Statistical Mechanics · Problem 3.15

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · extreme-relativistic-gas

Problem

Show that the partition function

Q_N(V, T)

of an extreme relativistic gas consisting of

N

monatomic molecules with energy--momentum relationship

\varepsilon = pc

c

being the speed of light, is given by

Q_N(V,T) = \frac{1}{N!}\left\{8\pi V\left(\frac{kT}{hc}\right)^3\right\}^N

Study the thermodynamics of this system, checking in particular that

PV = \tfrac{1}{3}U, \quad U/N = 3kT \quad \text{and} \quad \gamma = \tfrac{4}{3}.

Next, using the inversion formula (3.4.7), derive an expression for the density of states

g(E)

of this system.