7.1 Definition

Consider an isolated system of \(N\) particles with energy \(E\) in a volume \(V\). By definition it exchanges neither particles nor energy with its surroundings. The microcanonical ensemble is the theoretical tool for describing such a system: every member of the ensemble has the same \(N\), \(V\) and \(E\). Schematically, the ensemble can be pictured as an array of cells, each cell a microstate with the same energy, volume and number of particles (Figure 7.1).

Uniform distribution of microstates. In thermal equilibrium the phase-space distribution function \(\rho(q,p)\) does not change with time. If \(\rho\) depends on \((q,p)\) only through the Hamiltonian, \(\rho=\rho(H(q,p))\), then

\[\frac{d\rho}{dt}=\frac{\partial\rho}{\partial H}\dot H=0,\]

because \(H\) is conserved. The simplest choice consistent with the ergodic hypothesis and the postulate of equal a priori probabilities is a constant \(\rho\) on the energy surface.

Energy shell. Since \(E\) can only be known to within some \(\Delta\), take a thin shell \(E<H<E+\Delta\):

\[\rho(q,p)=\begin{cases}\dfrac{1}{\Gamma(E,V,N)}, & E<H(q,p)<E+\Delta\\[4pt] 0, & \text{otherwise}\end{cases}\qquad \Gamma(E,V,N)=\int_{E<H<E+\Delta}d^{3N}q\,d^{3N}p .\]

Here \(\Gamma\) is the phase-space volume of the shell between the energy surfaces \(E\) and \(E+\Delta\); its dependence on \(V\) comes from the limits of the coordinate integrals.