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Physical Review

Statistical Mechanics of an Assembly of Quasiparticles

Marvin D. Girardeau

Published Nov 15, 1965• 9 citations
Publisher

Abstract

The expressions obtained by Morita and Tanaka for the exact thermodynamic functions of an assembly of quasiparticles with diagonal interactions are rederived by a much simpler method due to Bogoliubov, Zubarev, Tserkovnikov, Wentzel, and others.
DOI
10.1103/physrev.140.a1139

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Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., nnn quasiparticles are of $O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})$, respectively, where VVV is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.
The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.
A general variational approach to the quantum statistics of many-particle systems is developed, based on the Gibbs-Bogoliubov variational principle and the ``thermodynamically equivalent Hamiltonian'' method first applied to the BCS model by Bogoliubov, Zubarev, and Tserkovnikov [Sov. Phys.---Dokl. 2, 535 (1957)] and subsequently generalized by Wentzel [Phys. Rev. 120, 1572 (1960)]. The method retains a judiciously chosen discrete set of momentum transfers in the second-quantized interaction Hamiltonian. It is expected to be applicable to a variety of systems including both crystalline solids and quantum fluids such as liquids and dense plasma phases.