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Journal of Modern Physics

On the Quantum Statistical Distributions Describing Finite Fermions and Bosons Systems

Elsayed K. Elmaghraby

Published Jan 1, 2011• 4 citations
Publisher

Abstract

A century old methodology for deriving statistical distribution using approximate Stirling’s formulation of the factorial becomes questionable. By avoiding the use of exaggerated approximations, a new picture of the energy distribution of fermions and bosons are presented. Energy distribution among fermions (or bosons) in systems with finite degeneracy are found to be degeneracy dependent. The presented point of view explains, successfully, presence of degeneracy pressure in ultra-cooled Fermi gas and predicts the minimum accessible temperature for finite degeneracy fermions system.

DOI
10.4236/jmp.2011.211154

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A century old methodology for deriving statistical distribution using approximate Stirling’s formulation of the factorial becomes questionable. By avoiding the use of exaggerated approximations, a new picture of the energy distribution of fermions and bosons are presented. Energy distribution among fermions (or bosons) in systems with finite degeneracy are found to be degeneracy dependent. The presented point of view explains, successfully, presence of degeneracy pressure in ultra-cooled Fermi gas and predicts the minimum accessible temperature for finite degeneracy fermions system.
The derivations of the most probable distribution are not self-consistent in most statistical physics textbooks since the Stirling’s approximation adopted in the derivations is not valid for systems with very few particles. Any improvement by taking more higher terms of the Stirling series is insufficient due to erroneous divergence for small number. In this paper, we derive the equilibrium distribution by extrapolating the factorial function by the gamma function so that the Stirling’s approximation or the Stirling formula is avoided. For the ideal Bose system in a one-dimensional harmonic trap, we find that the equilibrium distribution given by this paper coincide with the rigorous result perfectly even for very small system size.
It has been suggested by Timmermans [Phys. Rev. Lett. 87, 240403 (2001)] that loss of fermions in a degenerate system causes strong heating. We address the fundamental limit imposed by this loss on the temperature that may be obtained by sympathetic cooling of fermions by bosons. Both a quantum Boltzmann equation and a quantum Boltzmann master equation are used to study the evolution of the occupation number distribution. It is shown that, in the thermodynamic limit, the Fermi gas cools to a minimal temperature ${k}_{\mathrm{B}}T∕\ensuremath{\mu}\ensuremath{\propto}{({\ensuremath{\gamma}}_{\text{loss}}∕{\ensuremath{\gamma}}_{\text{coll}})}^{0.44}$, where ${\ensuremath{\gamma}}_{\text{loss}}$ is a constant loss rate, ${\ensuremath{\gamma}}_{\text{coll}}$ is the bare fermion-boson collision rate not including the reduction due to Fermi statistics, and $\ensuremath{\mu}\ensuremath{\sim}{k}_{\mathrm{B}}{T}_{\mathrm{F}}$ is the chemical potential. It is demonstrated that, beyond the thermodynamic limit, the discrete nature of the momentum spectrum of the system can block cooling. The unusual nonthermal nature of the number distribution is illustrated from several points of view: the Fermi surface is distorted, and in the region of zero momentum the number distribution can descend to values significantly less than unity. Our model explicitly depends on a constant evaporation rate, the value of which can strongly affect the minimum temperature.
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 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.
The laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.