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arXiv:hep-th/9304120 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

The free quon gas suffers Gibbs' paradox

Reinhard F. Werner

Published Sep 15, 1993• 21 citations
arXivHTMLPDF

Abstract

We consider the statistical mechanics of systems of particles satisfying the $q$-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (Fermi) relations for $q\ensuremath{\rightarrow}1$ ($q\ensuremath{\rightarrow}\ensuremath{-}1$), the partition functions of free gases are independent of $q$ in the range $\ensuremath{-}1<q<1$. The partition functions exhibit Gibbs' paradox in the same way as a classical gas without a correction factor $\frac{1}{N}!$ for the statistical weight of the $N$-particle phase space; i.e., the statistical mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.

DOI
10.1103/physrevd.48.2929

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We consider the statistical mechanics of systems of particles satisfying the qqq-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (Fermi) relations for q\ensuremath→1q\ensuremath{\rightarrow}1q\ensuremath→1 (q\ensuremath→\ensuremath−1q\ensuremath{\rightarrow}\ensuremath{-}1q\ensuremath→\ensuremath−1), the partition functions of free gases are independent of qqq in the range \ensuremath−1<q<1\ensuremath{-}1<q<1\ensuremath−1<q<1. The partition functions exhibit Gibbs' paradox in the same way as a classical gas without a correction factor 1N!\frac{1}{N}!N1​! for the statistical weight of the NNN-particle phase space; i.e., the statistical mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.
We calculate the canonical partition function Z_N for a system of N free particles obeying so-called `quon' statistics where q is real and satisfies |q|<1 by using simple counting arguments. We observe that this system is afflicted by the Gibbs paradox and that Z_N is independent of q. We demonstrate that such a system of particles obeys the ideal gas law and that the internal energy U ( and hence the specific heat capacity C_V ) is identical to that of a system of N free particles obeying Maxwell-Boltzmann statistics.
Since the grand partition function Zq for the so-called q-particles (i.e., quons), q∈(−1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q∈[−1,1]. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., q=0) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor 1/n! in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons q∈(0,1).
The Gibbs paradox in statistical mechanics is often taken to indicate that already in the classical domain particles should be treated as fundamentally indistinguishable. This paper shows, on the contrary, how one can recover the thermodynamical account of the entropy of mixing, while treating states that only differ by permutations of similar particles as distinct. By reference to the orthodox theory of thermodynamics, it is argued that entropy differences are only meaningful if they are related to reversible processes connecting the initial and final state. For mixing processes, this means that processes should be considered in which particle number is allowed to vary. Within the context of statistical mechanics, the Gibbsian grandcanonical ensemble is a suitable device for describing such processes. It is shown how the grandcanonical entropy relates in the appropriate way to changes of other thermodynamical quantities in reversible processes, and how the thermodynamical account of the entropy of mixing is recovered even when treating the particles as distinguishable.
We revisit recent discussions concerning the Gibbs paradox—the apparent discrepancy between the entropy change upon mixing identical gases as evaluated from the statistical mechanics of classical distinguishable particles and macroscopic thermodynamics. Contrary to what is often stated, we show that thermodynamics does not require this entropy of mixing to be zero. A zero value follows from the implicit assumption that the identical gas particles are indistinguishable. If the identical particles are explicitly assumed to be distinguishable, thermodynamics yields the same entropy of mixing as classical statistical mechanics.
A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from “yes” to “no” is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S=klnW or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.
Identical classical particles are distinguishable. This distinguishability affects the number of ways W a macrostate can be realized on the microlevel, and from the relation S=k ln W leads to a nonextensive expression for the entropy. This result is usually considered incorrect because of its inconsistency with thermodynamics. It is sometimes concluded from this inconsistency that identical particles are fundamentally indistinguishable and that quantum mechanics is indispensable for making sense of this inconsistency. In contrast, we argue that the classical statistics of distinguishable particles and the resulting nonextensive entropy function are perfectly acceptable from both a theoretical and an experimental perspective. The inconsistency with thermodynamics can be removed by taking into account that the entropy concept in statistical mechanics is not completely identical to the thermodynamical one. We observe that even identical quantum particles are in some cases distinguishable, and conclude that quantum mechanics is irrelevant to the Gibbs paradox.