Statistical Mechanics · Problem 2.8

The Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · hypersphere-volume-relativistic-gas

Problem

Following the method of Appendix C, replacing eqn. (C.4) by the integral ∫0∞e−rr2 dr=2,\int_0^\infty e^{-r} r^2\, dr = 2, show that V3N=∫…∫0≤∑i=1Nri≤R∏i=1N(4πri2 dri)=(8πR3)N/(3N)!.V_{3N} = \int \ldots \int_{0 \leq \sum_{i=1}^N r_i \leq R} \prod_{i=1}^N (4\pi r_i^2\, dr_i) = (8\pi R^3)^N/(3N)!. Using this result, compute the ``volume'' of the relevant region of the phase space of an extreme relativistic gas (ε=pc\varepsilon = pc) of NN particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7.

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