Brownian Motion Simulations in Python

This repository contains a single Jupyter Notebook—brownian motion.ipynb—that demonstrates simulations of 1-D and 2-D Brownian motion using Python's numpy, matplotlib, scipy, and random libraries.

Overview

The Notebook covers the following topics:

• 1-D Brownian Motion:

  • Simulates a 1-D random walk.
  • The process stops when it reaches the absorption point at 50.
  • The resulting path is plotted to visualize the trajectory.

• 2-D Brownian Motion (Initial Domain [-50, 50]):

  • Simulates a 2-D random walk where both the domain and range are defined as [-50, 50].
  • The process is designed to stop once it reaches the absorption point at (50, 0).
  • A single run is plotted to show the path taken in 2-D space.

• Multiple Runs for 2-D Simulation:

  • Performs 100 runs and 1000 runs of the 2-D Brownian motion.
  • Histograms of the endpoint distributions are plotted.
  • Key statistics—mean, standard deviation, median, mode, and variance—are calculated to assess the behavior of the system.

• Modified Domain in 2-D Simulation:

  • The simulation domain is adjusted from [-50, 50] to [-25, 25].
  • The Notebook includes plots to demonstrate how changing the domain affects the random walk behavior.

Getting Started

Prerequisites

Ensure you have the following:

• Python 3.7+
• Jupyter Notebook or JupyterLab

The Notebook requires the following Python libraries:

• numpy
• matplotlib
• scipy
• Python’s built-in random module

You can install the necessary libraries using pip:

pip install numpy matplotlib scipy

Running the Notebook

1. Clone the repository:

   git clone https://github.com/ian0671/Brownian-Motion.git
   cd brownian-motion-notebook

2. Launch Jupyter Notebook or JupyterLab:

   jupyter notebook

   or

   jupyter lab

3. Open the Notebook: Open the file brownian motion.ipynb to start exploring and running the simulations interactively.

Notebook Structure

The Notebook is organized into the following sections:

1. Introduction:
   Provides background on Brownian motion and outlines the objectives of the simulations.

2. 1-D Brownian Motion Simulation:

   • Code to simulate a 1-D random walk.
   • The simulation stops once the process reaches an absorption point at 50.
   • A plot is generated to visualize the path.

3. 2-D Brownian Motion Simulation (Single Run):

   • Simulates the 2-D Brownian motion in a domain of [-50, 50].
   • Stops when the process reaches the absorption point at (50, 0).
   • A plot shows the resulting 2-D trajectory.

4. Statistical Analysis with Multiple Runs:

   • The Notebook performs 100 runs and 1000 runs of the 2-D simulation.
   • Histograms of endpoints (or other designated metrics) are generated.
   • Statistical measures—mean, standard deviation, median, mode, and variance—are extracted from the simulation data, offering insights into the distribution.

5. Modified Domain Analysis:

   • Adjusts the 2-D simulation domain from [-50, 50] to [-25, 25].
   • Includes updated plots to compare the behavior under the new domain.

Results

Upon running the Notebook, you will obtain:

• A visualization of the 1-D Brownian motion with an absorption point at 50.
• A detailed 2-D simulation where the process stops at (50, 0) for a single-run scenario.
• Histograms and computed statistics (mean, standard deviation, median, mode, variance) for 100 and 1000 runs of the 2-D simulations.
• Comparative visualizations showing the effect of reducing the simulation domain on the 2-D random walk.

License

This project is licensed under the MIT License. See the LICENSE file for more details.

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