testing

1999 files changed, 408 insertions, 0 deletions

diff --git a/hw6/.DS_Store b/hw6/.DS_Store
new file mode 100644
index 0000000..cda0c2f
--- /dev/null
+++ b/hw6/.DS_Store
diff --git a/hw6/8-1.jl b/hw6/8-1.jl
new file mode 100644
index 0000000..dc79580
--- /dev/null
+++ b/hw6/8-1.jl
@@ -0,0 +1,69 @@
+# FOR PROBLEM 8.1
+# author: sotech117
+
+# parameters
+z = 4
+
+function equation(s, T)
+    return s - tanh(z * s / T)
+end
+function derivative(s, T)
+    return 1 - z / T * sech(z * s / T)^2
+end
+
+
+# apply newton solver to find the solution
+function newton_solver(f, df, x0, tol, max_iter, T = 1)
+    x = x0
+    for i in 1:max_iter
+        if abs(df(x, T)) < 1e-6
+            println("derivative is too small")
+            return x
+        end
+        x = x - f(x, T) / df(x, T)
+        if abs(f(x, T)) < tol
+            return x
+        end
+    end
+    return x
+end
+
+# initial guess
+s0 = .5
+
+# solve the equation
+t = newton_solver(equation, derivative, s0, 1e-6, 100, 1)
+println("s(T) = ", t)
+
+# go over Ts from 0 - 8 and plot the solutions
+using Plots
+Ts = .001:0.001:8
+ss = [newton_solver(equation, derivative, s0, 1e-6, 100, T) for T in Ts]
+scatter(
+    Ts, ss, xlabel="T (units of J / k_b)", ylabel="Magnetization", label="Numerical Solution", title="Magnetization vs Temp (Mean Field Theory)",
+    msw=0, ms=1.5, mc=:blue, lc=:red, lw=1.5
+)
+
+# plot the analytical solution
+function analytical_solution(T)
+    n1 = (3 / T) * (T / z) ^ 3
+    n2 = z - T
+    if n1 < 0 || n2 < 0
+        return 0
+    end
+    return sqrt(n1) * sqrt(n2)
+end
+plot!(
+    Ts, [analytical_solution(T) for T in Ts], label="Analytical Solution",
+    lc=:green, lw=2
+)
+
+# make a plot of the difference between the analytical and numerical solutions
+diffs = [abs(analytical_solution(T) - newton_solver(equation, derivative, s0, 1e-6, 100, T)) for T in Ts]
+plot!(Ts, diffs, label="Difference in Solutions", lc=:red, lw=1.5, ls=:dot)
+
+# plot a vertical line where T = 3.75
+vline!([3.9], label="T = 3.9", lc=:orange, lw=1.5, ls=:dash)
+
+# save the plot
+savefig("8-1.png")
\ No newline at end of file
diff --git a/hw6/8-1.png b/hw6/8-1.png
new file mode 100644
index 0000000..c9d25eb
--- /dev/null
+++ b/hw6/8-1.png
diff --git a/hw6/8-11.jl b/hw6/8-11.jl
new file mode 100644
index 0000000..e8a0b2e
--- /dev/null
+++ b/hw6/8-11.jl
@@ -0,0 +1,198 @@
+# FOR PROBLEM 8.11
+# author: sotech117
+
+using Statistics 
+using Plots
+
+function wrap_index(i::Int, l::Int)::Int
+	wrap = (i - 1) % l + 1
+	return (wrap <= 0) ? l + wrap : wrap
+end
+
+mutable struct Ising2D
+	l::Int 
+	n::Int 
+	temperature::Float64
+	w::Vector{Float64} # Boltzmann weights
+	state::Matrix 
+	energy::Int
+	magnetization::Int
+	mc_steps::Int 
+	accepted_moves::Int 
+	energy_array::Vector{Int}
+	magnetization_array::Vector{Int}
+end 
+
+Ising2D(l::Int, temperature::Float64) = begin	
+	n = l^2 
+	w = zeros(9)
+	w[9] = exp(-8.0 / temperature)
+	w[5] = exp(-4.0 / temperature)
+	state = ones(Int, l, l) # initially all spins up
+	energy = -2 * n
+	magnetization = n
+	return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, 
+				   Int[], Int[])
+end
+
+function reset!(ising::Ising2D)
+	ising.mc_steps = 0 
+	ising.accepted_moves = 0 
+	ising.energy_array = Int[] 
+	ising.magnetization_array = Int[] 
+end
+
+function mc_step!(ising::Ising2D) 
+	l::Int = ising.l 
+	n::Int = ising.n 
+	w = ising.w 
+
+	state = ising.state 
+	accepted_moves = ising.accepted_moves
+	energy = ising.energy 
+	magnetization = ising.magnetization
+
+	random_positions = l * rand(2 * n)
+	random_array = rand(n) 
+
+	for k in 1:n
+		i = trunc(Int, random_positions[2 * k - 1]) + 1 
+		j = trunc(Int, random_positions[2 * k]) + 1 
+
+		de = 2 * state[i, j] * (state[i % l + 1, j] + 
+				state[wrap_index(i - 1, l), j] + state[i, j % l + 1] + 
+				state[i, wrap_index(j - 1, l)])
+
+		if de <= 0 || w[de + 1] > random_array[k]
+			accepted_moves += 1 
+			new_spin = - state[i, j] # flip spin
+			state[i, j] = new_spin 
+			energy += de 
+			magnetization += 2 * new_spin
+		end
+
+	end
+
+	ising.state = state 
+	ising.accepted_moves = accepted_moves
+	ising.energy = energy 
+	ising.magnetization = magnetization 
+
+	append!(ising.energy_array, ising.energy) 
+	append!(ising.magnetization_array, ising.magnetization)
+	ising.mc_steps = ising.mc_steps + 1
+end
+
+function steps!(ising::Ising2D, num::Int=100)
+	for i in 1:num
+		mc_step!(ising)
+	end 
+end
+
+function mean_energy(ising::Ising2D)
+	return mean(ising.energy_array) / ising.n
+end 
+
+function specific_heat(ising::Ising2D)
+	return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n
+end
+
+function mean_magnetization(ising::Ising2D)
+	return mean(ising.magnetization_array) / ising.n
+end
+
+function susceptibility(ising::Ising2D)
+	return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n)	
+end
+
+function observables(ising::Ising2D)
+	printstyled("Temperature\t\t", bold=true)
+	print(ising.temperature); print("\n")
+
+	printstyled("Mean Energy\t\t", bold=true)
+	print(mean_energy(ising)); print("\n")
+
+	printstyled("Mean Magnetiz.\t\t", bold=true)
+	print(mean_magnetization(ising)); print("\n")
+
+	printstyled("Specific Heat\t\t", bold=true)
+	print(specific_heat(ising)); print("\n")
+
+	printstyled("Susceptibility\t\t", bold=true)
+	print(susceptibility(ising)); print("\n")
+
+	printstyled("MC Steps\t\t", bold=true)
+	print(ising.mc_steps); print("\n")
+	printstyled("Accepted Moves\t\t", bold=true)
+	print(ising.accepted_moves); print("\n")
+end
+
+
+function plot_ising(state::Matrix{Int})
+	pos = Tuple.(findall(>(0), state))
+	neg = Tuple.(findall(<(0), state))
+	scatter(pos, markersize=5)
+	scatter!(neg, markersize=5)
+end
+
+
+function get_magnetization(T, n=1000)
+    m = Ising2D(64, T)
+    steps!(m, n)
+
+    println("done with T = $T")
+    return mean_magnetization(m)
+end
+
+Ts = 0:.1:5
+ms = [abs(get_magnetization(T)) for T in Ts]
+
+println("done with calculating magnetizations")
+
+function linear_regression(x, y)
+    n = length(x)
+    x̄ = sum(x) / n
+    ȳ = sum(y) / n
+    a = sum((x .- x̄) .* (y .- ȳ)) / sum((x .- x̄).^2)
+    b = ȳ - a * x̄
+    return (a, b)
+end
+
+# plot M^(8) over T
+betas = .001:.001:1
+residuals = []
+for i in 1:length(betas)
+    b = betas[i]
+    m = ms .^ (1 / b)
+    # filter out the zero values
+
+
+    s = scatter(p,
+        Ts, m, xlabel="T (units of J / k_b)", ylabel="Magnetization", label="$b-beta", title="Magnetization vs Temp (Ising Monte Carlo)",
+        msw=0, ms=1.5, mc=:red, lc=:red, lw=1.5, legend=:bottomleft
+    )
+
+
+    # do a linear regression
+    a, b = linear_regression(Ts, m)
+
+    # plot a linear regression line
+    plot!(s, Ts, a*Ts .+ b, label="Linear Regression", lw=1.2, color=:red, linestyle=:dash)
+
+    # calculate the residuals
+    push!(residuals, sum((m .- (a*Ts .+ b)).^2))
+
+
+    savefig(s, "hw6/b/8-2-$i.png")
+end
+
+# plot the residuals over beta
+plot(betas, residuals, xlabel="beta", ylabel="Squared Distance", label="Residuals", title="Error from Linear Regression of M^(1/Beta)", lw=1.2, lc=:red, legend=:topright)
+# find the min on the first half of the residuals
+min_residuali = argmin(residuals[1:div(length(residuals), 2)])
+min_residual = betas[min_residuali]
+println("Minimum Residual: ", min_residual)
+vline!([min_residual], label="Minimum Point @ Beta = $min_residual", lc=:orange, lw=1.5, ls=:dash)
+# plot the analityical beta of .125
+vline!([.125], label="Analytical Minimum Beta = .125", lc=:green, lw=1.5, ls=:dot)
+savefig("hw6/8-2-residuals-100.png")
\ No newline at end of file
diff --git a/hw6/8-2-residuals-100.png b/hw6/8-2-residuals-100.png
new file mode 100644
index 0000000..2731d33
--- /dev/null
+++ b/hw6/8-2-residuals-100.png
diff --git a/hw6/8-2-residuals-2.png b/hw6/8-2-residuals-2.png
new file mode 100644
index 0000000..20b366a
--- /dev/null
+++ b/hw6/8-2-residuals-2.png
diff --git a/hw6/8-2-residuals.png b/hw6/8-2-residuals.png
new file mode 100644
index 0000000..dba70e5
--- /dev/null
+++ b/hw6/8-2-residuals.png
diff --git a/hw6/IsingModel.jl b/hw6/IsingModel.jl
new file mode 100644
index 0000000..d1e04de
--- /dev/null
+++ b/hw6/IsingModel.jl
@@ -0,0 +1,141 @@
+#!/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia
+
+using Statistics 
+using Plots
+
+function wrap_index(i::Int, l::Int)::Int
+	wrap = (i - 1) % l + 1
+	return (wrap <= 0) ? l + wrap : wrap
+end
+
+mutable struct Ising2D
+	l::Int 
+	n::Int 
+	temperature::Float64
+	w::Vector{Float64} # Boltzmann weights
+	state::Matrix 
+	energy::Int
+	magnetization::Int
+	mc_steps::Int 
+	accepted_moves::Int 
+	energy_array::Vector{Int}
+	magnetization_array::Vector{Int}
+end 
+
+Ising2D(l::Int, temperature::Float64) = begin	
+	n = l^2 
+	w = zeros(9)
+	w[9] = exp(-8.0 / temperature)
+	w[5] = exp(-4.0 / temperature)
+	state = ones(Int, l, l) # initially all spins up
+	energy = -2 * n
+	magnetization = n
+	return Ising2D(l, n, temperature, w, state, energy, magnetization, 0, 0, 
+				   Int[], Int[])
+end
+
+function reset!(ising::Ising2D)
+	ising.mc_steps = 0 
+	ising.accepted_moves = 0 
+	ising.energy_array = Int[] 
+	ising.magnetization_array = Int[] 
+end
+
+function mc_step!(ising::Ising2D) 
+	l::Int = ising.l 
+	n::Int = ising.n 
+	w = ising.w 
+
+	state = ising.state 
+	accepted_moves = ising.accepted_moves
+	energy = ising.energy 
+	magnetization = ising.magnetization
+
+	random_positions = l * rand(2 * n)
+	random_array = rand(n) 
+
+	for k in 1:n
+		i = trunc(Int, random_positions[2 * k - 1]) + 1 
+		j = trunc(Int, random_positions[2 * k]) + 1 
+
+		de = 2 * state[i, j] * (state[i % l + 1, j] + 
+				state[wrap_index(i - 1, l), j] + state[i, j % l + 1] + 
+				state[i, wrap_index(j - 1, l)])
+
+		if de <= 0 || w[de + 1] > random_array[k]
+			accepted_moves += 1 
+			new_spin = - state[i, j] # flip spin
+			state[i, j] = new_spin 
+			energy += de 
+			magnetization += 2 * new_spin
+		end
+
+	end
+
+	ising.state = state 
+	ising.accepted_moves = accepted_moves
+	ising.energy = energy 
+	ising.magnetization = magnetization 
+
+	append!(ising.energy_array, ising.energy) 
+	append!(ising.magnetization_array, ising.magnetization)
+	ising.mc_steps = ising.mc_steps + 1
+end
+
+function steps!(ising::Ising2D, num::Int=100)
+	for i in 1:num
+		mc_step!(ising)
+	end 
+end
+
+function mean_energy(ising::Ising2D)
+	return mean(ising.energy_array) / ising.n
+end 
+
+function specific_heat(ising::Ising2D)
+	return (std(ising.energy_array) / ising.temperature) ^ 2 / ising.n
+end
+
+function mean_magnetization(ising::Ising2D)
+	return mean(ising.magnetization_array) / ising.n
+end
+
+function susceptibility(ising::Ising2D)
+	return (std(ising.magnetization_array)) ^ 2 / (ising.temperature * ising.n)	
+end
+
+function observables(ising::Ising2D)
+	printstyled("Temperature\t\t", bold=true)
+	print(ising.temperature); print("\n")
+
+	printstyled("Mean Energy\t\t", bold=true)
+	print(mean_energy(ising)); print("\n")
+
+	printstyled("Mean Magnetiz.\t\t", bold=true)
+	print(mean_magnetization(ising)); print("\n")
+
+	printstyled("Specific Heat\t\t", bold=true)
+	print(specific_heat(ising)); print("\n")
+
+	printstyled("Susceptibility\t\t", bold=true)
+	print(susceptibility(ising)); print("\n")
+
+	printstyled("MC Steps\t\t", bold=true)
+	print(ising.mc_steps); print("\n")
+	printstyled("Accepted Moves\t\t", bold=true)
+	print(ising.accepted_moves); print("\n")
+end
+
+
+function plot_ising(state::Matrix{Int})
+	pos = Tuple.(findall(>(0), state))
+	neg = Tuple.(findall(<(0), state))
+	scatter(pos, markersize=5)
+	scatter!(neg, markersize=5)
+end
+
+
+m = Ising2D(32, 2.3)
+steps!(m, 1000)
+observables(m)
+plot_ising(m.state)
diff --git a/hw6/b-old/8-2-1.png b/hw6/b-old/8-2-1.png
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+++ b/hw6/b-old/8-2-1.png
diff --git a/hw6/b-old/8-2-10.png b/hw6/b-old/8-2-10.png
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diff --git a/hw6/b-old/8-2-100.png b/hw6/b-old/8-2-100.png
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diff --git a/hw6/b-old/8-2-11.png b/hw6/b-old/8-2-11.png
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diff --git a/hw6/b-old/8-2-110.png b/hw6/b-old/8-2-110.png
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diff --git a/hw6/b-old/8-2-112.png b/hw6/b-old/8-2-112.png
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diff --git a/hw6/b-old/8-2-114.png b/hw6/b-old/8-2-114.png
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diff --git a/hw6/b-old/8-2-120.png b/hw6/b-old/8-2-120.png
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diff --git a/hw6/b-old/8-2-122.png b/hw6/b-old/8-2-122.png
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diff --git a/hw6/b-old/8-2-128.png b/hw6/b-old/8-2-128.png
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