Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 36A (Statistical Physics)

The microcanonical ensemble describes an isolated system at fixed energy $E$, volume $V$, and particle number $N$, assigning equal weight to accessible states in a narrow energy shell.

The canonical ensemble describes a system that can exchange energy with a heat reservoir at fixed temperature $T$, while $V$ and $N$ remain fixed. A microstate $s$ has probability

The grand canonical ensemble allows exchange of both energy and particles at fixed $T$, $V$, and chemical potential $\mu$. Its state weights are

The choice is irrelevant for bulk thermodynamic observables in the thermodynamic limit when the system is additive, has sufficiently short-range interactions, and lies away from singular phase-coexistence regimes. Then relative energy and particle-number fluctuations vanish as the system size grows. These are the standard conditions for equivalence of statistical ensembles.

The classical canonical partition function uses one phase-space cell $h^2$ per state. In polar coordinates,Equivalently, using $h=2\pi \hslash$,This is the particle in a finite two-dimensional harmonic trap partition function.

PutThe thermodynamic derivatives of the canonical partition function giveThe two momentum coordinates are unrestricted quadratic degrees of freedom, so the equipartition theorem gives mean kinetic energy $1/\beta$. HenceThe same result follows from $⟨V⟩=-(\kappa /\beta ){\partial }_{\kappa }\log Z_1$.

Ifthen $e^{\beta a}$ is large and the wall correction is exponentially small:This is exactly the equipartition theorem result from the two kinetic and two harmonic-potential quadratic terms.

If $a\ll k_{B}T$, useThenThe leading term is the kinetic energy. The particle is nearly uniform in the disk, so $⟨r^2⟩\to R^2/2$ and $⟨V⟩\to \kappa R^2/4$, much smaller than $k_{B}T$. The hard wall therefore suppresses the positional equipartition contribution, as in the hard-wall correction to harmonic equipartition.

For $N$ identical, noninteracting classical particles, the one-particle phase-space integrals factorize and indistinguishability supplies the Gibbs factor. Thus the classical ideal-gas partition function rule gives