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arXiv:1001.5365 · Physical Review C

Level density of a Fermi gas and integer partitions: A Gumbel-like finite-size correction

Jérôme Roccia, P. Lebœuf

Published Apr 5, 2010• 5 citations
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Abstract

We investigate the many-body level density of a gas of noninteracting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics, and differences with respect to the Bose gas.

DOI
10.1103/physrevc.81.044301

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We investigate the many-body level density of a gas of noninteracting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics, and differences with respect to the Bose gas.
We establish a connection between the level density of a gas of noninteracting bosons and the theory of extreme value statistics. Depending on the exponent that characterizes the growth of the underlying single-particle spectrum, we show that at a given excitation energy the limiting distribution function for the number of excited particles follows the three universal distribution laws of extreme value statistics, namely, the Gumbel, Weibull, and Fréchet distributions. Implications of this result, as well as general properties of the level density at different energies, are discussed.
We present a theory that accurately describes the counting of excited states of a noninteracting fermionic gas. At high excitation energies the results reproduce Bethe's theory. At low energies oscillatory corrections to the many-body density of states, related to shell effects, are obtained. The fluctuations depend nontrivially on energy and particle number. Universality and connections with Poisson statistics and random matrix theory are established for regular and chaotic single-particle motion.
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By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
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