Statistical Mechanics · Problem 8.18

Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions — R. K. Pathria

Physics — Statistical Mechanics · relativistic-fermi-gas-ground-state-energy

Problem

Show that the ground-state energy

E_0

of a relativistic gas of electrons is given by

E_0 = \frac{\pi V m^4 c^5}{3h^3} B(x),

where

B(x) = 8x^3\{(x^2+1)^{1/2} - 1\} - A(x),

A(x)

and

x

being given by eqns (8.4.13) and (8.4.14). Check that the foregoing result for

E_0

and eqn. (8.4.12) for

P_0

satisfy the thermodynamic relations

E_0 + P_0 V = N\mu_0 \quad \text{and} \quad P_0 = -(\partial E_0/\partial V)_N.