The Microcanonical Ensemble and the Postulate of Equal a Priori Probabilities

The microcanonical ensemble is the conceptual foundation of the entire subject, but it is also a live computational tool. Molecular-dynamics simulations, which integrate Newton's equations for thousands to millions of particles, run in the microcanonical (NVE) ensemble by construction — energy is conserved, so the trajectory samples a constant-energy shell, and thermodynamic properties are extracted exactly as this article describes, by counting or time-averaging over accessible states. The NVE ensemble is the baseline against which thermostatted (canonical) simulations are validated. The postulate of equal a priori probabilities also underlies modern discussions of thermalization and the foundations of statistical mechanics. The eigenstate thermalization hypothesis, a central idea in quantum many-body physics, is essentially a quantum statement of why an isolated system reaches microcanonical equilibrium — why a single quantum state can look thermal. Experiments with isolated ultracold-atom systems now test these foundations directly, watching whether and how an isolated quantum gas equilibrates to the microcanonical prediction, and probing exotic cases (many-body localization) where it fails to. And the sharpness result — that equilibrium fluctuations vanish as $1/\sqrt{N}$ — is why thermodynamics, a theory of a handful of macroscopic variables, is so astonishingly accurate for macroscopic matter: the microcanonical count concentrates overwhelmingly on the equilibrium macrostate.

Start with the hardest possible problem: an isolated gas of $10^{23}$ molecules, its microstate a single point wandering forever on a $6N$-dimensional constant-energy surface, following Newton's (or Schrödinger's) equations exactly. Predicting that trajectory is hopeless. The microcanonical ensemble sidesteps it entirely with one assumption: over long times the system visits all accessible microstates with equal frequency, so we may replace the time-average of any quantity with a simple average over all accessible microstates, weighted equally. Dynamics is traded for democracy. Why "equal"? Because there is no reason to prefer any accessible microstate over any other — they all have the same energy, and the underlying dynamics (Liouville's theorem) preserves phase-space volume, so a uniform distribution stays uniform. This is the postulate of equal a priori probabilities. It is an assumption, tested by the spectacular success of everything built on it, and it turns statistical mechanics into an exercise in counting.

Once you accept the postulate, the entropy is simply $S = k_B\ln\Omega$: the logarithm of how many microstates the macrostate contains. The logarithm is what makes entropy additive (two independent systems have $\Omega_{12} = \Omega_1\Omega_2$, so $S_{12} = S_1 + S_2$), and $k_B$ sets the units. Now the second law becomes a triviality: release a constraint, and the number of accessible states can only grow, so $S$ can only increase. Equilibrium is the macrostate with the most microstates. The final, staggering fact is how dominant that equilibrium macrostate is. Split an isolated system in two and ask how energy divides between the halves. The number of microstates is peaked at the equal-temperature split, and the peak is fantastically sharp — its relative width shrinks as $1/\sqrt{N}$. The plot below shows this for increasing $N$: with $N = 10^{23}$ the distribution of the energy split is a spike of relative width $\sim 10^{-12}$, which is why an isolated system in equilibrium never spontaneously develops a hot half and a cold half.

The postulate of equal a priori probabilities assigns each accessible microstate the same probability:

where: $\Omega(E,V,N)$ — the number of microstates with energy in $[E, E+\delta E]$, volume $V$, particle number $N$; all others have $p_i = 0$. This uniform distribution is the microcanonical ensemble. The entropy is the Boltzmann relation — the logarithm of the count:

where: $k_B = 1.381\times10^{-23}\ \mathrm{J/K}$ — Boltzmann's constant. The logarithm makes $S$ extensive; equivalently $S = -k_B\sum_i p_i\ln p_i$ (Gibbs entropy) evaluated on the uniform distribution gives exactly $k_B\ln\Omega$. For a classical ideal gas the count is a geometric object — a thin shell in $6N$-dimensional phase space:

where: the $V^N$ counts spatial configurations, the $E^{3N/2}$ is the volume of a $3N$-dimensional momentum ball of radius $\sqrt{2mE}$, $N!$ is the Gibbs indistinguishability correction, and $h^{3N}$ makes the count dimensionless. $g(E)$ is the density of states. Differentiating the entropy with respect to energy gives the temperature, and the ideal-gas count regenerates equipartition and the equation of state:

where: the energy slope of $\ln\Omega \sim \tfrac{3N}{2}\ln E$ gives $1/T$, and inverting yields equipartition — $\tfrac12 k_BT$ per quadratic degree of freedom — derived from counting alone, with no dynamics. The sharpness of equilibrium is captured by the relative width of the energy-sharing distribution:

where: for two subsystems sharing energy, the most probable split is at equal temperature, and fluctuations around it scale as $1/\sqrt{N}$ — utterly negligible for macroscopic $N$.

How the microcanonical ensemble behaves:

1 · Counting microstates of a small system, then taking N large (confidence builder)

Consider $N$ non-interacting spin-$\tfrac12$ particles in a magnetic field, each with energy $0$ (down) or $\epsilon$ (up), total energy $E = n\epsilon$ with $n$ spins up. Count the microstates, find the entropy, and compute the temperature — then check the equal-a-priori peak for $N = 100$.

Step 1. Count: the number of ways to choose $n$ up-spins from $N$ is $\Omega(n) = \dbinom{N}{n} = \dfrac{N!}{n!(N-n)!}$. This is the microcanonical count at energy $E = n\epsilon$ — a direct enumeration, no dynamics.

Step 2. Entropy (Stirling for large $N$): $S/k_B = \ln\Omega = N\ln N - n\ln n - (N-n)\ln(N-n)$. For $N = 100$, $n = 50$: $\Omega = \binom{100}{50} \approx 1.01\times10^{29}$, so $S/k_B = \ln(1.01\times10^{29}) = \mathbf{66.8}$.

Step 3. Temperature: $\dfrac1T = \dfrac{\partial S}{\partial E} = \dfrac{1}{\epsilon}\dfrac{\partial S}{\partial n} = \dfrac{k_B}{\epsilon}\ln\dfrac{N-n}{n}$. At half-filling ($n = N/2$) this vanishes, giving $T \to \infty$ — the maximum-entropy, hottest state; beyond it $T$ goes negative (population inversion), a signature of the bounded spectrum.

Step 4. Sharpness: the peak of $\binom{N}{n}$ is at $n = N/2$ with width $\sigma_n = \tfrac12\sqrt{N} = 5$ for $N = 100$. The relative width $\sigma_n/(N/2) = \sqrt{N}/N = 1/\sqrt{N} = \mathbf{0.1}$ — already narrow, and it would be $10^{-11}$ for a mole. The equilibrium (equal up/down) is the most probable macrostate, exactly as the postulate predicts.

2 · Equipartition and the temperature of a gas from counting (exam level)

For $1\ \mathrm{mol}$ of monatomic ideal gas with total energy $E = 3740\ \mathrm{J}$, use the microcanonical count $\Omega \propto V^N E^{3N/2}$ to find the temperature, and verify the average energy per particle.

Step 1. Entropy energy-dependence: $S = k_B\big[\tfrac{3N}{2}\ln E + \dots\big]$, so $\dfrac1T = \dfrac{\partial S}{\partial E} = k_B\dfrac{3N}{2E}$. With $N = N_A$ and $E = 3740\ \mathrm{J}$: $\dfrac1T = \dfrac{3(8.314)}{2(3740)} = \dfrac{24.94}{7480} = 3.335\times10^{-3}\ \mathrm{K^{-1}}$.

Step 2. Temperature: $T = 1/(3.335\times10^{-3}) = \mathbf{300\ K}$. Purely from counting states — no thermometer, no assumption about kinetic energy, just the geometry of the phase-space shell.

Step 3. Equipartition: rearranging, $E = \tfrac32 Nk_BT = \tfrac32 RT$ per mole. Energy per particle: $E/N = \tfrac32 k_BT = \tfrac32(1.381\times10^{-23})(300) = \mathbf{6.21\times10^{-21}\ J}$, i.e. $\tfrac12 k_BT$ for each of the three momentum components.

Step 4. Consistency: the average kinetic energy per particle is $\tfrac32 k_BT = 6.21\times10^{-21}\ \mathrm{J}$, matching the direct kinetic-theory result $\tfrac12 m\langle v^2\rangle$. The microcanonical count, through $S = k_B\ln\Omega$, reproduces both the temperature and the energy per degree of freedom — equipartition emerges from counting, and the ideal-gas thermodynamics is complete.

References. Pathria & Beale, Statistical Mechanics, 3rd ed., ch. 1–2 (the microcanonical ensemble) · Reif, Fundamentals of Statistical and Thermal Physics, ch. 2–3 · Kardar, Statistical Physics of Particles, ch. 4 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, ch. 3 · Callen, Thermodynamics, ch. 15 · Gibbs, Elementary Principles in Statistical Mechanics (1902).

Provenance. The postulate of equal a priori probabilities, $S = k_B\ln\Omega$, and the ideal-gas count $\Omega \propto V^N E^{3N/2}$ are standard (Pathria & Beale ch. 1–2; Reif ch. 2). A verifier should reproduce $\Phi(E) \propto E^{3N/2}$, the equations of state $E = \tfrac32 Nk_BT$ and $PV = Nk_BT$ by differentiating $\ln\Omega$, and the $1/\sqrt{N}$ energy-split width (numerically $1.2\times10^{-12}$ at $N = 10^{24}$). Numerical instances: spin system $\binom{100}{50} \approx 1.01\times10^{29}$, $S/k_B = 66.8$; ideal-gas $T = 2E/3R = 300\ \mathrm{K}$ at $E = 3740\ \mathrm{J}$, energy per particle $\tfrac32 k_BT = 6.21\times10^{-21}\ \mathrm{J}$. All in verify_aq-sm-microcanonical-ensemble-equal-probabilities.py. $k_B = 1.381\times10^{-23}\ \mathrm{J/K}$.

Builds on: First Law of Thermodynamics and Internal Energy · Second Law of Thermodynamics and Entropy · Zeroth Law of Thermodynamics and Empirical Temperature · Temperature, Pressure, and Chemical Potential as Entropy Derivatives. Leads to: Boltzmann Entropy and the Density of States · Phase-Space Volume and the Liouville Measure · Equilibrium Conditions Between Subsystems · Canonical Ensemble.