Physica Scripta

An asymptotic formula for the free energy density of ideal quantum gases

Jan F. Maćkowiak

Published Oct 1, 1988• 7 citations

Abstract

It is shown that the expressions for the free energy density of ideal quantum gases in the canonical and grand canonical ensembles, are identical up to additive terms which vanish in the thermodynamic limit.

DOI: 10.1088/0031-8949/38/4/001

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Physica Scripta

An asymptotic formula for the free energy density of ideal quantum gases

Jan F. Maćkowiak

Published Oct 1, 1988• 7 citations

Publisher

Abstract

DOI: 10.1088/0031-8949/38/4/001

Related papers

Quantum gases in the grand microcanonical ensemble near the thermodynamic limit

Viktor Begun, M. I. Gorenstein, A. Kostyuk +1 more

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.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.

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In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit

L

," i.e., in the limit in which the number of particles

N

and the volume

V

of the system are allowed to tend to infinity so as to keep

\frac{N}{V}

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

L

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.

N

particles. The pair distribution function

g^{(2)}(r)

of fermions clearly exhibits antibunching with

g^{(2)}(0)=0

due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with

g^{(2)}(0)\approx 2

, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature

T_0

, an off-diagonal long-range order develops in the one-particle density matrix to reach

g^{(1)}(r)=1

T=0

, and the pair correlation starts to decrease towards

g^{(2)}(r)\approx 1

T=0

. The results for

N\rightarrow \infty

are seen to converge to those of the grand canonical ensemble obtained by assuming the average

\langle\hatψ({\bf r})\rangle

of the field operator

\hatψ({\bf r})

below

T_0

. This fact justifies the introduction of the "anomalous" average

\langle\hatψ({\bf r})\rangle\neq 0

below

T_0

in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.