Count physical states in phase space

A point in the phase space of N particles contains three positions and three momenta per particle. Integrating over that space gives units of action to the power three N, not a pure number. A logarithm used for entropy or free energy must have a dimensionless argument. Dividing by Planck's constant to the same power supplies a semiclassical counting cell. It does not assert that a classical particle follows a grid of discrete positions and momenta.

For identical particles, assigning labels to otherwise identical trajectories counts permutations repeatedly. In the dilute classical limit the symmetric Hamiltonian integral is therefore divided by N factorial. This correction is distinct from the phase-space cell. For several species, divide by a factorial for each identical species rather than the factorial of their combined population. Particles bound to distinct identifiable sites require a different counting problem; the gas factorial must not be inserted into every collection of degrees of freedom.

For a noninteracting monatomic gas, every position integral gives the box volume and every momentum component gives a Gaussian integral. The one-particle partition function is volume divided by thermal wavelength cubed. Raising it to N counts independent labeled contributions; dividing by N factorial then accounts for identical particles. The wavelength packages mass, temperature, and the phase-space normalization. It is a statistical scale, not the wavelength of every individual particle in a thermal sample.

The dimensionless density times thermal wavelength cubed measures whether exchange effects are small. If internal degeneracy supplies several equally accessible states, the relevant occupancy per internal state is reduced by that degeneracy. Dilution makes Maxwell–Boltzmann statistics accurate; it does not turn identical atoms into fundamentally distinguishable quantum particles. When occupancies are appreciable, a single global factorial does not reproduce all Bose enhancement or Pauli exclusion effects, and the quantum occupation counting must replace the dilute approximation.

Using Stirling's approximation on the factorial makes ideal-gas free energy extensive when volume and particle number grow together at fixed density. Omitting that factorial leaves an artificial entropy of mixing for equal gases and a chemical potential with the wrong size dependence. Pressure at fixed particle number can nevertheless look correct because the missing factorial has no volume derivative. A successful equation of state alone is therefore an incomplete audit of the state-counting convention.

A useful spatial counting check divides a classical uniform ideal gas into two equal observation regions. For a fixed total number, the probability of a given count in the left region is binomial. The combinatorial factor counts choices of positions without requiring experimentally trackable atom labels. This gives a concrete lab connection: the probability instrument computes the exact count distribution for independent positions, while Planck cells, momentum integrals, and quantum exchange corrections remain outside that model.