Specific heat from energy fluctuations

In equilibrium statistical mechanics, the constant-volume heat capacity can be expressed directly in terms of equilibrium energy fluctuations. This is one of the most important examples of a fluctuation–response relation.

Setup

Work in the canonical ensemble at fixed volume $V$ and particle number $N$, with Hamiltonian (energy) $H$. Let $\beta$ be the inverse temperature and $k_B$ the Boltzmann constant .

The mean energy is the ensemble average $\langle H\rangle$. The constant-volume heat capacity (total, not per particle) is the thermodynamic derivative

matching the thermodynamic notion in $C_V$ .

Fluctuation formula

Let $Z(\beta)$ be the canonical partition function . Then

where $\mathrm{Var}(H)$ is the ensemble variance of the energy.

Using $\beta = 1/(k_B T)$, one finds

Equivalently,

This is a special case of the general machinery summarized in fluctuation formulas from $\log Z$ and observables from $\log Z$ .

Physical interpretation

• $C_V$ measures how strongly the mean energy reacts to a change in temperature .
• The identity above says: large energy fluctuations imply large heat capacity, and vice versa.
• In the canonical ensemble, $\mathrm{Var}(H)\ge 0$ forces $C_V\ge 0$. (Other ensembles, or systems with long-range interactions, can behave differently, but the canonical fluctuation identity itself is always nonnegative.)

For intensive versions, one often defines $c_V=C_V/N$ (per particle) or $C_V/V$ (per volume), depending on context.