="Residuals", title="Error from Linear Regression of M^(1/Beta)", lw=1.2, lc=:red, legend=:topright)
# find the min on the first half of the residuals
min_residuali = argmin(residuals[1:div(length(residuals), 2)])
min_residual = betas[min_residuali]
println("Minimum Residual: ", min_residual)
vline!([min_residual], label="Minimum Point @ Beta = $min_residual", lc=:orange, lw=1.5, ls=:dash)
# plot the analityical beta of .125
vline!([.125], label="Analytical Minimum Beta = .125", lc=:green, lw=1.5, ls=:dot)
savefig("hw6/8-2-residuals-100.png")
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