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arXiv:cond-mat/0109324 · AIP conference proceedings

Yet another resolution of the Gibbs paradox: an information theory approach

Chih‐Yuan Tseng

Published Jan 1, 2002• 18 citations
arXivHTMLPDF

Abstract

The “Gibbs Paradox” refers to several related questions concerning entropy in thermodynamics and statistical mechanics: whether it is an extensive quantity or not, how it changes when identical particles are mixed, and the proper way to count states in systems of identical particles. Several authors have recognized that the paradox is resolved once it is realized that there is no such thing as the entropy of a system, that there are many entropies, and that the choice between treating particles as being distinguishable or not depends on the resolution of the experiment. The purpose of this note is essentially pedagogical; we add to their analysis by examining the paradox from the point of view of information theory. Our argument is based on that ‘grouping’ property of entropy that Shannon recognized, by including it among his axioms, as an essential requirement on any measure of information. Not only does it provide the right connection between different entropies but, in addition, it draws our attention to the obvious fact that addressing issues of distinguishability and of counting states requires a clear idea about what precisely do we mean by a state.

DOI
10.1063/1.1477057

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The “Gibbs Paradox” refers to several related questions concerning entropy in thermodynamics and statistical mechanics: whether it is an extensive quantity or not, how it changes when identical particles are mixed, and the proper way to count states in systems of identical particles. Several authors have recognized that the paradox is resolved once it is realized that there is no such thing as the entropy of a system, that there are many entropies, and that the choice between treating particles as being distinguishable or not depends on the resolution of the experiment. The purpose of this note is essentially pedagogical; we add to their analysis by examining the paradox from the point of view of information theory. Our argument is based on that ‘grouping’ property of entropy that Shannon recognized, by including it among his axioms, as an essential requirement on any measure of information. Not only does it provide the right connection between different entropies but, in addition, it draws our attention to the obvious fact that addressing issues of distinguishability and of counting states requires a clear idea about what precisely do we mean by a state.
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.
Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistinguishability that is crucial for the treatment of the Gibbs’ problem and the consistency of its solution with conventional thermodynamics. Quantum mechanics provides a quantitative criterion, not possible in the classical picture, for the degree of indistinguishability in terms of visibility of quantum interference, or overlap of the states as pointed out by von Neumann, thereby endowing the entropy expression with mathematical continuity and physical reasonableness.
The Gibbs Paradox is an important question in thermodynamics. In this thesis I search for a solution for the paradox from a classical, philosophical and quantum mechanical point of view. A common way to solve the Gibbs Paradox is by using the reduced entropy. However, this definition of entropy should not be interpreted as an argument for the indistinguishability of identical particles, because all other particles which have unique trajectories through space-time are distinguishable in principle. The Gibbs paradox could be solved in a few ways, but each solution has certain constrains.
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.
The issue of the Gibbs paradox is that when considering the mixing of two gases within classical thermodynamics, the entropy of mixing is a discontinuous function of the difference between the gases: it is finite for no matter how small difference, but vanishes for identical gases. The resolution offered in literature with help of quantum mixing entropy was later on shown to be unsatisfactory precisely where it sought to resolve the paradox. Within quantum thermodynamics entropy looses its dominant place. It is argued that the target of the paradox is shifted to the decrease of the available work upon mixing (mixing work), which in contrast to entropy is an unambiguous quantity. For almost identical gases the mixing work continuously goes to zero, thus resolving the paradox. In this approach the concept of ``difference between the gases'' attains a clear operational meaning related to possibilities of controlling the involved quantum states. The difficulties which prevent the resolution of the paradox in its entropic formulation do not arise here. The mixing work can increase when less precise operations are allowed. In contrast to the classical situation, in the quantum case the amount of mixing work can also increase when decreasing the degree of mixing between the gases, or when decreasing their distinguishability. These points go against a direct quantitative association of physical irreversibility with lack of information.
In physics, there are two distinct paradoxes, which are both known as the “Gibbs paradox”. This article is concerned with only one of them: the false increase in entropy, which is calculated from the process of combining two gases of the same kind consisting of distinguishable particles. In the following, this paradox will be referred to as the Gibbs paradox of the first kind (GP1). (Two particles are said to be distinguishable if they are either non-identical, that is, if they have different properties, or if they are identical and there are microstates which change under transposition of the two particles.) The GP1 is demonstrated and subsequently analyzed. The analysis shows that, for (quantum or classical) systems of distinguishable particles, it is generally uncertain of which particles they consist. The neglect of this uncertainty is the root of the GP1. For the statistical description of a system of distinguishable particles, an underlying set of particles, containing all particles that in principle qualify for being part of the system, is assumed to be known. Of which elements of this underlying particle set the system is composed, differs from microstate to microstate. Thus, the system is described by an ensemble of possible particle compositions. The uncertainty about the particle composition contributes to the entropy of the system. Systems for which all possible particle compositions are equiprobable will be called harmonic . Classical systems of distinguishable identical particles are harmonic as a matter of principle; quantum or classical systems of non-identical particles are not necessarily harmonic, since for them the composition probabilities depend individually on the preparation of the system. Harmonic systems with the same underlying particle set are always correlated; hence, for harmonic systems, the entropy is no longer additive and loses its thermodynamic meaning. A quantity derived from entropy is introduced, the reduced entropy , which, for harmonic systems, replaces the entropy as thermodynamic potential. For identical classical particles, the equivalence (in particular with respect to the second law of thermodynamics) between distinguishability and indistinguishability is proved. The resolution of the GP1 is demonstrated applying the previously found results.