Random Walk Simulator

Simulate and visualize random walks in 1D, 2D, and 3D space

Overview

The Random Walk Simulator allows you to simulate and visualize different types of random walks in one, two, and three dimensions. Explore how randomness creates emergent patterns through simple walks, biased walks, Lévy flights, and Brownian motion. Run multiple walks simultaneously to see statistical distributions emerge, and analyze key metrics like displacement, efficiency, and mean squared displacement to understand fundamental concepts in probability theory and statistical physics.

Tips and Tricks

Getting Started

  1. Choose a dimension (1D, 2D, or 3D) to set the space for your simulation
  2. Select a walk type to explore different random behaviors
  3. Adjust the number of steps to see how walks evolve over time
  4. Run multiple walks to observe statistical distributions

Understanding Walk Types

Simple Random Walk - At each step, move in a random direction with fixed step size - Expected displacement grows as √n where n is the number of steps - Try 1,000+ steps to see the characteristic diffusion behavior

Biased Random Walk - Directional preference controlled by the bias parameter - Higher bias = more directional movement - Useful for modeling drift in diffusion processes

Lévy Flight - Occasional very long steps mixed with short ones - Efficient for exploration and search tasks - Models animal foraging behavior

Brownian Motion - Steps drawn from a Gaussian (normal) distribution - Models molecular motion and stock prices - More “natural-looking” than simple random walk

Key Statistics Explained

Displacement: Straight-line distance from start to end - For simple walks: average displacement ≈ √n - Larger values indicate directional bias or long jumps

Efficiency: Ratio of displacement to path length (0 to 1) - Values near 0: walker returns close to start - Values near 1: nearly straight path - Simple random walks typically have low efficiency

Mean Squared Displacement (MSD) - Average of squared displacements across multiple walks - For simple random walks, MSD grows linearly with time - Different growth rates indicate different types of diffusion

Practical Tips

  • Start simple: Begin with a 2D simple random walk to understand the basics
  • Multiple walkers: Run 10+ walks to see the statistical distribution clearly
  • Adjust step size: Smaller steps create smoother paths, larger steps show the randomness better
  • 3D visualization: Rotate the view to see the full structure of 3D walks
  • Compare types: Run the same number of steps for different walk types to compare behavior

Educational Applications

  • Central Limit Theorem: With many steps, final positions follow a normal distribution
  • Diffusion: Random walks model how particles spread through space
  • Scaling Laws: The √n relationship appears in many natural phenomena
  • Stochastic Processes: Learn about randomness in time-evolving systems