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Progress of Theoretical Physics

A Method in Quantum Statistical Mechanics, II

Hiroshi Ichimura

Published Apr 1, 1954• 0 citations
Publisher

Abstract

Using the method which has been developed by the author, we calculate the effect of the interparticle interaction of the thermodynamic potential per particle of the highly degenerated Fermi-Dirac gas. We show that under certain assumptions the specific heat of the system of conduction electrons in a metal maintain its linear temperature dependence in the presence of the interaction.

DOI
10.1143/ptp.11.385

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Using the method which has been developed by the author, we calculate the effect of the interparticle interaction of the thermodynamic potential per particle of the highly degenerated Fermi-Dirac gas. We show that under certain assumptions the specific heat of the system of conduction electrons in a metal maintain its linear temperature dependence in the presence of the interaction.
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Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
The method of auxiliary variables is generalized so as to be applicable to the systems having strong interparticle interactions or obeying Fermi-Dirac statistics. The ground state energy and the excitation energy specrum are calculated to the zeroth approximation. It is shown that Feynman's relation for liquid helium and Tomonaga's expression for a one-dimensional Fermi gas can be derived as the special cases of the present theory. The theory is applied also to the classical statistical mechanics.
The problem of the interaction between conduction electrons and the thermal vibration of the lattice ions in the metal is treated by a statistical mechanical method. We adopt the free electron and the phonon models as usual, using the Bloch-Bethe type interaction Hamiltonian. By the method of expanding the grand partition function in powers of the interaction parameter, we get a formula for the free energy of the system of conduction electrons in the presence of the interaction. A technique analogous to the renormalization method in the quantum mechanics is utilized in the calculation.
The momentum distribution in a low-temperature Fermi gas is investigated using the methods of quantum statistics developed by Lee and Yang together with the linked-pair expansions of the previous paper. It is shown that in order to determine the momentum distribution at very low temperatures two coupled integral equations must be considered, one in momentum variables and due to Lee and Yang, and the other in temperature variables. It is also shown that the dominant low-temperature behavior of the momentum distribution can be extracted in terms of a certain function ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$. For a low-density Fermi gas with strong, short-range, two-body interactions, it is shown to third order in the scattering parameters of the interaction that at T=0T=0T=0 the function ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$ is equal to the free particle momentum distribution. Also, the energy and other thermodynamic quantities are expressed in terms of ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$, so that the theory permits a generalization of perturbation theoretic results to nonzero temperatures. The ground-state energy, momentum distribution, and thermodynamic potential are calculated to third order in the scattering parameters.