MBDistribution

This repository contains a numerical routine developed in Python and R for calculating and visualizing the Maxwell-Boltzmann velocity distribution of an ideal gas. The code generates the probability density function $f(v)$ for a given temperature and molar mass, extracting the characteristic velocities of the thermodynamic system.

Authorship

Victor Moreira Acacio

Institute of Astronomy, Geophysics and Atmospheric Sciences of the University of São Paulo

GitHub: @OAkacio

ORCID: 0009-0007-4484-2129

Installation

Clone this repository and install the Python dependencies by running the following commands in your terminal. Ensure that an R environment is also installed on your system to execute the plotting routine.

git clone https://github.com/OAkacio/maxwell-boltzmann-distribution.git
cd maxwell-boltzmann-distribution
pip install -r requirements.txt

Usage

The workflow of this integrator is divided into two sequential steps: numerical data generation via Python and high-resolution visualization via R.

1. Generate the distribution data: Execute the main engine to calculate the probability density and characteristic velocities.

python main.py

2. Plot the results: Run the R script to process the generated .txt data and export the standardized scientific plot.

Rscript plot_main.R

Note: To change the macroscopic variables such as Temperature ($Tc$) and Molar Mass ($M$), edit the parameters inside src/parameters.py before running the integration engine.

Theoretical Background

The mathematical foundation of this routine is based on the classical kinetic theory of gases. The code calculates the following physical quantities:

1. Probability Density Function ($f(v)$) The probability of a particle having a velocity $v$ at a thermodynamic temperature $T$ for a gas of molar mass $M$ is given by:

$$f(v) = 4\pi \left( \frac{M}{2\pi R T} \right)^{3/2} v^2 \exp\left( -\frac{M v^2}{2 R T} \right)$$

2. Most Probable Velocity ($v_{mp}$) The velocity possessed by the largest fraction of molecules in the gas, corresponding to the peak of the distribution curve:

$$v_{mp} = \sqrt{\frac{2 R T}{M}}$$

3. Mean Velocity ($v_{avg}$) The arithmetic average of the velocities of all molecules within the thermodynamic system:

$$v_{avg} = \sqrt{\frac{8 R T}{\pi M}}$$

4. Root-Mean-Square Velocity ($v_{rms}$) The square root of the average of the squared velocities, which is directly related to the average kinetic energy of the gas particles:

$$v_{rms} = \sqrt{\frac{3 R T}{M}}$$

Project Structure

├── data/           
│   ├── dados.txt      # Distribution curve points (v, f(v))
│   └── dados_car.txt  # Characteristic velocities (v_mp, v_avg, v_rms)                 # Generated data files (.txt)
├── figures/        # High-resolution distribution plots (.png)
├── src/            # Core modules (core.py, parameters.py, utils.py)
├── main.py         # Python numerical integration engine
├── plot_main.R     # R plotting and visualization script
└── requirements.txt

Motivation

This repository was developed as an auxiliary computational tool for the study of Thermodynamics and Statistical Mechanics. The central objective is to provide a reproducible numerical routine to visualize how the statistical distribution of particle velocities shifts according to macroscopic thermodynamic variables, offering a clear analytical visualization of ideal gas behavior.