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Progress of Theoretical Physics

Even-Odd Effect on the Thermal Equilibrium State of the Pairing Model. I: Comparison between Canonical and Grand Canonical Ensembles

A. Kuriyama, João da Providência, Y. Tsue, M. Yamamura

Published Sep 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.108.533

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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.
We define a mean field approximation of the modified grand partition function of the pairing model, which has been defined on the modified grand canonical ensemble with even or odd particle number. Using numerical evaluations, it is shown that this approximation reproduces the even-odd effect exhibited in the case of the modified grand partition function at low temperature very well in a qualitative sense. The expectation values of one-body operators can be expressedsimply in terms of those obtainedwith the conventional mean field approximation when a Fermi-type distribution is replaced by a renormalized distribution. Although Wick’s theorm does not hold in this case, the dominant parts of the expectation values of two-body operators are also given using those obtained by the conventional mean fieldapproximation with the replacement mentionedabove.
In order to compare the approach of the mixed state representation of the pairing model with the exact evaluation of the grand partition function, we introduce the fermion space of a grand canonical ensemble. For the definition of the grand partition function, we adopt a quasi-spin space and evaluate its multiplicity. then we show how to derive the mixed state representation, which yields results that are approximately the same as those of the thermal Hartree-Fock-Bogoliubov theory for the thermal equilibrium state, from the grand partition function under the mean field approximation. We compare numerical results of the mean field approximation with those obtained from the exact evaluation of the grand partition function. We find that the mean field approximation gives good results. However, we find a distinctive difference between the two sets of results in the condensed phase.
Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.
We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number.
We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent y1y_1y1​ that depends on the temperature exponent yty_tyt​ and the dimensionality ddd as y1=−∣2yt−d∣y_1=-|2y_t-d|y1​=−∣2yt​−d∣. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.
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.