Turn a state sum into an energy integral

In a large box with periodic boundaries, allowed wavevectors occupy a cubic grid with spacing two pi divided by side length. The number of states in a thin spherical shell is its momentum-space volume divided by the cell volume, multiplied by any independent internal degeneracy. For a free nonrelativistic particle energy is momentum squared divided by twice the mass. Converting the shell thickness to an energy interval introduces a Jacobian that changes the power of energy.

In three dimensions the one-particle density of states is proportional to square root of energy. Its integral below an energy cutoff grows as energy to the three-halves power. These are different functions. Neither is a probability density until an ensemble weight and normalization are supplied. The continuum approximation neglects the discreteness of low box levels and boundary corrections. It works when many relevant levels lie within thermal energy scales, but a special ground state may still need separate treatment.

The one-particle canonical partition function is the integral of density of states times the Boltzmann factor. Dividing that same product by the partition function gives a normalized energy probability density. High energy has more available states but a smaller weight per state. The balance can place the most probable energy away from the ground even though every individual high-energy state is less likely than a lower-energy state. Confusing a state probability with an energy-band probability misses this degeneracy effect.

Suppose more generally that the relevant state density is a constant times energy to the power s minus one. The partition integral is a Gamma integral and scales as beta to the power minus s. The mean energy is s times thermal energy, and its variance is s times thermal energy squared. The assumptions are a continuum extending from zero upward and a positive s. Extra thresholds, gaps, finite upper limits, or energy-dependent internal structure modify the integral.

For one free particle in three dimensions s equals three halves. The mean energy is therefore three halves times thermal energy, while the most probable kinetic energy is only one half times thermal energy. The mode depends on the variable used: a momentum distribution and a speed distribution have different Jacobians and therefore different maxima. A most probable value cannot be transferred between variables without changing the probability measure.

For N independent dilute particles, the one-particle partition function builds the fixed-number many-particle function with its Gibbs factor. That factor affects free energy but not the temperature derivative of translational energy. The quadratic lab can illustrate the kinetic-energy mapping from one momentum component to energy, including its changing slope. It cannot count three-dimensional momentum shells or render the resulting Gamma energy distribution; those are supplied by the derivation and the partition integral.