Microcanonical Ensemble

The microcanonical ensemble describes an isolated system with fixed total energy $E$Particle number $N$And volume $V$.

14.1 Density of States

The number of microstates with energy between $E$ and $E + \delta E$ is:

$\Omega(E, V, N) = \int_{E < \mathcal{H} < E + \delta E} \frac{d^{3N}q\, d^{3N}p}{N!h^{3N}}$

The entropy (Boltzmann entropy):

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

The temperature is defined via:

$\frac{1}{T} = \frac{\partial S}{\partial E}$

14.2 The Ideal Gas in the Microcanonical Ensemble

For $N$ non-interacting particles in volume $V$ with total energy $E$:

$\Omega = \frac{V^N}{N!}\frac{(2\pi m E)^{3N/2}}{E\, \Gamma(3N/2)\, h^{3N}} \cdot \frac{\delta E}{E}$

Using Stirling”s approximation and the large-argument expansion of the Gamma function:

$S = Nk_B\left[\ln\!\left(\frac{V}{N}\right) + \frac{3}{2}\ln\!\left(\frac{4\pi m E}{3Nh^2}\right) + \frac{5}{2}\right]$

This is the Sackur—Tetrode equation, identical to the canonical ensemble result (as expected by ensemble equivalence).

From $1/T = \partial S/\partial E$:

$E = \frac{3}{2}Nk_B T$

Reproducing the equipartition theorem.

14.3 Classical Virial Theorem

For a system with Hamiltonian $\mathcal{H} = \sum_i p_i^2/(2m_i) + U(\mathbf{r}_1, \ldots, \mathbf{r}_N)$:

$\left\langle \sum_i \mathbf{p}_i \cdot \frac{\partial \mathcal{H}}{\partial \mathbf{p}_i} \right\rangle = 3Nk_B T$

$\left\langle \sum_i \mathbf{r}_i \cdot \frac{\partial \mathcal{H}}{\partial \mathbf{r}_i} \right\rangle = -3Nk_B T$

For a power-law potential $U \propto r^n$This gives:

$\langle K \rangle = \frac{n}{2}\langle U \rangle$

(For the harmonic oscillator, $n = 2$: $\langle K \rangle = \langle U \rangle$.)