Differential equations for the entropy density, energy density, and number density of statistical mechanics

A partial differential equation is derived for the number density as a function of position, temperature, and chemical potential. In the classical limit where Planck’s constant is zero, it is shown that the expression for the classical number density of statistical mechanics satisfies the resulting equation. Similar partial differential equations are derived for the entropy density and energy density. It is then shown that these partial differential equations can be used to derive easily general expressions for the semiclassical number density, entropy density, and energy density of statistical mechanics.