Density of states

Consider a classical thermodynamic system modeled by a microstate $x$ ranging over a classical phase space $\Gamma$, equipped with the Liouville phase-space volume element $d\Gamma$. Let $H(x)$ denote the Hamiltonian (total energy).

Definition (classical)

The cumulative phase-space volume below energy $E$ is

When $\Omega$ is differentiable (or in a distributional sense), the density of states is

Equivalently, using the Dirac delta to localize on the energy surface,

Energy shells and counting interpretation

For a thin microcanonical shell of width $\Delta E>0$,

so $g(E)$ is the phase-space “number of microstates per unit energy” (up to the conventional normalization of $d\Gamma$).

Quantum version (spectral density)

For a quantum Hamiltonian $\hat H$ with discrete spectrum $\{E_n\}$ (finite volume), one often encodes the density of states as a measure

or, equivalently, by the counting function $N(E)=\#\{n: E_n\le E\}$, which plays the role of $\Omega(E)$.

Link to entropy and temperature

The density of states controls the microcanonical notion of entropy via the Boltzmann entropy . In a thin shell,

with Boltzmann's constant $k_B$.

Differentiating $S(E)$ with respect to energy yields the microcanonical inverse temperature, implemented in practice by extracting temperature from entropy and matched to the thermodynamic inverse temperature $\beta$ in the thermodynamic limit .