Microcanonical entropy density

The entropy per unit volume (or per particle) in the microcanonical ensemble, derived from the density of states.

Microcanonical entropy density

In the microcanonical ensemble at energy $E$ , the microcanonical entropy is

S(E, V, N) = k_B \ln \Omega(E, V, N),

where $\Omega(E, V, N)$ is the number of microstates (or the phase space volume ) at energy $E$ .

Entropy density

The entropy density (entropy per unit volume) is

s = \frac{S}{V} = \frac{k_B}{V} \ln \Omega(E, V, N).

In the thermodynamic limit $V, N \to \infty$ with fixed density $\rho = N/V$ and energy density $u = E/V$ , one typically has

s(u, \rho) = \lim_{V \to \infty} \frac{S(uV, V, \rho V)}{V}.

The temperature is determined by

\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{V, N} = \frac{\partial s}{\partial u}.

This relates the microcanonical entropy to the canonical ensemble via Legendre transformation.