Boltzmann Distribution Lab

Interactive laboratory to verify Maxwell-Boltzmann statistics through particle velocity simulations with SQL database integration.

Physics Background

The Maxwell-Boltzmann distribution describes the velocity/speed distribution of particles in an ideal gas at thermal equilibrium. It's a fundamental result in statistical mechanics that connects microscopic particle motion to macroscopic temperature.

Key Results:

• Speed distribution: f(v) ∝ v² exp(-mv²/2kT)
• Most probable speed: v_p = √(2kT/m)
• Mean speed: = √(8kT/πm)
• RMS speed: v_rms = √(3kT/m)

Progress

Day 1 ✓

• ✓ Implemented particle system with velocity initialization
• ✓ Theoretical Maxwell-Boltzmann distributions (speed & energy)
• ✓ Visualization comparing simulation vs theory
• ✓ Statistical analysis tools

Day 2 ✓

• ✓ Complete SQLite database schema
• ✓ Database integration with particle system
• ✓ Storage of distributions, statistics, and metadata
• ✓ Advanced query capabilities
• ✓ Database visualization tools
• ✓ Temperature sweep functionality

Quick Start

# Install dependencies
pip install -r requirements.txt

# Run single test
python test_simulation.py

# Run database test (multiple temperatures)
python test_database.py

# Or use Jupyter notebook
jupyter notebook notebooks/demo.ipynb

Database Schema

Tables:

• simulations: Metadata (temperature, particles, timestamp)
• speed_distributions: Histogram data for speeds
• energy_distributions: Histogram data for energies
• statistics: Summary statistics (simulation vs theory)
• particles: Optional individual particle data

Example Queries:

from src.database import BoltzmannDatabase

with BoltzmannDatabase() as db:
    # Get all temperatures
    temps = db.get_all_temperatures()
    
    # Query by temperature range
    sims = db.get_simulations_by_temperature(250, 400)
    
    # Compare multiple temperatures
    comparison = db.compare_temperatures([300, 500, 1000])

Features

• ✓ Generate particle ensembles at any temperature
• ✓ Compare simulation with analytical theory
• ✓ Visualize speed and energy distributions
• ✓ Store unlimited simulation runs in SQLite database
• ✓ Query and compare different temperatures
• ✓ Statistical validation and error analysis
• ✓ Temperature sweep automation

Project Structure

boltzmann-distribution-lab/
├── src/
│   ├── particle_system.py        # Particle simulation
│   ├── boltzmann_theory.py       # Theoretical distributions
│   ├── visualization.py          # Basic plotting
│   ├── database.py              # SQLite database class
│   ├── simulation_runner.py     # High-level runner
│   └── database_visualization.py # Database-specific plots
├── data/
│   └── boltzmann.db             # SQLite database (created on first run)
├── notebooks/
│   └── demo.ipynb               # Interactive demo
├── test_simulation.py           # Single simulation test
└── test_database.py            # Database integration test

Next Steps

• Day 3: Interactive queries, chi-squared goodness-of-fit tests
• Day 4: Advanced analysis, phase space visualization
• Day 5: Documentation polish, final visualizations

Example Output

After running test_database.py, you'll have:

• A database with 6 temperature points (100K - 1000K)
• Beautiful overlaid distributions showing temperature dependence
• Statistical validation plots
• All data queryable via SQL

Requirements

• Python 3.8+
• NumPy
• Matplotlib
• SciPy