Microcanonical Ensemble

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

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:

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

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.

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, $n = 2$: $\langle K \rangle = \langle U \rangle$.)

In the thermodynamic limit ($N \to \infty$, $V \to \infty$, $N/V$ fixed), the microcanonical, canonical, and grand canonical ensembles produce identical thermodynamic predictions. This is a consequence of the fact that the energy fluctuations in the canonical ensemble scale as $\Delta E / E \sim 1/\sqrt{N}$, vanishing in the limit.

Proposition 14.1. For a system with Hamiltonian $\mathcal{H}$, the microcanonical entropy $S(E)$ and the canonical free energy $F(\beta) = -\beta^{-1} \ln Z(\beta)$ are related by the Legendre transform:

$F(\beta) = \inf_E [E - \beta^{-1} S(E)]$

Proposition 14.2 (Nernst’s Theorem). As $T \to 0$, the entropy of a system approaches a constant (typically zero for a non-degenerate ground state):

$\lim_{T \to 0} S(E, V, N) = k_B \ln g_0$

where $g_0$ is the degeneracy of the ground state.

In the microcanonical picture, at $T = 0$ the system occupies only the ground state microstate(s). If the ground state is unique, $\Omega = 1$ and $S = 0$.

Problem. Consider $N$ non-interacting spin-1/2 particles in a magnetic field $B$. Each spin has energy $\pm \mu B$. Find the microcanonical entropy and the equation of state.

Problem. Find the microcanonical entropy of an ideal gas confined to a two-dimensional area $A$ with $N$ particles and total energy $E$.

In the canonical ensemble, the energy fluctuates around its mean value $\langle E \rangle$. The variance is related to the heat capacity:

$\langle (\Delta E)^2 \rangle = k_B T^2 C_V$

The relative fluctuation $\sqrt{\langle (\Delta E)^2 \rangle}/\langle E \rangle \sim 1/\sqrt{N}$, vanishing in the thermodynamic limit. This justifies the equivalence of microcanonical and canonical ensembles for macroscopic systems.