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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum gas in an external field: Exact grand canonical expressions and numerical treatment

P. N. Vorontsov‐Velyaminov, Sergei D. Ivanov, R. I. Gorbunov

Published Jan 1, 1999• 12 citations
Publisher

Abstract

An exact Feynman-type presentation of the grand canonical partition function and averages as series over cycles for a system of noninteracting identical particles with a spin in an arbitrary external field is derived, and a numerical procedure for obtaining $\ensuremath{\mu}(\ensuremath{\beta})$ and other dependencies at constant N is developed. It is shown that the same series can be obtained also from the conventional form of the grand potential $\ensuremath{\Omega}$ (i.e., a sum over single-particle energy states). Numerical calculations at constant N are carried out for quantum gas of bosons and fermions in three-dimensional harmonic field and in the P\"oschl-Teller potential.

DOI
10.1103/physreve.59.168

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An exact Feynman-type presentation of the grand canonical partition function and averages as series over cycles for a system of noninteracting identical particles with a spin in an arbitrary external field is derived, and a numerical procedure for obtaining \ensuremathμ(\ensuremathβ)\ensuremath{\mu}(\ensuremath{\beta})\ensuremathμ(\ensuremathβ) and other dependencies at constant N is developed. It is shown that the same series can be obtained also from the conventional form of the grand potential \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ (i.e., a sum over single-particle energy states). Numerical calculations at constant N are carried out for quantum gas of bosons and fermions in three-dimensional harmonic field and in the P"oschl-Teller potential.
Previously developed approach [1 is extended to systems of noninteracting quantum particles in an external magnetic field. Feynman type series for grand potential, number of particles, internal energy and magnetization are derived. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in presence of the homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single particle states. Particular attention is paid to low temperature behavior of fermionic systems.
Feynman-type series for grand potential , number of particles, internal energy, magnetization and susceptibility are derived for systems of noninteracting quantum particles in an external magnetic field. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in the presence of a homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single-particle states. N -convergence to universal curves for chemical potential and energy is presented. At low temperatures magnetization of the Fermi gas exhibits a pronounced quantum behaviour.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
This chapter develops the basic equations that will be used to analyse the Fermi-Dirac and Bose-Einstein gases. The representation of many-particle states in terms of products of single-particle states is presented. The reasons for using the quantum grand canonical ensemble are given, and a general expression for the grand canonical partition function is derived. The essential equations for fermions, bosons, and distinguishable particles are developed, and the basic strategy for using them to solve problems is given.
The ideal Bose-gas with finite number NNN of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case N≫1N\gg1N≫1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential μμμ of the ideal Bose-gas can lie in the range $-\infty