testing

1 files changed, 69 insertions, 0 deletions

diff --git a/hw6/8-1.jl b/hw6/8-1.jl
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+# 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")
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