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Acta Physica Polonica B

Inequivalence of Canonical and Grand Canonical Ensembles for Bosonic Systems

S.K. Mehdi, N. Daoudi, S. Kessal

Published Jan 1, 2013• 1 citation
Publisher

Abstract

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.

DOI
10.5506/aphyspolb.44.1949

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For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.
We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
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.
It is demonstrated in many thermodynamic textbooks that the equivalence of the different ensembles is achieved in the thermodynamic limit. In this present work we discuss the inequivalence of microcanonical and canonical ensembles in a finite ultracold system at low energies. We calculate the microcanonical momentum distribution function (MDF) in a system of identical fermions (bosons). We find that the microcanonical MDF deviates from the canonical one, which is the Fermi-Dirac (Bose-Einstein) function, in a finite system at low energies where the single-particle density of states and its inverse are finite.
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.
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.
Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E_{S} is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z_{S} is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E_{S}=-d(ln Z_{S})/dbeta is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.