Ludwig Boltzmann Statistical Mechanics & Energy Distribution Simulation

1. 2D Hard-Sphere Molecular Chamber Real-Time Kinetic Motion

Low Energy ($E < \langle E \rangle$) Moderate Energy ($\approx \langle E \rangle$) High Energy Tail ($E > 2\langle E \rangle$)

2. Live Energy Population Distribution Continuous Binned Counts vs. Theoretical Curve

Observed Bins $N(E)$ Boltzmann Curve: $N(E) \propto e^{-E/k_B T}$

Physical Principles of Ludwig Boltzmann’s Distribution (1877)

• Probability Density: P(E) = (1 / k_B T) · e^(-E / k_B T)
• Boltzmann Multiplicity: S = k_B · ln(Ω)
• Conservation Law: Total Energy E_total = ∑ (1/2 · m · v_i²) = Constant

Why Lower Energies Always Dominate: For a single particle to reach an extreme kinetic energy state ($E \gg k_B T$), many other surrounding particles must surrender their momentum and slow down. Because there are vastly fewer microscopic configurations (microstates $\Omega$) capable of concentrating energy into a few chosen particles, the probability decays exponentially.
Fast Relaxation of High-Energy Bursts: When you click "Inject 30 Fast Particles", watch the histogram. The sudden high-energy peak vanishes within seconds as fast particles undergo frequent collisions, dispersing their excess momentum and returning the ensemble to the classic Boltzmann curve.