Number density, entropy density, and energy density of statistical mechanics for D-dimensional grand canonical ensembles
Morton D. Kostin · Journal of Mathematical Physics · 1995
A partial differential equation for the number density of a D-dimensional grand canonical ensemble is used to derive an explicit expression for the semiclassical number density. This is accomplished by first using the Bloch equation and its solutions to derive the classical number density and then combining the classical result with a Planck’s constant expansion of the partial differential equation to obtain the semiclassical formula. Similar results are also derived for the entropy density, energy density, and grand potential density of a D-dimensional grand canonical ensemble.