Random Walk Diffusion Simulation (3D Stochastic Model)

This work presents a computational simulation of a 3D stochastic random walk system combined with a density-based visualization model. A set of particles evolves inside a bounded cubic space, where each particle moves randomly along one of the three axes at each iteration. The system visualizes both spatial distribution and local density using a real-time 3D scatter plot with color mapping.

Introduction

This work implements a discrete random walk simulation in a bounded 3D grid with the following objectives:

• Simulate stochastic movement of multiple particles in 3D space.
• Constrain motion inside a finite cubic domain using wrap-around behavior.
• Compute local spatial density of particle distributions.
• Visualize particle evolution in real time using Matplotlib 3D rendering.
• Encode density information into color intensity for visual interpretation.

The system is designed as an exploratory tool for understanding diffusion-like processes and emergent spatial clustering in stochastic systems.

Problem Representation

The simulation operates on a discrete 3D lattice defined as:

$$ (x, y, z) \in [0, G]^3 $$

Where:

• $G \in \mathbb{N}$ : grid size (spatial boundary)
• $N \in \mathbb{N}$ : number of particles

Each particle $i$ is represented by:

$$ p_i = (x_i, y_i, z_i) $$

The system state at time $t$ is:

$$ S(t) = {p_1(t), p_2(t), \dots, p_N(t)} $$

Random Walk Dynamics

At each iteration, every particle performs a stochastic update:

1. A random axis is selected: $x$, $y$, or $z$.
2. A random step is chosen: $+1$ or $-1$.
3. Position is updated using modular arithmetic:

$$ p_i(t+1) = (p_i(t) + \text{step}) \pmod{G + 1} $$

This ensures:

• Constrained motion within bounds.
• Toroidal (wrap-around) space topology.
• No particle loss at boundaries.

Density Computation

To analyze spatial clustering, a density metric is computed:

$$ \text{density}_i = \frac{\text{count}(p_i)}{\max(\text{counts})} $$

Where:

• $\text{count}(p_i)$ is the number of occurrences of position $p_i$.
• $\max(\text{counts})$ is the highest frequency among all positions.

This produces a normalized density value in the range $[0, 1]$ used for color mapping in visualization.

Results

The system produces:

• Emergent clustering patterns in bounded space.
• Density heatmaps over 3D scatter distributions.
• Dynamic diffusion-like behavior over time.

Higher iteration counts reveal:

• Region concentration effects.
• Random equilibrium distribution.
• Local clustering persistence due to discrete collisions.

Limitations

• Density computation is $\mathcal{O}(n^2)$.
• Visualization re-creates scatter plot each frame.
• No physical forces (purely stochastic motion).
• Discrete grid approximation may introduce aliasing effects.