1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
|
#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia
using Statistics
using Plots
function wrap_index(i::Int, l::Int)::Int
wrap = (i - 1) % l + 1
return (wrap <= 0) ? l + wrap : wrap
end
mutable struct Ising2D
l::Int
n::Int
temperature::Float64
w::Vector{Float64} # Boltzmann weights
state::Matrix
energy::Float64
magnetization::Int
mc_steps::Int
accepted_moves::Int
energy_array::Vector{Float64}
magnetization_array::Vector{Int}
H::Float64
end
Ising2D(l::Int, temperature::Float64, H=1.0) = begin
n = l^2
w = zeros(9)
w[9] = exp(-8.0 / temperature)
w[5] = exp(-4.0 / temperature)
state = ones(Int, l, l) # initially all spins up
energy = Float64(-2 * n + 2 * H * n)
magnetization = n
return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0,
Int[], Int[], H)
end
function reset!(ising::Ising2D)
ising.mc_steps = 0
ising.accepted_moves = 0
ising.energy_array = Int[]
ising.magnetization_array = Int[]
end
function mc_step!(ising::Ising2D)
l::Int = ising.l
n::Int = ising.n
w = ising.w
state = ising.state
accepted_moves = ising.accepted_moves
energy = ising.energy
magnetization = ising.magnetization
random_positions = l * rand(2 * n)
random_array = rand(n)
for k in 1:n
i = trunc(Int, random_positions[2 * k - 1]) + 1
j = trunc(Int, random_positions[2 * k]) + 1
changed_spins = state[i, j] * (state[i % l + 1, j] +
state[wrap_index(i - 1, l), j] + state[i, j % l + 1] +
state[i, wrap_index(j - 1, l)])
de = 2 * changed_spins + 2 * ising.H * state[i, j]
if de <= 0 || rand() < exp(-de / ising.temperature)
accepted_moves += 1
new_spin = - state[i, j] # flip spin
state[i, j] = new_spin
# add the effects of the new spin
energy += de
magnetization += 2 * new_spin
end
end
ising.state = state
ising.accepted_moves = accepted_moves
ising.energy = energy
ising.magnetization = magnetization
append!(ising.energy_array, ising.energy)
append!(ising.magnetization_array, ising.magnetization)
ising.mc_steps = ising.mc_steps + 1
end
function steps!(ising::Ising2D, num::Int=100)
for i in 1:num
mc_step!(ising)
end
end
function mean_energy(ising::Ising2D)
return mean(ising.energy_array) / ising.n
end
function specific_heat(ising::Ising2D)
return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n
end
function mean_magnetization(ising::Ising2D)
return mean(ising.magnetization_array) / ising.n
end
function susceptibility(ising::Ising2D)
return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n)
end
function observables(ising::Ising2D)
printstyled("Temperature\t\t", bold=true)
print(ising.temperature); print("\n")
printstyled("Mean Energy\t\t", bold=true)
print(mean_energy(ising)); print("\n")
printstyled("Mean Magnetiz.\t\t", bold=true)
print(mean_magnetization(ising)); print("\n")
printstyled("Specific Heat\t\t", bold=true)
print(specific_heat(ising)); print("\n")
printstyled("Susceptibility\t\t", bold=true)
print(susceptibility(ising)); print("\n")
printstyled("MC Steps\t\t", bold=true)
print(ising.mc_steps); print("\n")
printstyled("Accepted Moves\t\t", bold=true)
print(ising.accepted_moves); print("\n")
end
function plot_ising(state::Matrix{Int})
pos = Tuple.(findall(>(0), state))
neg = Tuple.(findall(<(0), state))
scatter(pos, markersize=5)
scatter!(neg, markersize=5)
end
function find_m(H::Float64, l::Int, num::Int, T::Float64)
m = Ising2D(l, T, H)
steps!(m, num)
print("T = $T\n")
print("H = $H\n")
print("Mean Energy: $(mean_energy(m))\n")
print("Mean Magnetization: $(mean_magnetization(m))\n\n")
return mean_magnetization(m)
end
function map_h_to_m(H_range::Vector{Float64}, l::Int, num::Int, T::Float64)
m = []
for H in H_range
push!(m, find_m(H, l, num, T))
end
return m
end
function do_linear_regression(x::Vector{Float64}, y::Vector{Float64})
n = length(x)
x̄ = mean(x)
ȳ = mean(y)
Σxy = sum((x .- x̄) .* (y .- ȳ))
Σx² = sum((x .- x̄) .^ 2)
b = Σxy / Σx²
a = ȳ - b * x̄
return a, b
end
function plot_log_of_m_and_h(H_range::Vector{Float64}, l::Int, num::Int, T=2.27)
m = map_h_to_m(H_range, l, num, T)
p = scatter(H_range, m, label="M vs H", xlabel="H", ylabel="M", title="Magnetization (M) vs Field (B) for Ising Model at T_c", scale=:ln)
# get the linear regression of the log
log_h = log.(H_range)
log_m = log.(m)
a, b = do_linear_regression(log_h, log_m)
println("a: $a, b: $b")
# plot the linear regression
plot!(p, H_range, exp.(a) .* H_range .^ b, label="linear regression = $(round(a, digits=3)) + $(round(b, digits=3))x", line=:dash, color=:red)
return p
end
# textbook rec
h_range = .02:.02:.2
h_range = collect(h_range)
T = 2.27 # T_c for this system
side = 64
steps = 5000
plot_log_of_m_and_h(h_range, side, steps)
|
