Statistical Mechanics · Problem 8.13

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

Physics — Statistical Mechanics · fermi-gas-general-low-temperature-behavior

Problem

Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat and the entropy of an ideal Fermi gas is given by μ≃εF[1−π26(∂ln⁡a(ε)∂ln⁡ε)ε=εF(kTεF)2]\mu \simeq \varepsilon_F\left[1 - \frac{\pi^2}{6}\left(\frac{\partial \ln a(\varepsilon)}{\partial \ln \varepsilon}\right)_{\varepsilon=\varepsilon_F}\left(\frac{kT}{\varepsilon_F}\right)^2\right] and Cv≃S≃π23k2Ta(εF),C_v \simeq S \simeq \frac{\pi^2}{3}k^2 T a(\varepsilon_F), where a(ε)a(\varepsilon) is the density of (the single-particle) states in the system. Examine these results for a gas with energy spectrum ε∝ps\varepsilon \propto p^s, confined to a space of nn dimensions, and discuss the special cases: s=1s = 1 and 22, with n=2n = 2 and 33. [Hint: Use eqn. (E. 16).]

Related theory

Browse the problem library