Microcanonical Ensemble | Physics

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 (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.

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    A[14_Microcanonical Ensemble] --> B[Key Concepts]
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    D --> G[Real-world usage]

The microcanonical ensemble describes an isolated system with fixed energy, volume, and particle number. The central idea is that all accessible microstates are equally probable, and the number of these microstates determines the entropy through Boltzmann’s formula. Temperature emerges as the rate of change of entropy with energy: adding energy to a system with many available microstates raises the entropy slowly, giving a high temperature. The entropy is maximized at equilibrium because that is the macrostate compatible with the most microstates. For an ideal gas, the Sackur-Tetrode equation shows that entropy increases with volume and energy, capturing the logarithmic counting of phase space volumes.

Mistake 1: Confusing entropy with multiplicity Entropy $S = k_B \ln \Omega$ is a logarithmic function of the multiplicity $\Omega$. A system with twice the multiplicity does not have twice the entropy; it has $S + k_B \ln 2$. Students often treat entropy and multiplicity as interchangeable, but they differ by a logarithm and a constant. The entropy is what appears in thermodynamic relations, not the multiplicity directly.

Mistake 2: Assuming microcanonical and canonical ensembles always give the same results The ensembles are equivalent only in the thermodynamic limit ($N \to \infty$, $V \to \infty$, $N/V$ fixed). For small systems, the canonical ensemble (which allows energy exchange with a reservoir) can give different predictions than the microcanonical ensemble (fixed energy). The relative energy fluctuation in the canonical ensemble scales as $1/\sqrt{N}$, which is negligible for macroscopic systems but not for small ones.

Mistake 3: Forgetting the $1/N!$ factor in the classical partition function For indistinguishable particles, the phase space volume must be divided by $N!$ to avoid overcounting microstates that differ only by particle labels. Omitting this factor leads to the Gibbs paradox: the entropy of mixing two identical gases would be nonzero, which is physically incorrect. The $1/N!$ factor ensures extensivity of the entropy.

• Statistical Mechanics: The canonical ensemble is the fixed-temperature counterpart to the microcanonical ensemble, with equivalent predictions in the thermodynamic limit.

• The Laws of Thermodynamics: The microcanonical entropy $S = k_B \ln \Omega$ provides the microscopic foundation for the second law of thermodynamics.

• The Grand Canonical Ensemble: The grand canonical ensemble further extends the framework to systems that exchange both energy and particles with a reservoir.

• Calculus

• Linear Algebra

• Vector Calculus