Statistical Mechanics · Problem 2.9

The Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · extreme-relativistic-gas-one-dimension

Problem

(a) Solve the integral

\int \ldots \int_{0 \leq \sum_{i=1}^{3N} |x_i| \leq R} (dx_1 \ldots dx_{3N})

and use it to determine the ``volume'' of the relevant region of the phase space of an extreme relativistic gas (

\varepsilon = pc

) of

3N

particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy

E

among this system of particles and show that, asymptotically,

\omega_0 = h^{3N}

. (b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.