Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 175

where the upper sign in (S.4.68.1) andbelow correspondsto Fermistatistics

and the lower to Bose Using weobtain

The totalenergy isgiven by

On the otherhand,using the grand canonical potential where

and replacing the sum by an integral, using (S.4.68.2), we obtain

Integrating(S.4.68.5) byparts, we have

Comparing this expression with (S.4.68.3), we find that

However, Therefore, we obtain the equation
of state, which is valid both for Fermi and Bose gases (and is, of course,
also true for a classicalBoltzmanngas):

Note that(S.4.68.8) was derivedunder theassumption ofa particulardis-

persion law for relativistic particles or photonswith

(S.4.68.8) becomes (see Problem 4.67). From (S.4.68.8) and

(S.4.68.3), weobtain