Classical gases and equipartition

Take $N$ identical, nonrelativistic, structureless particles in a three-dimensional box, with negligible interaction energy. Their Hamiltonian is the sum of $\mathbf p_i^2/(2m)$. Each position integral contributes the volume. Each momentum component contributes a Gaussian integral. Dividing by $h^{3N}$ makes the semiclassical phase-space count dimensionless. Dividing by $N!$ removes overcounting from permuting identical particles; omitting it gives an unphysical entropy of mixing for identical dilute gases.

The resulting expression is the Maxwell–Boltzmann approximation to the quantum partition function. Its validity requires low phase-space density, $n\lambda_T^3\ll1$, and sufficiently weak interaction corrections, where $n=N/V$. It does not require particles to be distinguishable in principle: exchange effects must merely be negligible at the occupied densities and temperatures. Finite-size boundary corrections also become small when the box dimensions are large compared with the thermal wavelength. An external potential would replace the simple volume factor with a position-dependent integral.

Use $F=-k_BT\ln Z_N$. Differentiation with respect to volume at fixed temperature and particle number gives pressure, while differentiation of $\ln Z_N$ with respect to inverse temperature gives energy. The volume factor yields $pV=Nk_BT$. The temperature dependence of the three Gaussian momentum integrals yields $U=3Nk_BT/2$. These are complementary consequences of one partition function, rather than independent empirical substitutions pasted together.

The constant-volume heat capacity is $3Nk_B/2$ for this monatomic translational model. It is extensive because independent particle contributions add. Pressure is intensive: doubling both particle number and volume at fixed temperature leaves it unchanged. Finite particle number does not prevent writing these idealized derivatives, but macroscopic pressure measurement additionally averages collisions over a suitable area and time. Rarefied gases can have an equilibrium equation of state while failing the local-equilibrium assumptions needed for ordinary fluid dynamics.

For a classical equilibrium variable entering a quadratic energy with an effectively unrestricted integration range, its mean contribution is $k_BT/2$. One particle therefore has three translational contributions. A classical harmonic oscillator has both kinetic and potential quadratic terms, giving total mean energy $k_BT$. A molecule can also rotate and vibrate, but those motions have discrete quantum levels. A mode with a level spacing much larger than $k_BT$ is typically frozen near its ground state and does not contribute its classical heat capacity.

Anharmonic potentials, constrained coordinates, strong interactions, relativistic dispersion, and insufficient equilibration all demand renewed analysis. Changing the coefficient of a quadratic term changes its amplitude distribution, not its classical equipartition energy. Taking that statement to arbitrarily high-frequency field modes leads to the classical ultraviolet catastrophe, which quantum statistics resolves. The right lesson is to identify the Hamiltonian, allowed state space, and equilibrium regime before counting quadratic terms. A remembered degrees-of-freedom count cannot replace those physical checks.