Expand about a filled Fermi sea

For a uniform nonrelativistic ideal gas in three dimensions, each allowed momentum and internal state can hold at most one identical fermion. At zero temperature every state below chemical potential is filled and every state above it is empty. Counting a sphere of occupied wavevectors relates Fermi wavevector to number density. Its energy then follows from quadratic dispersion. The spin or internal degeneracy must be included once in the count; it does not multiply the energy of each occupied state.

Integrating energy over the filled sphere gives mean energy three fifths of Fermi energy per particle. Differentiating the ground-state energy with respect to volume at fixed number gives a nonzero pressure even at zero temperature. This degeneracy pressure is not caused by a thermal distribution of distinguishable particles and does not vanish merely because temperature does. The ideal formula assumes negligible interactions and nonrelativistic momenta, so applying it to strongly interacting matter or relativistic stars requires further physics.

The Fermi occupation smooths the step over an energy width of order thermal energy. Far below chemical potential, holes are exponentially rare; far above it, particles are exponentially rare. For an integral of a smooth function times the occupation, thermal changes from the two sides mostly cancel at first order in temperature. The leading surviving correction is proportional to temperature squared and the derivative of the smooth function at chemical potential.

This is an asymptotic expansion requiring chemical potential well above the lower spectral edge and the integrand smooth across the thermal window. Gaps, sharp density-of-states singularities, or temperatures comparable with Fermi temperature invalidate the simplest formula. The error notation refers to an appropriate smooth-integrand expansion, not a universal numerical bound for arbitrary spectra. One should inspect the energy dependence near the Fermi surface before treating the first correction as a controlled prediction.

At fixed density, insert the square-root state density into the number integral and solve for the chemical potential shift that keeps number unchanged. Chemical potential falls slightly below its zero-temperature Fermi energy. Substituting that shift into the energy integral gives a quadratic increase in total energy and a heat capacity linear in temperature. Holding chemical potential at Fermi energy instead changes number and yields different intermediate coefficients; that is another ensemble path, not the fixed-density result.

The linear heat capacity reflects the small fraction of particles within the thermal surface layer, each changing energy by roughly thermal energy. The probability lab can illustrate exclusion-limited occupation fluctuations for equal-energy fermionic states at a prescribed occupation probability. It cannot display a Fermi surface or carry out the Sommerfeld integral. Its probability must be supplied from the Fermi function, and its finite equal-energy count must not be mistaken for the entire dispersive gas.