Boltzmann entropy in the microcanonical ensemble

Definition (Boltzmann microcanonical entropy).
A classical microstate is a point $x=(q,p)$ in phase space , equipped with a Hamiltonian $H(x)$. Fix macroscopic constraints such as $(E,V,N)$ and consider the microcanonical energy shell (in practice, an energy window of width $\Delta E$). Using the phase-space volume element $d\Gamma$, define the (dimensionless) phase-space volume of accessible states by

where $d$ is the spatial dimension, and the factor $h^{dN}N!$ is the standard classical normalization (Planck cell and indistinguishability).

The Boltzmann entropy is

where $k_B$ is the Boltzmann constant .

Density-of-states form.
If one introduces the density of states

then for small $\Delta E$ one has $\Omega(E,\Delta E)\approx \omega(E)\,\Delta E$, so $S_B(E)$ is (up to an additive constant $k_B\ln\Delta E$) the logarithm of $\omega(E)$.

Physical interpretation.
$S_B$ measures “how many” microstates are compatible with a given macrostate constraint (here, fixed energy). The associated microcanonical measure is uniform over the energy shell/window, so $\Omega$ is literally the volume being sampled.

Thermodynamic structure.
In the thermodynamic limit , $S_B$ becomes extensive and agrees with thermodynamic entropy (for equilibrium states), while the dependence on the particular choice of $\Delta E$ drops out at the level of entropy density.

A key consequence is the microcanonical definition of temperature:

equivalently $\,\beta = (\partial S_B/\partial E)/k_B$, where $\beta$ is inverse temperature and the construction is summarized in temperature from entropy .