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Physical Review

Fermion Ensembles of Maximum Entropy

Martin C. Gutzwiller

Published Mar 1, 1962• 3 citations
Publisher

Abstract

Ensembles of particles obeying Fermi-Dirac statistics are considered from a point of view which is analogous to some recent work of Mayer on classical particle statistics. The density functions ${\ensuremath{\rho}}_{n}$ are defined for $n$ much smaller than the average number of particles in the ensemble. Since the knowledge of only a few ${\ensuremath{\rho}}_{n}$ is sufficient for the computation of most averages of physical interest, it is important to compare different ensembles with the same density functions, say ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$. The ensemble with the largest entropy is constructed as being the most significant. A number of simple examples are briefly considered.

DOI
10.1103/physrev.125.1455

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Ensembles of particles obeying Fermi-Dirac statistics are considered from a point of view which is analogous to some recent work of Mayer on classical particle statistics. The density functions \ensuremathρn{\ensuremath{\rho}}_{n}\ensuremathρn​ are defined for nnn much smaller than the average number of particles in the ensemble. Since the knowledge of only a few \ensuremathρn{\ensuremath{\rho}}_{n}\ensuremathρn​ is sufficient for the computation of most averages of physical interest, it is important to compare different ensembles with the same density functions, say \ensuremathρ1{\ensuremath{\rho}}_{1}\ensuremathρ1​ and \ensuremathρ2{\ensuremath{\rho}}_{2}\ensuremathρ2​. The ensemble with the largest entropy is constructed as being the most significant. A number of simple examples are briefly considered.
We developed a reasonably detailed and comprehensive understanding of boson and fermion statistics from the previous chapter. While this is interesting physics in itself, more important for us is an understand of how the symmetry properties of these particles impact on the entropy associated with them. In this chapter, we start by considering situations where the number of particles in a system is not fixed. Next, the theory developed is used to derive quantum statistics again, but this time without recourse to various approximations. Access to the requisite probability functions enables the Gibbs entropy to be calculated for fermions and bosons .
Equilibrium distribution functions are obtained for boson and fermion ensembles with a limited number of particles. It is shown that the number-of-particle distribution functions in different quantum states are statistically dependent; this dependence disappears only for a large number of particles in the ensemble. The distributions are transformed into the Boltzmann distribution at a high temperature and into the Bose-Einstein and Fermi-Dirac distributions for a large number of particles in the ensemble.
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
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
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' ϕϕϕ-entropy.