Statistical mechanics counts microstates. Three principal ensembles:

• Microcanonical $(N,V,E)$: $S = k_B \ln\Omega$.
• Canonical $(N,V,T)$: $Z = \sum_n e^{-\beta E_n}$, $\beta = 1/k_BT$.
• Grand canonical $(\mu,V,T)$: $\Xi = \sum_N e^{\beta\mu N} Z_N$.

Helmholtz free energy $F = -k_BT \ln Z$, mean energy $\langle E\rangle = -\partial_\beta \ln Z$. Relative fluctuations vanish as $1/\sqrt N$, so all ensembles agree in the thermodynamic limit.

Interactive: Boltzmann distribution

Probability $p(E) \propto e^{-\beta E}$ for energy levels $E_n = n$. Adjust temperature to see the population shift.