🧭Random Walk Simulations in 1D, 2D and 3D

A comprehensive collection of random walk simulations in 1D, 2D, and 3D with interactive visualizations, animations, and statistical analysis.

1D Random Walk	2D Random Walk	3D Random Walk

📚 Table of Contents

• 📘 Introduction
• ✨ Features
• 📁 Folder Structure
• 🛠️ Setup and Installation
• 🚀 Usage
• ⚠️ Important Notes
• 📓 Notebook Content Overview
• 🎞️ Animation & Visual Assets
• 📊 Statistics and Analysis
• 🧩 Interactive Features
• 🔬 Advanced Analysis
• 🔗 References
• 🤝 Contributing
• 📄 License

📘 Introduction

A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers. This notebook provides a hands-on approach to understanding random walks by:

• Simulating paths in 1D, 2D, and 3D.
• Analyzing statistical properties like mean displacement and standard deviation.
• Visualizing trajectories and distributions using matplotlib and plotly.
• Creating interactive animations to observe the walk's evolution over time.
• Exploring the effects of bias on the walker's path.

✨ Features

• Multi-Dimensional Simulations: Explore random walks in 1D, 2D, and 3D space.
• Statistical Analysis: Calculate and visualize mean position, standard deviation, and final position distributions over multiple trials.
• Biased Walks: Simulate and analyze random walks where steps in certain directions are more probable.
• Interactive Visualizations: Utilize plotly and ipywidgets for dynamic plots.
• Animations: Generate .mp4 or .gif animations of random walks using matplotlib.animation.
• Quadrant Analysis (2D): Track the walker's time in each quadrant.
• Step Size Comparison (2D): Analyze the effect of different step sizes.
• Simultaneous Walks (3D): Visualize multiple walkers starting from different positions.

📁 Folder Structure

.
├── 🎞️ ANIMATION/
│   ├── 🎞️ 1D/         # Animations for 1D walks (.mp4, .gif)
│   ├── 🎞️ 2D/         # Animations for 2D walks (.mp4, .gif)
│   └── 🎞️ 3D/         # Animations for 3D walks (.mp4, .gif)
├── 🖼️ IMAGES/
│   ├── 🖼️ 1D/         # Static plots for 1D walks (.png, .jpg)
│   ├── 🖼️ 2D/         # Static plots for 2D walks (.png, .jpg)
│   └── 🖼️ 3D/         # Static plots for 3D walks (.png, .jpg)
├── 📊 STATISTICS/     # Optional: CSV or other files with statistical results
├── 🧪 RandomWalk.ipynb  # The main Jupyter Notebook file
├── 📦 requirements.txt # List of required Python packages
└── 📄 README.md       # This file

🛠️ Setup and Installation

You can follow these steps to set up your environment and run the notebook.

✅ Prerequisites

• Python 3.7 or higher
• pip (Python package installer)
• git (for cloning the repository)

🧬 Cloning the Repository

git clone https://github.com/PuspenduPH/Random-Walk-Simulations
cd Random-Walk-Simulations

🧰 Setting Up a Virtual Environment (Recommended)

Using a virtual environment prevents dependency conflicts.

1. Create a virtual environment (e.g., named randomwalk_env):

   python -m venv randomwalk_env

2. Activate the environment:
   • Windows:

     randomwalk_env\Scripts\activate

   • Linux/macOS:

     source randomwalk_env/bin/activate

3. (Optional) Upgrade pip:

   python -m pip install --upgrade pip

📦 Installing Dependencies

Install all required Python libraries using the requirements.txt file:

pip install -r requirements.txt

Alternatively, install them manually:

pip install numpy scipy scikit-learn matplotlib plotly ipywidgets pandas tqdm ipympl

Note: FFmpeg is necessary to save animations in .mp4 format and must be installed manually (see Important Notes below).

🚀 Usage

▶️ Running the Notebook

1. Ensure your virtual environment is activated.
2. Navigate to the repository directory in your terminal.
3. Launch Jupyter Notebook or JupyterLab:

   jupyter notebook
   # OR
   jupyter lab

4. Open the random_walk_notebook.ipynb (or your actual notebook file name) in the Jupyter interface.
5. Run the cells sequentially or explore specific sections.

⚠️ Important Notes

This notebook contains interactive visualizations, animations, and data manipulation tools. Follow these instructions to avoid common errors and ensure smooth execution:

• Animation in Jupyter:

  • Install the ipympl package with:

    pip install ipympl

  • Use the following magic commands at the top of your notebook:

    %matplotlib ipympl  # For displaying animations
    %matplotlib widget  # For interactive widgets

  • Tip: Use only one magic command in a cell, depending on your requirements. If you need any help installing ipympl yo can follow this link

• Saving Animations as .mp4:

  • Install FFmpeg:
    • Windows:
      • Download FFmpeg from ffmpeg.org.
      • Extract and add the bin folder to your system's PATH.
    • macOS/Linux:

      brew install ffmpeg     # For macOS with Homebrew
      sudo apt install ffmpeg # For Ubuntu/Debian

• General Tips:

  • Use FuncAnimation from matplotlib.animation for creating animations.
  • Utilize PillowWriter if you prefer saving animations as .gif.
  • Use tqdm for progress tracking in loops or simulations.
  • Suppress unnecessary warnings:

    import warnings
    warnings.filterwarnings("ignore")

  • Use Python’s traceback module for inline error logging during debugging.

Reminder: Always restart the kernel after switching %matplotlib backends or installing new Jupyter-related packages to avoid rendering issues.

📓 Notebook Content Overview

The notebook is divided into sections exploring random walks in increasing dimensions:

1. One-Dimensional Random Walk

• Simulation: Basic simulation of a single 1D random walk.

  

• Analysis: Calculating displacement and distance from the origin.

  

• Multiple Trials: Running simulations multiple times to observe average behaviour.

  

• Biased Random Walk: Introducing probabilities for moving left or right.

Probability p=0.7	Probability p=0.3

• Mean and Standard Deviation Analysis: Examining how mean position and standard deviation evolve.

  

• Distribution of Walker's Position: Visualizing the probability distribution of the walker's position after N steps.

  

• Animation: Animating the 1D random walk.

• Animation with Position Distribution: Animating the walk alongside its evolving position distribution.

  rdwalk_1D_dist_s50_t40_p0.5.mp4

2. Two-Dimensional Random Walk

• Simulation & Plotting: Simulating and plotting a 2D walk using matplotlib and plotly.

  

• Contour Plot: Visualizing the density of visited locations.

• 

• Quadrant Analysis: Tracking the proportion of time spent in each quadrant.

  

• Quadrant Analysis with Animation: Animating the walk and the evolving quadrant distribution.

  

• Analysis with Contour Plot and Heatmap: Combined visualisation of path and visit frequency.

  

• Distribution of Walker's Position: Analyzing the 2D distribution of positions over multiple trials.

  

• Distribution of Final Position: Visualising where walkers end up after N steps.

  

• Animation: Basic animation of the 2D random walk path.

• Animation with Position Frequency: Animating the walk alongside a heatmap of visited positions.

• Step Size Comparison: Analysing walks with different step lengths.

  

• 2D Biased Random Walk: Implementing bias in movement directions (e.g., tendency to move North-East).

  2D Random Walk with Angular Bias	2D Random Walk with Target Bias

• Comparison with/without Bias: Visual comparison of biased vs. unbiased walks.

  

• Biased Walk with Different Probabilities: Exploring various bias configurations.

  rdwalk_2D_with_counts_s100_t20_p0.4_q0.1_r0.25_s0.25.mp4

3. Three-Dimensional Random Walk

• Simulation (Single Trial): Simulating and visualizing a single walk in 3D space using matplotlib or plotly.

  random_walk_3d_s100.mp4

• Simulation (Multiple Trials): Running and analyzing multiple 3D walks.

  random_walk_3d_multi_s50_t20_1.1.mp4

• Multiple Trials with Different Initial Positions: Simulating walkers starting from various points.

  random_ip_walk_3d_multi_s50_t20.mp4

• Distribution of Walker's Final Position: Visualizing the 3D distribution of endpoints.

• Simultaneous Random Walk: Animating multiple walks concurrently.

• Biased Random Walk: Introducing directional bias in 3D steps.

  • 3D Random Walk with Target Bias

    distance_biased_rdwalk_3d_s50_t15.mp4

  • 3D Random Walk with Angular Bias

    angular_biased_rdwalk_3d_s50_t15.mp4

• Visualise Biased Walks with IpyWidgets: Using widgets to explore different bias parameters interactively.

• Comparison with/without Bias: Comparing trajectories of biased and unbiased 3D walks.

  final_walk_comparison_3d_s50_t15.mp4

🎞️ Animation & Visual Assets

Animations and visual outputs are organized into separate directories by dimensionality:

• ANIMATION/1D/ – Animations for 1D random walk.
• ANIMATION/2D/ – Animations for 2D random walk.
• ANIMATION/3D/ – Animations for 3D random walk.
• IMAGES/1D/, IMAGES/2D/, IMAGES/3D/ – Supporting images and figures.

Please add, update, or replace assets as your experiments change.

📊 Statistics and Analysis

The STATISTICS folder is intended for:

• Storing detailed statistical reports and datasets related to random walk experiments.
• Tracking parameters such as mean, variance, skewness, and other performance metrics of the simulations.

This modular organization makes maintaining a historical record of simulation outcomes and analysis easier.

🧩 Interactive Features

The notebook includes several interactive components:

• Adjustable parameters for walk length, bias, and step size
• Real-time visualization updates
• Animated simulations with playback controls
• Parameter exploration with ipywidgets
• Statistical analysis with immediate feedback

🔬 Advanced Analysis

Statistical investigations included in the notebook:

• Mean square displacement analysis
• First return times
• Probability distributions
• Central limit theorem demonstrations
• Quadrant occupation statistics
• Bias effects on displacement

🔗 References

• Matplotlib Animation Guide
• Plotly Documentation
• Jupyter ipympl Documentation
• FFmpeg Installation
• Wikipedia: Random Walk
• Lawler, G. F., & Limic, V. (2010). Random walk: a modern introduction (Vol. 123). Cambridge University Press.
• Mörters, P., & Peres, Y. (2010). Brownian motion (Vol. 30). Cambridge University Press.
• Spitzer, F. (2013). Principles of random walk (Vol. 34). Springer Science & Business Media.
• Bhattacharyya, Arka et al. “A Brief Study on Simple Random Walk in 1D.” Resonance 28 (2023): 945 - 958.

🤝 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

1. Fork the repository.
2. Create your feature branch (git checkout -b amazing-feature).
3. Commit your changes (git commit -m 'Add some amazing feature').
4. Push to the branch (git push origin amazing-feature).
5. Open a Pull Request.

📄 License

Distributed under the MIT License. See LICENSE for more information.