Microcanonical measure

Fix a classical phase space $\Gamma$ with Liouville volume element $d\Gamma$, and a Hamiltonian $H(x)$. For energy $E$ and window $\Delta E>0$, let $\Sigma_{E,\Delta E}$ be the microcanonical shell .

Definition (finite-thickness shell)

The microcanonical measure is the normalized restriction of Liouville measure to the shell:

where

is the shell volume. This makes $\mu_{E,\Delta E}$ a probability measure on $\Gamma$.

Equivalently, the microcanonical density with respect to $d\Gamma$ is

Definition (delta-function energy surface)

Formally, one can write a “surface” version concentrated on $H(x)=E$:

where $g(E)$ is the density of states ensuring normalization.

Ensemble averages

Given an observable $A:\Gamma\to\mathbb{R}$ (a function of the microstate ), its microcanonical ensemble average is

This is the same object as the expectation of $A$ under the probability distribution $\mu_{E,\Delta E}$.

Invariance and equilibrium meaning

Because Hamiltonian time evolution preserves Liouville volume (Liouville’s theorem) and conserves energy, the microcanonical measure is stationary under the dynamics restricted to the shell. In the equilibrium picture, it encodes “equal a priori probability” among accessible microstates at fixed energy.

Entropy connection

The associated microcanonical entropy is captured by the Boltzmann entropy

with Boltzmann's constant $k_B$.