Numerical Simulation of a Random Walk as a Stochastic Model of Diffusion

Papadopol Lucian Ioan
l.i.papadopol@gmail.com

1. Introduction

In this notebook we study a simple stochastic system: the one-dimensional random walk.

Unlike the deterministic models studied previously, in a stochastic system the evolution is not completely determined by the initial conditions. Even starting from the same initial state, two different simulations may produce different trajectories, because the model contains a random component.

The 1D random walk describes a particle moving along a line. At each time step, the particle can move to the right or to the left with a certain probability.

2. Libraries and parameters

import numpy as np
import matplotlib.pyplot as plt

# Simulation parameters

n_steps = 500          # number of time steps
p_right = 0.5          # probability of a step to the right
x0 = 0                 # initial position
step_scale = 1.0       # spatial step size

3. Random walk of a single particle

The following function simulates a one-dimensional random walk. At each step, the particle moves by either +step_scale or -step_scale. The final trajectory is obtained by taking the cumulative sum of all random steps.

def random_walk(n_steps, step_scale=1.0, seed=None):
    rng = np.random.default_rng(seed)

    steps = step_scale * rng.choice(
        [-1, 1],
        size=n_steps
    )

    positions = np.zeros(n_steps + 1)
    positions[1:] = np.cumsum(steps)

    return positions

4. Two-dimensional representation of a single particle

To visualize the motion in a plane, we use two independent one-dimensional random walks: one for the horizontal coordinate and one for the vertical coordinate.

# 2D random walk of a single particle

x = random_walk(n_steps, step_scale=step_scale, seed=1)
y = random_walk(n_steps, step_scale=step_scale, seed=2)

plt.figure(figsize=(7, 7))

plt.plot(x, y, linewidth=1.5)

# Starting point
plt.scatter(x[0], y[0], s=50, label="Start")

# Final point
plt.scatter(x[-1], y[-1], s=70, marker="x", label="End")

plt.xlabel("Position x")
plt.ylabel("Position y")
plt.title("2D random walk of a single particle")
plt.grid(True)
plt.axis("equal")
plt.legend()
plt.show()

5. Test with many particles

We now repeat the same stochastic process for many particles. Each particle starts from the same origin, but follows a different trajectory because the random choices are different.

n_particles = 20
step_scale = 1.0

time = np.arange(n_steps + 1)

plt.figure(figsize=(7, 7))

for i in range(n_particles):
    x = random_walk(n_steps, step_scale=step_scale, seed=2*i)
    y = random_walk(n_steps, step_scale=step_scale, seed=2*i + 1)

    plt.plot(x, y, linewidth=1.2, alpha=0.9)

    # Starting point
    plt.scatter(x[0], y[0], s=20)

    # Final point
    plt.scatter(x[-1], y[-1], s=45, marker="x")

plt.xlabel("Position x")
plt.ylabel("Position y")
plt.title("2D random walk of many particles")
plt.grid(True)
plt.axis("equal")
plt.show()

A single trajectory is irregular and cannot be predicted deterministically. However, when many independent particles are simulated, a collective statistical behavior emerges