Brownian Motion Pattern using Python

 

import numpy as np

import matplotlib.pyplot as plt

N=10

T=500

step_size=1

x=np.zeros((N,T))

Y=np.zeros((N,T))

for i in range(1, T):

    angle = 2 * np.pi * np.random.rand(N) 

    x[:, i] = x[:, i-1] + step_size * np.cos(angle)

    y[:, i] = y[:, i-1] + step_size * np.sin(angle)

plt.figure(figsize=(8,6))

for i in range(N):

    plt.plot(x[i],y[i],lw=1.5,alpha=0.7)

plt.scatter(x[:,-1],y[:,-1],c='red',marker='o',label="Final position")

plt.title("Brownian motion pattern")

plt.xlabel("X Position")

plt.ylabel("Y Position")

plt.legend()

plt.grid()

plt.show()

#source code --> clcoding.com 

Code Explanation:

1. Importing Necessary Libraries

import numpy as np

import matplotlib.pyplot as plt

NumPy (np): Used for efficient numerical operations, including random number generation and array manipulations.

Matplotlib (plt): Used for plotting and visualizing the Brownian motion paths.

2. Defining Parameters

N = 10   # Number of particles

T = 500  # Number of time steps

step_size = 1  # Step size for each move

N = 10 → The number of particles that will undergo Brownian motion.

T = 500 → The number of time steps, meaning each particle moves 500 times.

step_size = 1 → The fixed distance a particle moves at each time step.

3. Initializing Position Arrays

x = np.zeros((N, T))

y = np.zeros((N, T))

x and y arrays:

These are N × T matrices (10 × 500 in this case), initialized with zeros.

Each row represents a different particle, and each column represents a time step.

Initially, all particles start at (0,0).

4. Simulating Brownian Motion

for i in range(1, T):  # Loop over time steps (excluding the first step)

    angle = 2 * np.pi * np.random.rand(N)  # Generate N random angles (0 to 2π)

    x[:, i] = x[:, i-1] + step_size * np.cos(angle)  # Update x-coordinates

    y[:, i] = y[:, i-1] + step_size * np.sin(angle)  # Update y-coordinates

Breaking it Down

for i in range(1, T):

Loops through T-1 time steps (from 1 to 499) because the initial position (t=0) is at (0,0).

angle = 2 * np.pi * np.random.rand(N)

Generates N random angles between 0 and 2π (full circle) for random movement in any direction.

Updating Particle Positions:

X-direction:

x[:, i] = x[:, i-1] + step_size * np.cos(angle)

The next x-coordinate is determined by adding cos(angle) (step movement in x-direction).

Y-direction:

y[:, i] = y[:, i-1] + step_size * np.sin(angle)

The next y-coordinate is determined by adding sin(angle) (step movement in y-direction).

Since angles are random at each step, particles move in completely unpredictable directions.

5. Plotting the Brownian Motion Paths

plt.figure(figsize=(8, 6))

for i in range(N):  

    plt.plot(x[i], y[i], lw=1.5, alpha=0.7)

plt.figure(figsize=(8, 6)) → Sets the figure size to 8 inches by 6 inches.

for i in range(N): → Loops through each particle (N=10).

plt.plot(x[i], y[i], lw=1.5, alpha=0.7)

Plots each particle’s path using lines.

lw=1.5 → Line width is set to 1.5 for better visibility.

alpha=0.7 → Makes lines slightly transparent for better visualization.

6. Highlighting the Final Positions

plt.scatter(x[:, -1], y[:, -1], c='red', marker='o', label="Final Positions")

plt.scatter(0, 0, c='black', marker='x', label="Starting Point")

Final Positions (x[:, -1], y[:, -1])

plt.scatter(x[:, -1], y[:, -1], c='red', marker='o', label="Final Positions")

Marks the last position of each particle with red circles (o).

Starting Position (0,0)

plt.scatter(0, 0, c='black', marker='x', label="Starting Point")

Marks the starting point with a black ‘X’.

7. Customizing the Plot

plt.title("Brownian Motion of Particles")

plt.xlabel("X Position")

plt.ylabel("Y Position")

plt.legend()

plt.grid()

plt.show()

Title & Labels

plt.title("Brownian Motion of Particles") → Sets the title of the plot.

plt.xlabel("X Position") → Labels the X-axis.

plt.ylabel("Y Position") → Labels the Y-axis.

Legend

plt.legend() → Displays the labels for the final positions and the starting point.

Grid

plt.grid() → Adds a grid for better visualization.

Show Plot

plt.show() → Displays the final plot.