finished task 1

I.py

@@ -0,0 +1,61 @@
+import numpy as np
+import scipy as scp
+
+from matplotlib import pyplot as plt
+
+from RubberBand import RubberBand
+
+USE_SAVED = True
+
+NUM_BANDS = int(1E6)
+N = 100
+
+if USE_SAVED:
+    lengths = np.load("data/lengths.npy")
+else:
+    lengths = np.zeros((NUM_BANDS,))
+    
+    for i in range(NUM_BANDS):
+        band = RubberBand(N, a=1)
+        lengths[i] = band.length/band.a
+
+# histogram
+values, bin_edges = np.histogram(lengths, bins=int(2*N), range=(-N, N))
+p_mc = values / values.sum() # normalise histogram
+
+bin_centers = bin_edges[:-1] + np.diff(bin_edges)/2
+
+def P(N, L_over_a):
+    """
+    P(L/a) = Omega(N, n^+) / 2^N
+    where
+    n^+ = (L/a + N)/2 (see eq. 1)
+    """
+    return scp.special.binom(N, (N+L_over_a)/2) / 2**N
+
+p_true = P(N, bin_centers-.5)
+
+chi2 = np.sum((p_mc - p_true)**2 / p_true)
+
+# plot hist vs true dist.
+plt.plot(bin_centers-.5, p_true, label="$P(L)$")
+plt.step(bin_centers, p_mc, color='r', alpha = 0.5, label="$\hat{P}(L)$")
+plt.text(-45, 0.07, f"$\chi^2 = {round(chi2, 3)}$\n({len(lengths):.0E} samples)")
+plt.legend()
+plt.xlabel("$L/a$")
+plt.ylabel("P($L/a$)")
+xlim = (-50, 50)
+plt.xlim(xlim)
+
+# ratios
+p_ratio = p_mc / p_true
+plt.figure()
+plt.scatter(bin_centers, p_ratio, marker='x')
+plt.hlines(1, *xlim, color='r', linestyle='--')
+plt.xlim(xlim)
+plt.ylim(0.5, 1.5)
+plt.xlabel("$L/a$")
+plt.ylabel("$\hat{P}(L)/P(L)$")
+plt.show()
+
+np.save("data/lengths.npy", lengths)