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Journal of Physics A Mathematical and General

The d-dimensional Landsberg gas

Jeremy Dunning-Davies

Published Nov 1, 1981• 7 citations
Publisher

Abstract

Expressions based on a recently proposed approximate density of states function are derived for thermodynamic functions of an ideal relativistic Bose gas in d dimensions. The results obtained are compared with those found using the exact density of states and agreement is found to be good in the three-dimensional case but less good in higher dimensions. It is noted that, in the extreme relativistic and non-relativistic limits, results agree exactly and, in the vicinity of the state of zero kinetic energy, the two densities of states exhibit smaller behaviour. Mention is made also of the application of this approximate density of states function to a consideration of the ideal relativistic Fermi gas.

DOI
10.1088/0305-4470/14/11/021

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Expressions based on a recently proposed approximate density of states function are derived for thermodynamic functions of an ideal relativistic Bose gas in d dimensions. The results obtained are compared with those found using the exact density of states and agreement is found to be good in the three-dimensional case but less good in higher dimensions. It is noted that, in the extreme relativistic and non-relativistic limits, results agree exactly and, in the vicinity of the state of zero kinetic energy, the two densities of states exhibit smaller behaviour. Mention is made also of the application of this approximate density of states function to a consideration of the ideal relativistic Fermi gas.
The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. The limitations and errors of previous calculations which relied on the binomial expansion of the density of states are pointed out. It is shown that the behaviour of the system in the limit of zero temperature is dominated by the non-relativistic effects.
The properties of an ideal Fermi gas in an external potential in any dimensional space are studied, based on the semiclassical (Thomas-Fermi) approximation. The general analytical expressions of the total particle number N,N,N, density of states $D(\ensuremath{\varepsilon}),$ Fermi energy EF,{E}_{F},EF​, total energy E,E,E, and heat capacity CCC have been derived, where N,N,N, E,E,E, and CCC are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy E,E,E, chemical potential \ensuremath{\mu}, and heat capacity CCC in the high- and low-temperature approximations are given. From these results, how the characteristics of the Fermi gas depend on an external potential and the dimension of space is discussed.
An ideal relativistic Bose-gas model with pair production is solved exactly in ddd dimensions. The critical behavior is shown to be the same as the usual nonrelativistic Bose gas. Scaling functions (including nonuniversal parameters) for the equation of state and the correlation functions are obtained in a closed form. The scale-factor universality is verified by the formation of the expected universal ratios. Various limiting cases of the model depending on the relative magnitudes of the three basic length scales, i.e., Compton wave-length, thermal wavelength, and interparticle distance are discussed. In these limiting cases, previously known results are recovered, wherever these are available.
Abstract We present analytical formulae for the first and second derivatives of the Helmholtz free energy of non‐relativistic ideal Fermi gas. Important thermodynamic quantities such as heat capacity, sound velocity, heat capacity ratio, and others are explicitly expressed through the derivatives. We demonstrate correct ideal Boltzmann gas and low‐temperature Fermi gas asymptotes and derive corrections to thermodynamic functions for these limiting cases. Numerical computations of thermodynamic properties of ideal Fermi gas can be accurately performed using the developed freely available Python module ifg.
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
We investigate ground-state and excitation spectrum of a system of non-relativistic bosons in one-dimension interacting through repulsive, two-body contact interactions in a self-consistent Gaussian mean-field approximation. The method consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. There are mainly two advantages in working with one-dimension. First, the existence of an exact solution for the ground-state and excitation energies. Second, neither in the perturbative results nor in the Gaussian approximation itself we do not have to deal with the three-dimensional patologies of the contact interaction . So that this scheme provides a clear comparison between these three different results.