🎲 Gibbs Ensemble — Entropy & Microstates

Distribute energy quanta among quantum harmonic oscillators and count microstates. The simulation randomly samples configurations, builds an energy histogram, and shows how entropy S = k·ln W emerges from counting accessible microstates.

System Setup

Oscillators N 10 Energy quanta q 20

Oscillator energies

Click to reassign randomly

Statistics

Microstates W—

Entropy S/k—

Mean energy ⟨ε⟩—

Temperature kT/ε—

Samples collected0

W = (N+q-1)! / ((N-1)! · q!)
S = k·ln W
⟨ε⟩ = q/N · ε₀
P(ε) ∝ e^-ε/(kT)

Statistical Mechanics

In the microcanonical ensemble every microstate (specific distribution of energy quanta among N oscillators) has equal probability. The number of ways to distribute q indistinguishable quanta among N distinguishable oscillators is W = C(N+q−1, q) = (N+q−1)!/((N−1)!·q!). Boltzmann entropy S = k·ln W grows with both N and q. The most probable distribution — the one observed macroscopically — is the Boltzmann distribution P(ε_i) ∝ exp(−ε_i/kT), which emerges naturally from random sampling even without assuming it a priori. This simulation lets you verify that claim by collecting many random microstates and plotting the resulting energy histogram.