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WORLD SCIENTIFIC eBooks

Canonical and Grand–Canonical Ensembles for Trapped Bose Gases

Abel Camacho, Alfredo Macı́as, A. Camacho-Galván

Published May 1, 2010• 0 citations
Publisher
DOI
10.1142/9789814307543_0001

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This chapter focuses on the applications involving finite temperature theories. Technically, the grand canonical partition function is easiest to handle and in general it is the one used for the analysis of the thermodynamical properties. In fact, for properties like the energy and the specific heat of the gas this is justified because in the thermodynamical limit the different ensembles are known to give the same answer. The chapter uses the grand canonical ensemble to obtain properties like critical temperature, energy and specific heat. It summarizes the grand canonical description of ideal Bose gases trapped by magnetic fields and describes the formulation of Bose-Einstein condensation as a symmetry-breaking phenomenon.

Chapman & Hall/CRC eBooks•Dec 13, 2001•0 citations
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 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.
This chapter focuses on the applications involving finite temperature theories. Technically, the grand canonical partition function is easiest to handle and in general it is the one used for the analysis of the thermodynamical properties. In fact, for properties like the energy and the specific heat of the gas this is justified because in the thermodynamical limit the different ensembles are known to give the same answer. The chapter uses the grand canonical ensemble to obtain properties like critical temperature, energy and specific heat. It summarizes the grand canonical description of ideal Bose gases trapped by magnetic fields and describes the formulation of Bose-Einstein condensation as a symmetry-breaking phenomenon.