Statistical Mechanics · Problem 8.10

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

Physics — Statistical Mechanics · fermi-gas-n-dimensions-s-spectrum

Problem

Consider an ideal Fermi gas, with energy spectrum ε∝ps\varepsilon \propto p^s, contained in a box of ``volume'' VV in a space of nn dimensions. Show that, for this system, ((i)) PV=snUPV = \dfrac{s}{n}U; ((ii)) CVNk=ns(ns+1)f(n/s)+1(z)fn/s(z)−(ns)2fn/s(z)f(n/s)−1(z)\dfrac{C_V}{Nk} = \dfrac{n}{s}\left(\dfrac{n}{s}+1\right)\dfrac{f_{(n/s)+1}(z)}{f_{n/s}(z)} - \left(\dfrac{n}{s}\right)^2 \dfrac{f_{n/s}(z)}{f_{(n/s)-1}(z)}; ((iii)) CP−CVNk=(sCVnNk)2f(n/s)−1(z)fn/s(z)\dfrac{C_P - C_V}{Nk} = \left(\dfrac{sC_V}{nNk}\right)^2 \dfrac{f_{(n/s)-1}(z)}{f_{n/s}(z)}; ((iv)) the equation of an adiabat is: PV1+(s/n)=const.PV^{1+(s/n)} = \text{const.}, ((v)) the index (1+(s/n))(1 + (s/n)) in the foregoing equation agrees with the ratio (CP/CV)(C_P/C_V) of the gas only when T≫TFT \gg T_F. On the other hand, when T≪TFT \ll T_F, the ratio (CP/CV)≃1+(π2/3)(kT/εF)2(C_P/C_V) \simeq 1 + (\pi^2/3)(kT/\varepsilon_F)^2, irrespective of the values of ss and nn.
Intuitive explanation

The key is recognizing that scaling arguments—how energy and pressure respond to volume changes—unlock the thermodynamic relations for a Fermi gas. When you compress the box, the density of states and Fermi energy both shift in predictable ways determined by the dispersion relation ε∝ps\varepsilon \propto p^s and dimensionality nn. Use dimensional analysis and the virial theorem to connect pressure to internal energy, then leverage the Fermi-Dirac distribution's temperature dependence to extract heat capacities and adiabatic behavior. The low-temperature limit reveals a universal result independent of ss and nn because the heat capacity becomes dominated by the linear electronic contribution near the Fermi surface.

Intuitive explanation

The key is recognizing that scaling arguments—how energy and pressure respond to volume changes—unlock the thermodynamic relations for a Fermi gas. When you compress the box, the density of states and Fermi energy both shift in predictable ways determined by the dispersion relation ε∝ps\varepsilon \propto p^s and dimensionality nn. Use dimensional analysis and the virial theorem to connect pressure to internal energy, then leverage the Fermi-Dirac distribution's temperature dependence to extract heat capacities and adiabatic behavior. The low-temperature limit reveals a universal result independent of ss and nn because the heat capacity becomes dominated by the linear electronic contribution near the Fermi surface.

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