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Revista brasileira de fisica

Boltzmann, Gibbs, and Darwin-Fowler Approaches in Parastatistics

Roberto Leon Ponczek, C. C. Yan

Published Jan 1, 1976• 0 citations

Abstract

There are essentially three methods used in equilibrium statistical mechanics, namely, the Boltzmann method of identifying the equilibrium sta te with the most probable one; the Gibbs ensemble method of postulating a canonical distribution; and the Darwin-Fowler method' of identifying the equilibrium state with the average state. Most textbooks on statis tical mechanics include discussions of the first two methods. The in clusion into the text of the last method, which schrudinger2 seems to advocate, is less usual . The equivalence of those methods is usual 1 y demonstrated for the i deal Boltzmann, Bose - Einstein, and Fermi-Dirac sta tistics. However, the equivalence is not guaranteed when the methodsare applied to different statistics such as parastatistics3, where eachener gy leve1 can afford up to some number k of particles, and can be regar ded as a generalization of the above mentioned statistics. It seems desirable to confirm t he equivalence of those methods for parastatistics. In most textbooks however discussions on parastatistics are r ather rare.

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There are essentially three methods used in equilibrium statistical mechanics, namely, the Boltzmann method of identifying the equilibrium sta te with the most probable one; the Gibbs ensemble method of postulating a canonical distribution; and the Darwin-Fowler method' of identifying the equilibrium state with the average state. Most textbooks on statis tical mechanics include discussions of the first two methods. The in clusion into the text of the last method, which schrudinger2 seems to advocate, is less usual . The equivalence of those methods is usual 1 y demonstrated for the i deal Boltzmann, Bose - Einstein, and Fermi-Dirac sta tistics. However, the equivalence is not guaranteed when the methodsare applied to different statistics such as parastatistics3, where eachener gy leve1 can afford up to some number k of particles, and can be regar ded as a generalization of the above mentioned statistics. It seems desirable to confirm t he equivalence of those methods for parastatistics. In most textbooks however discussions on parastatistics are r ather rare.

Abstract : The Fermi-Dirac, Bose-Einstein and, for completeness the Maxwell-Boltzmann, distributions are obtained respectively from considerations of binomial, negative binomial, and Poisson assemblies. The method used has the simplicity of the traditional derivations that are based on combinatorial considerations but involves neither the identification of most probable values with means values nor the invocation of large numbers of particles involved in the use of Stirling's approximation for factorials. The method thereby also relaxes the requirement for large numbers of particles needed in other available derivations that also use mean values, namely, the Darwin-Fowler method of steepest descents and the Khinchin method that employs limit theorems of the theory of probability.

Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.

The reply by Sands & Dunning-Davies [3] to my paper [1] is acknowledged concerning the distribution functions. Keywords: Maxwell-Boltzmann distribution, Bose-Einstein distribution, classical ensembles, state functions In my paper ‘Are there physical systems obeying the Maxwell-Boltzmann statistics?’[1] I have written down the well-known probabilities of the 4 possible results of one fair toss of 2 fair coins according to the Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. The two results coin 1 = head, coin 2 = tail and coin 2 = head, coin 1 = tail are counting as different in MB and as one and the same in BE and in FD. Then, I have quoted Gibbs [2], that the interchange of two © 2011 C. Roy Keys Inc. — http://redshift.vif.comApeiron, Vol. 18, No. 1, January 2011 16

We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts'

ϕ

-entropy.

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 is a preprint of a theoretical study in statistical mechanics. We propose a generalized statistical framework based on para-statistics,introducing a modified grand partition function and generalized distributionfunctions. The formulation systematically extends conventional Bose–Einsteinand Fermi–Dirac statistics. This work is independent of the authors’ previously published studies andpresents a new theoretical framework. This preprint is intended for submission to arXiv in the category ofstatistical mechanics.