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Commit 8a70e1b4 authored by jung_42's avatar jung_42
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Update sheet3

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with 56 additions and 48 deletions
Jaslo/sheet3.py
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import numpy as np import numpy as np
from scipy.constants import Boltzmann as k_B from scipy.constants import Boltzmann as k_B
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from tqdm import tqdm
def verlet(force, x0, p0, m, dt, N): def verlet(force, x0, p0, m, dt, N):
assert(x0.shape == p0.shape) assert(x0.shape == p0.shape)
...@@ -11,7 +13,7 @@ def verlet(force, x0, p0, m, dt, N): ...@@ -11,7 +13,7 @@ def verlet(force, x0, p0, m, dt, N):
x[0] = x0 x[0] = x0
p[0] = p0 p[0] = p0
for i in range(1, N): for i in tqdm(range(1, N)):
p[i] = p[i-1] + 1/2 * force(x[i-1]) * dt p[i] = p[i-1] + 1/2 * force(x[i-1]) * dt
x[i] = x[i-1] + p[i] / m * dt x[i] = x[i-1] + p[i] / m * dt
p[i] = p[i] + 1/2 * force(x[i]) * dt p[i] = p[i] + 1/2 * force(x[i]) * dt
...@@ -43,81 +45,87 @@ p0[-1] = -p0[0] ...@@ -43,81 +45,87 @@ p0[-1] = -p0[0]
dt = 0.1 dt = 0.1
t = np.arange(0, 300, dt) t = np.arange(0, 300, dt)
x, p = verlet(force, x0, p0, m, dt, len(t)) # x, p = verlet(force, x0, p0, m, dt, len(t))
# fig, ax = plt.subplots() # # fig, ax = plt.subplots()
for i in range(x.shape[1]): # for i in range(x.shape[1]):
plt.plot(t, x[:,i], label=f"Bead {i+1}") # plt.plot(t, x[:,i], label=f"Bead {i+1}")
plt.xlabel(r"$t / \tau$") # plt.xlabel(r"$t / \tau$")
plt.ylabel(r"$x / l$") # plt.ylabel(r"$x / l$")
plt.legend() # plt.legend()
plt.show() # plt.show()
# E = np.sum(p**2 / (2 * m), axis=1) + 1/2 * k * np.sum((x[:,1:] - x[:,:-1])**2, axis=1) # # E = np.sum(p**2 / (2 * m), axis=1) + 1/2 * k * np.sum((x[:,1:] - x[:,:-1])**2, axis=1)
# K = np.sum(p**2 / (2 * m), axis=1) # # K = np.sum(p**2 / (2 * m), axis=1)
# plt.plot(t, K) # # plt.plot(t, K)
# plt.show() # # plt.show()
# b) # # b)
for N in (2, 3, 11, 100): # for N in (2, 3, 11, 100):
m = np.full(N, 1) # m = np.full(N, 1)
I = np.arange(1, N+1) # I = np.arange(1, N+1)
x0 = l * I # x0 = l * I
p0 = np.zeros(x0.shape) # p0 = np.zeros(x0.shape)
p0[0] = 1/2 * np.sqrt(m[0] * eps) # p0[0] = 1/2 * np.sqrt(m[0] * eps)
p0[-1] = -p0[0] # p0[-1] = -p0[0]
x, p = verlet(force, x0, p0, m, dt, len(t)) # x, p = verlet(force, x0, p0, m, dt, len(t))
K = np.sum(p**2 / (2 * m), axis=1) # K = np.sum(p**2 / (2 * m), axis=1)
K_l = np.average(K) # K_l = np.average(K)
T = 2 * K_l / N # T = 2 * K_l / N
print(f"<K> = {K_l}") # print(f"<K> = {K_l}")
print(f"k_B T = {T}") # print(f"k_B T = {T}")
# c) # # c)
for N in (2, 3, 11, 100): # for N in (2, 3, 11, 100):
for i in range(3): # for i in range(3):
m = np.full(N, 1) # m = np.full(N, 1)
I = np.arange(1, N+1) # I = np.arange(1, N+1)
x0 = l * I # x0 = l * I
p0 = np.random.normal(0, np.sqrt(m[0] * eps), N) # p0 = np.random.normal(0, np.sqrt(m[0] * eps), N)
x, p = verlet(force, x0, p0, m, dt, len(t)) # x, p = verlet(force, x0, p0, m, dt, len(t))
R_l = np.abs(x[:,0] - x[:,-1]) # R_l = np.abs(x[:,0] - x[:,-1])
print(f"N = {N}, i = {i}: R_l = {np.average(R_l)}") # print(f"N = {N}, i = {i}: R_l = {np.average(R_l)}")
plt.plot(t, R_l) # plt.plot(t, R_l)
plt.xlabel(r"$t / \tau$") # plt.xlabel(r"$t / \tau$")
plt.ylabel(r"$\overline{R_l} / l$") # plt.ylabel(r"$\overline{R_l} / l$")
plt.show() # plt.show()
# d) # d)
def d(x1, p1, x2, p2): def d(x1, p1, x2, p2):
return np.sqrt(1 / (6 * x1.shape[0]) * np.sum(k * (x1 - x2)**2 + (p1 - p2)**2 / m)) return np.sqrt(1 / (6 * x1.shape[0]) * np.sum(k * (x1 - x2)**2 + (p1 - p2)**2 / m))
dt = 0.1
t = np.arange(0, 1*10e3, dt)
t = np.arange(0, 2e4, dt) N_vals = [10,20]
for N in N_vals:
for N in (10, 20):
m = np.full(N, 1) m = np.full(N, 1)
I = np.arange(1, N+1) I = np.arange(1, N+1)
x0 = l * I x0 = l * I
p0 = np.random.normal(0, np.sqrt(m[0] * eps), N) # p0 = np.random.normal(0, np.sqrt(m[0] * eps), N)
p0 = np.zeros(x0.shape)
p0[0] = 1/2 * np.sqrt(m[0] * eps)
p0[-1] = -p0[0]
x, p = verlet(force, x0, p0, m, dt, len(t)) x, p = verlet(force, x0, p0, m, dt, len(t))
ts = t[int(10 / dt):] ts = t[int(10 / dt):]
ds = np.zeros(ts.shape) ds = np.zeros(ts.shape)
for i in range(len(ds)): for i in tqdm(range(len(ds))):
ds[i] = d(x[int(10 / dt) + i], p[int(10 / dt) + i], x[int(10 / dt)], p[int(10 / dt)]) ds[i] = d(x[int(10 / dt) + i], p[int(10 / dt) + i], x[int(10 / dt)], p[int(10 / dt)])
print(f"N = {N}: d_min = {np.min(ds)}") print(f"N = {N}: d_min = {np.min(ds)}")
plt.figure(figsize=(48, 6))
plt.xlabel(r"$t / \tau$") plt.xlabel(r"$t / \tau$")
plt.ylabel(r"$d / \sqrt{k l^2}$") plt.ylabel(r"$d / \sqrt{k l^2}$")
plt.plot(ts, ds) plt.plot(ts, ds)
plt.show() # plt.tight_layout()
plt.savefig(f'4c_N={N}.png')
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