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Journal of Experimental and Theoretical Physics

Statistics of mesoscopic ensembles of bosons and fermions

В. А. Алексеев

Published Jun 1, 2011• 4 citations
Publisher

Abstract

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.

DOI
10.1134/s1063776111050013

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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
This chapter reviews the techniques of the grand canonical ensemble to give an exact treatment for both Bose and Fermi particles. It looks at the results of Maxwell-Boltzmann statistics for the average number of particles in sufficiently high temperatures and in a single-particle state for both fermions and bosons. It also examines the exclusion principle in Fermi particles, which means that there cannot be two particles in the same single-particle state, while there is no such restriction for Bose particles. The chapter clarifies that the resulting formulae for the average number of particles in a single-particle state are different at low temperatures. It describes Fermi particles at low temperatures that have a large kinetic energy, an energy which can be much larger than the interaction between particles.
In the classical mechanical limit in which quantum cells of volume (2πħ)2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ)−3N (N!)−1 f. f d3r1 d3p1…d3rN d3pN exp (–βH)  Zc. The two factors, (2πħ)−3N and (N!)−1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction.
In nature, two different classes of particles are found, called fermi particles, or fermions, and bose particles, or bosons, respectively. The MB statistics is therefore an approximation to the true particle statistics and is applicable only in situations when most states are empty with some states occupied by a single particle and very few states, in the case of bosons, occupied by more than one particle. At high temperatures, the chemical potential decreases and the distribution changes its form dramatically. In discussing the quantum distribution functions, the authors have for convenience chosen the lowest state to have energy zero. The quantum distribution functions derived in this chapter are extremely useful in many areas of physics whenever what are called degenerate Fermi or Bose fluids are considered.
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
The statistical properties of quantum particles moving between two heat reservoirs at different temperatures are studied by solving the quantum master equation using a generating function technique. Bosons and fermions satisfy the same fluctuation theorem: Logarithm of the ratio of the probabilities of forward and backward transports reaches a constant value at long times, with an asymptotic $\ensuremath{\sim}1∕t$ correction. Non-Poissonian transfer statistics (bunching for bosons and antibunching for fermions) are examined using the Mandel parameter. These come primarily from the tail of the distribution corresponding to transfer of large number of particles, k⪢⟨k⟩k⪢⟨k⟩k⪢⟨k⟩.