Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 217

Similarly, for a Bose gas

we have

b) First solution: Since a classical ideal gas is a limiting case of both Fermi
and Bose gases at we get, from (S.4.94.3) or (S.4.94.6),

Alternatively, we can take the distribution function for an ideal classical
gas,

and use (S.4.94.2) to get the same result. Since all the numbers of
particles in each state are statistically independent, we can write

Second solution: In Problem 4.93 we derived the volume fluctuation

This gives the fluctuation of a system containing N particles. If we divide
(S.4.94.9) by we find the fluctuation of the volume per particle:

This fluctuation should not depend on which is constant, the volume or
the number of particles. If we consider that the volume in (S.4.94.10) is
constant, then

Substituting (S.4.94.11) into (S.4.94.10) gives