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Annalen der Physik

Quantum statistics and discreteness. Differences between the canonical and grand canonical ensembles for a fermionic lattice gas

Ryszard Kutner, R. Przeniosło, M. Kwiatkowski

Published Jan 1, 1995• 2 citations
Publisher

Abstract

Abstract Studying the electronic properties of small metallic particles Kubo and subsequently Denton, Mühlschlegel and Scalapino recognized the surprising effect that the mean occupation numbers of noninteracting fermions (electrons) are not given within the canonical ensemble by the usual expression of Fermi‐Dirac statistics. The fermions behave as if they are colder and observed deviations are most significant in the vicinity of the Fermi level. The results of our Monte Carlo simulations, as well as our complementary numerical and analytical results are in very good agreement with the predictions of Denton et al. We found that the effect vanishes for very small and very large spacing between the energy levels in comparison to the thermal energy kBT. Associated with this effect are the particle‐hole correlations. Extending the method used by Denton et al. we derive an analytical formula for the particle‐hole correlation functions whose predictions are again in very good agreement with our numerical results. The low‐temperature aspects of the effect are considered by applying a two‐state model. The role of the finite‐size effect is also discussed.

DOI
10.1002/andp.19955070704

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Abstract Studying the electronic properties of small metallic particles Kubo and subsequently Denton, Mühlschlegel and Scalapino recognized the surprising effect that the mean occupation numbers of noninteracting fermions (electrons) are not given within the canonical ensemble by the usual expression of Fermi‐Dirac statistics. The fermions behave as if they are colder and observed deviations are most significant in the vicinity of the Fermi level. The results of our Monte Carlo simulations, as well as our complementary numerical and analytical results are in very good agreement with the predictions of Denton et al. We found that the effect vanishes for very small and very large spacing between the energy levels in comparison to the thermal energy kBT. Associated with this effect are the particle‐hole correlations. Extending the method used by Denton et al. we derive an analytical formula for the particle‐hole correlation functions whose predictions are again in very good agreement with our numerical results. The low‐temperature aspects of the effect are considered by applying a two‐state model. The role of the finite‐size effect is also discussed.
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
Exact results for the thermodynamic properties and mean occupation numbers of a system of noninteracting fermions with equidistant level spacing are presented for an arbitrary number of particles. It is discussed quantitatively how the results converge to the corresponding ones in the grand canonical ensemble when the thermodynamic limit is reached. From the simple calculations it also follows that the thermodynamics of an infinite two-dimensional electron gas is identical to that of a one-dimensional harmonic chain with linearized dispersion. The results generalize and simplify previous approaches to this model published in this journal.
We consider equilibrium level occupation numbers in a Fermi gas with a fixed number of particles, n, and finite level spacing. Using the method of generating functions and the cumulant expansion we derive a recurrence relation for canonical partition function and an explicit formula for occupation numbers in terms of single-particle partition function at n different temperatures. We apply this result to a model with equidistant non-degenerate spectrum and obtain close-form expressions in terms of q-polynomials and Rogers-Ramanujan partial theta function. Deviations from the standard Fermi-Dirac distribution can be interpreted in terms of a gap in the chemical potential between the particle and the hole excitations with additional correlations at temperatures comparable to the level spacing.
The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.
In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit LLL," i.e., in the limit in which the number of particles NNN and the volume VVV of the system are allowed to tend to infinity so as to keep NV\frac{N}{V}VN​ a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit LLL the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.
To facilitate the study of thermal properties of the pairing model, an approach based on the grand canonical ensemble is numerically compared with one based on the canonical ensemble. At low temperature in the phase of pair condensation, their difference becomes significant, in particular with regard to the behavior depending on whether the total number of fermions is even or odd. We separate the grand canonical ensemble into two pieces: one containing systems with even numbers of particles and one containing systems with odd numbers of particles. The grand partition functions defined on such grand canonical ensembles faithfully reproduce the even-odd effect.