Random Walk and Brownian Motion Simulation on the Sphere

This project provides a simulation framework for modeling various types of random walks, including Brownian motion of a point on the sphere and the random walk of a coordinate system on the sphre. Additional configurable parameters such as walls and different observables are implemented. The simulation supports dynamic configurations via a TOML file, allowing users to easily control the simulation parameters and output settings.

Download the project

Clone the git repo:

git clone https://github.com/romainljsimon/sphere_walk

Then install it with pip in the directory where you cloned the repo. Thus you will be able to launch the program in every directory of your computer.

pip install .

Requirements

• Python 3.x
• NumPy: For numerical computations
• Numba: For JIT compilation to optimize performance
• PyArrow: For handling Parquet file storage
• TOML: To load configuration from a config.toml file
• Julia: To solve numerically SDEs

Install dependencies with:

pip install numpy numba pyarrow toml

Project Structure

• z_functions.py: Defines mathematical functions used as observables in the simulation, including trigonometric, polynomial, and step functions. These observables track and analyze specific properties of the walker's position at each time step.

• brownian_motion.py: Implements a PointWalk class for simulating 2D Brownian motion with optional wall constraints. The walker’s movement and boundary interactions are determined based on user-specified parameters.

• random_walk.py: Contains the XYZWalk class, designed for 3D random walks with customizable rotation and boundary configurations. The random walk can be configured for unrestricted or bounded movement within a specified space.

Configuration

The config.toml file specifies parameters for the simulation. Required fields include:

• n_walkers: Number of walkers to simulate.
• walls: Boolean flag to enable boundary constraints.
• type: Simulation type, either "brownian" or "walk".

Additional fields depend on the chosen simulation type:

For brownian:

• tmax: Maximum time for simulation.
• dt: Time step increment.
• a: Boundary constraint parameter (if walls is enabled).

For walk:

• n_iter: Number of iterations.
• theta: Angle for rotation in each step.

Usage

To run the simulation, ensure you have a properly configured config.toml file in the working directory. Then, execute:

python -m sphere_walk

The simulation performs a specified number of iterations for each walker, storing data snapshots as Parquet files in the out/ directory.

Output

The output files are stored in Parquet format in the out/ folder. Each file corresponds to a specific time step and includes the positions and observable values for all walkers. Metadata, such as the current time, is stored in each Parquet file for easy post-processing and analysis.

Example Configuration (config.toml)

n_walkers = 100
walls = true
type = 'walk'
n_iter = 100
theta = 0.2

License

This project is licensed under the MIT License.