Statistical Mechanics · undergrad

Canonical Ensemble

The equilibrium ensemble at fixed temperature, volume, and particle number.

Published Apr 26, 2026 · 1 min

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Definition

The canonical ensemble fixes $T,V,N$ and assigns probabilities

p_i=\frac{e^{-\beta E_i}}{Z}.

The Helmholtz free energy is

F=-k_BT\ln Z.

The average energy is obtained from

\langle E\rangle=-\frac{\partial}{\partial\beta}\ln Z.

Energy fluctuates, but relative fluctuations become small for macroscopic systems.

The equilibrium probability weight for states in contact with a heat bath.

The response of internal energy or enthalpy to temperature change.

The statistical object from which thermodynamic quantities are derived.