The Microcanonical Ensemble and Statistical Entropy

An isolated system exchanges neither energy nor matter with its surroundings. Its total energy $E$, volume $V$, and particle number $N$ are constants of the motion, and every microscopic configuration the system visits is confined to the surface in phase space on which the Hamiltonian equals $E$. The microcanonical ensemble is the probability distribution appropriate to this situation: a flat distribution over that surface, expressing the one postulate on which equilibrium statistical mechanics rests. From the flat distribution and a count of how much phase space it covers, the entropy follows, and with it the entire thermodynamics of the isolated system.

The fundamental postulate

A classical system of $N$ particles has a microstate specified by the $3N$ canonical coordinates $q=(q_1,\dots ,q_{3N})$ and the $3N$ conjugate momenta $p=(p_1,\dots ,p_{3N})$. This point $(q,p)$ lives in the $6N$-dimensional phase space $\Gamma$, and it moves under Hamilton's equations along a trajectory confined to the constant-energy surface $H(q,p)=E$. For an isolated system the only information available is that the state lies somewhere on the accessible part of this surface. Equilibrium statistical mechanics posits that no accessible microstate is preferred over any other.

The postulate is not derived from mechanics; it is the statistical input that makes the theory predictive.¹ Its justification is partly the ergodic expectation that a trajectory spends equal time in equal accessible volumes, and mostly the overwhelming empirical success of the results it generates. Because the constraint $H=E$ is one condition on a $6N$-dimensional space, the accessible set is a surface of dimension $6N-1$. Measurements never resolve energy to a mathematical point, and a surface has zero volume, so it is convenient to admit a thin shell of energies between $E$ and $E+\delta E$ with $\delta E\ll E$. The final thermodynamics will not depend on $\delta E$.

Because $\rho$ depends on $(q,p)$ only through the conserved energy $H$, it is stationary under the Hamiltonian flow: $\partial \rho /\partial t=0$ by Liouville's theorem. A distribution built on a constant of the motion describes an equilibrium that does not evolve, which is the property a candidate equilibrium ensemble must have.

Phase-space volume, surface, and shell count

Three closely related measures of the accessible phase space appear throughout the subject. Each is made dimensionless by dividing the raw coordinate–momentum volume by $h^{3N}$, and made correct for identical particles by dividing by $N!$; both factors are justified below.

The delta-function form of $\omega (E)$ follows from differentiating the step function $\Theta (E-H)$ inside the integral for $\Gamma$. Physically $\omega (E)$ counts microstates per unit energy at energy $E$, and $\Omega (E)$ counts the microstates actually accessible to a system whose energy is known to lie within $\delta E$. The normalization constant in the microcanonical density is exactly this shell count.

For a typical system the enclosed volume is a steeply rising power of the energy. Writing $\Gamma (E)\propto E^{\alpha N}$ with $\alpha$ a number of order unity — the monatomic ideal gas has $\alpha =3/2$, since $3N/2$ momentum dimensions each contribute a half-power — gives

The density of states is the enclosed volume multiplied by the enormous factor $\alpha N/E$, so $\omega$ rises even more steeply with energy than $\Gamma$ does.

Why the three measures share a logarithm

The choice among $\Gamma$, $\omega E$, and $\Omega =\omega \delta E$ looks arbitrary, and the arbitrariness matters because entropy will be a logarithm of one of them. The resolution is that for large $N$ the three logarithms differ by terms of order $\ln N$, which are negligible beside a logarithm of order $N$.

Take $\Gamma (E)\propto E^{\alpha N}$, so $\ln \Gamma =\alpha N\ln E+\text{const}$, a quantity of order $N$. Then

The first correction is $\ln (\alpha N)\sim \ln N$, and the second is a finite number set by the experimental energy resolution. Dividing by $N$,

For $N\sim 10^{22}$ the correction is smaller than the leading term by a factor of about $10^{21}$. The same estimate applies to $\omega E$. All three prescriptions therefore give the same entropy per particle in the thermodynamic limit, and the value of $\delta E$ drops out. This insensitivity is what lets the microcanonical count be defined loosely and still yield sharp thermodynamics.

The geometric content is that almost all of the volume of a high-dimensional body sits in a thin skin near its surface. In $6N$ dimensions the shell of relative thickness $\delta E/E$ holds essentially the entire enclosed volume, because the volume of a $D$-ball scales as $R^D$ and the fraction within a skin of relative thickness $ϵ$ is $1-(1-ϵ{)}^D\to 1$ as $D\to \infty$. Volume and shell count coincide not by coincidence but because dimension is astronomically large.

The Boltzmann entropy

With the accessible count in hand, the entropy of the isolated system is defined as its logarithm.

By the equivalence just established, $S=k\ln \Gamma$ and $S=k\ln (\omega E)$ give the same value to leading order in $N$, so the definition is unambiguous. Two properties make this the right microscopic object to call entropy.

• Additivity. For two independent subsystems the number of joint microstates is the product $\Omega ={\Omega }_1{\Omega }_2$, because any microstate of the first can be combined with any microstate of the second. The logarithm turns the product into a sum, so $S=k\ln ({\Omega }_1{\Omega }_2)=S_1+S_2$. Entropy is extensive across independent parts precisely because it is a logarithm of a count.
• The second law as counting. When an internal constraint is relaxed — a partition removed, a chemical reaction allowed — the system explores a larger accessible region, so $\Omega$ can only grow or stay the same, and $S$ never decreases. The approach to equilibrium is the drift toward the macrostate of largest multiplicity, which the next lesson makes quantitative.

The Boltzmann constant $k$ is a units conversion, fixing the size of the entropy quantum so that $S=k\ln \Omega$ matches the thermodynamic entropy measured in $\mathrm{J}{\mathrm{K}}^{-1}$. Were entropy measured in bits, $k$ would be replaced by $1/\ln 2$ and the formula would read $S={\log }_2\Omega$, the Shannon content of a uniform distribution over $\Omega$ outcomes.

The measure factors and extensivity

The raw integral over coordinates and momenta carries dimensions of $(\text{action}{)}^{3N}$ and treats the particles as labelled. Two factors correct both defects, and though their full justification is quantum, they are fixed here by the requirements the classical theory must meet.

• The $h^{3N}$ factor divides the phase-space volume by Planck's constant once per conjugate coordinate–momentum pair, rendering $\Gamma$, $\omega$, and $\Omega$ pure numbers. Its value $h=6.626070\times 10^{-34}\mathrm{J}\mathrm{s}$ sets the size of the elementary phase-space cell $\Delta q\Delta p\sim h$ demanded by the uncertainty principle: a classical cell finer than $h$ has no quantum meaning, so states are counted in units of $h$ per degree of freedom.³ Any other constant with the units of action would shift $\ln \Omega$ by an $N$-dependent constant and alter the entropy by an additive term; matching the classical entropy to the quantum count of states in a box fixes the constant to be $h$.
• The $N!$ factor divides out the permutations of identical particles. Labelled counting treats an exchange of two identical atoms as a distinct microstate, overcounting each physical configuration by the $N!$ ways of permuting the labels. Dividing by $N!$ is correct Boltzmann counting. Without it the entropy of a classical ideal gas fails to be extensive and produces a spurious entropy of mixing for identical gases — the Gibbs paradox, taken up when the Sackur–Tetrode entropy is derived.

Extensivity is the concrete test. A thermodynamic entropy must satisfy $S(\lambda E,\lambda V,\lambda N)=\lambda S(E,V,N)$: doubling the system doubles the entropy. The enclosed volume of an ideal gas without the $N!$ is $\Gamma \propto V^N{E}^{3N/2}$, whose logarithm contains $N\ln V$, a term that grows faster than linearly in $N$ at fixed density $V/N$ and so is not extensive. Inserting $1/N!$ and using Stirling's approximation $\ln N!\approx N\ln N-N$ converts $N\ln V$ into $N\ln (V/N)$, a function of the intensive density that scales linearly in $N$. The two measure factors are what make $S=k\ln \Omega$ a genuine thermodynamic entropy rather than a labelled counting artifact.

The microcanonical recipe

The microcanonical ensemble reduces the thermodynamics of an isolated system to a counting problem with a fixed sequence of steps.

Step 4 is the content of the next lesson: the derivatives of the entropy with respect to its natural variables $E$, $V$, and $N$ define temperature, pressure, and chemical potential, and reproduce the fundamental relation of thermodynamics. The microcanonical program is thus complete in principle — every equilibrium property of an isolated system is a derivative of a single counting function — even where the counting integral is too hard to do in closed form.

Summary

• The fundamental postulate assigns equal probability to every microstate on the energy shell $E\le H\le E+\delta E$ of an isolated system, giving the uniform microcanonical density $\rho =1/\Omega$ inside the shell.
• Three measures of accessible phase space — the enclosed volume $\Gamma (E)$, the density of states $\omega (E)=\mathbf{d}\Gamma /\mathbf{d}E$, and the shell count $\Omega =\omega \delta E$ — have logarithms that agree to $O(\ln N)$, so the entropy is insensitive to which is chosen and to the value of $\delta E$.
• The Boltzmann entropy $S=k\ln \Omega$ is additive across independent systems and non-decreasing when constraints are relaxed, recovering the second law as the growth of multiplicity.
• The measure factors $h^{3N}$ (dimensionless counting, one cell of size $h$ per degree of freedom) and $N!$ (indistinguishability) make $S$ extensive; without the $N!$ the ideal-gas entropy is non-extensive and the Gibbs paradox appears.