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arXiv:nucl-th/0505069 · Journal of Physics G Nuclear and Particle Physics

Quantum gases in the grand microcanonical ensemble near the thermodynamic limit

Viktor Begun, M. I. Gorenstein, A. Kostyuk, O. S. Zozulya

Published May 19, 2006• 16 citations
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Abstract

A new method is proposed for a treatment of ideal quantum gases in the grand microcanonical ensemble near the thermodynamic limit. The method allows rigorous asymptotic calculations of the average number of particles and particle number fluctuations in the system with exactly fixed total energy. It also gives the finite-volume corrections for the total average number of particles and for higher moments of the particle number distribution in a grand microcanonical system approaching the thermodynamic limit. The present consideration confirms our previous findings that the scaled variance for particle number fluctuations in the grand microcanonical ensemble is different from that in the grand canonical ensemble even in the thermodynamic limit.

DOI
10.1088/0954-3899/32/7/004

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A new method is proposed for a treatment of ideal quantum gases in the grand microcanonical ensemble near the thermodynamic limit. The method allows rigorous asymptotic calculations of the average number of particles and particle number fluctuations in the system with exactly fixed total energy. It also gives the finite-volume corrections for the total average number of particles and for higher moments of the particle number distribution in a grand microcanonical system approaching the thermodynamic limit. The present consideration confirms our previous findings that the scaled variance for particle number fluctuations in the grand microcanonical ensemble is different from that in the grand canonical ensemble even in the thermodynamic limit.
We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.
We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.
The ideal Bose-gas with finite number NNN of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case N≫1N\gg1N≫1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential μμμ of the ideal Bose-gas can lie in the range $-\infty
For a non-self-interacting Bose gas with a fixed, large number of particles confined to a trap, as the ground-state occupation becomes macroscopic, the condensate number fluctuations remain microscopic. However, this is the only significant aspect in which the grand canonical description differs from canonical or microcanonical in the thermodynamic limit. General arguments and estimates including some of the vanishingly small quantities are compared to explicit, fixed-number calculations for 102{10}^{2}102--106{10}^{6}106 particles. \textcopyright{} 1996 The American Physical Society.
The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.