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Application in Statistical Mechanics: 2D Random Walk Simulation
Application in Statistical Mechanics
Study of Random Numbers and Time Series
Generate Different Variate from Uniform Variate
Coin Tossing Simulation
Nuclear Decay Simulation
1D Random Walk Simulation
2D Random Walk Simulation
Monte Carlo Method
Specific Heat of Solids (Einstein’s Theory)
Specific Heat of Solids (Debye’s Theory)
Plots of Different Distribution Functions
2D Random Walk Simulation: Program 1
2D RW Simulation
import numpy as np from random import choice import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation # Initialize the random walk x, y = 0, 0 X, Y = [], [] # Create the figure and axis fig, ax = plt.subplots() line, = ax.plot([], [], lw=2) ax.set_xlim(-20, 20) ax.set_ylim(-20, 20) # Function to initialize the plot def init(): line.set_data([], []) return line, # Function to update the plot for each frame def update(frame): dx, dy = choice([(1,0), (-1, 0), (0, 1), (0, -1)]) global x, y x, y = x + dx, y + dy X.append(x) Y.append(y) line.set_data(X, Y) return line, # Create the animation ani = FuncAnimation(fig, update, frames=range(100), init_func=init, blit=True, interval=1) # Save the animation as a GIF ani.save('random_walk.gif')
Run Code
Output 1
2D Random Walk Simulation: Program 2
RMS distance in 2D RW
import numpy as np from random import choice import matplotlib.pyplot as plt import numpy.polynomial.polynomial as poly def walk(step): # return square of distance in each step x, y = 0, 0 X, Y, pos = [], [], [] for i in range(steps): dx, dy = choice([(1,0), (-1, 0), (0, 1), (0, -1)]) x, y = x + dx, y + dy pos.append(x**2 + y**2) return pos steps, config = 500, 2000 R2 = np.array([walk(steps) for i in range(config)]) mwalk = np.mean(R2, axis = 0) rms = np.sqrt(mwalk) t = range(1, steps + 1) t = np.log(t) rms = np.log(rms) coeff = poly.polyfit(t, rms, 1) rmsfit = poly.polyval(t, coeff) print(coeff) intercept = np.round(coeff[1], 4) plt.plot(t, rms, 'o', t, rmsfit, '-', lw = 3) plt.title('Exponent of Random walk in 2D = '+str(intercept)) plt.xlabel('log(time)') plt.ylabel('log($R_{rms}$)') plt.savefig('plot.png')
Run Code
Output 2
2D Random Walk Simulation: Program 3
3D RW Simulation
import numpy as np from random import choice import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation from mpl_toolkits.mplot3d import Axes3D # Initialize the random walk x, y, z = 0, 0, 0 X, Y, Z = [], [], [] # Create the figure and 3D axis fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.set_xlim(-10, 10) ax.set_ylim(-10, 10) ax.set_zlim(-10, 10) # Function to initialize the plot def init(): ax.clear() ax.set_xlim(-10, 10) ax.set_ylim(-10, 10) ax.set_zlim(-10, 10) return [] # Function to update the plot for each frame def update(frame): global x, y, z dx, dy, dz = choice([(0,0,1), (0,1,0), (0,-1,0), (0,0,-1), (1,0,0), (-1,0,0)]) x, y, z = x + dx, y + dy, z + dz X.append(x) Y.append(y) Z.append(z) ax.clear() ax.set_xlim(-10, 10) ax.set_ylim(-10, 10) ax.set_zlim(-10, 10) ax.plot(X, Y, Z, alpha=0.6) ax.scatter(X[-1], Y[-1], Z[-1], color='red') return [] # Create the animation ani = FuncAnimation(fig, update, frames=range(100), init_func=init, blit=False, interval=1) # Save the animation as a GIF ani.save('random_walk_3d.gif')
Run Code
Output 3
2D Random Walk Simulation: Program 4
RMS distance in 3D RW
import numpy as np from random import choice import matplotlib.pyplot as plt import numpy.polynomial.polynomial as poly def walk(step): x, y, z = 0, 0, 0 X, Y, Z, pos = [], [], [], [] for i in range(steps): dx, dy, dz = choice([(0,0,1), (0,1,0), (0,-1,0), (0,0,-1), (1,0,0),(-1,0,0)]) x, y, z = x + dx, y + dy, z + dz pos.append(x**2 + y**2 + z**2) return pos steps, config = 1000, 1000 R2 = np.array([walk(steps) for i in range(config)]) mwalk = np.mean(R2, axis = 0) rms = np.sqrt(mwalk) t = range(1, steps + 1) t = np.log(t) rms = np.log(rms) coeff = poly.polyfit(t, rms, 1) rmsfit = poly.polyval(t, coeff) print(coeff) intercept = np.round(coeff[1], 4) plt.plot(t, rms, 'o', t, rmsfit, '-', lw = 3) plt.title('Exponent of Random walk in 3D = '+str(intercept)) plt.xlabel('log(time)') plt.ylabel('log($R_{rms}$)') plt.savefig('plot.png')
Run Code
Output 4