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Resolution of the Gibbs paradox via quantum thermodynamics

ARMEN E. ALLAHVERDYAN, Theo M. Nieuwenhuizen

Published Jul 15, 2005• 0 citations
OpenAlex

Abstract

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.

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Explanation of the Gibbs paradox within the framework of quantum thermodynamics

ARMEN E. ALLAHVERDYAN, Theo M. Nieuwenhuizen

The issue of the Gibbs paradox is that when considering mixing of two gases within classical thermodynamics, the entropy of mixing appears to be a discontinuous function of the difference between the gases: it is finite for whatever small difference, but vanishes for identical gases. The resolution offered in the literature, with help of quantum mixing entropy, was later shown to be unsatisfactory precisely where it sought to resolve the paradox. Macroscopic thermodynamics, classical or quantum, is unsuitable for explaining the paradox, since it does not deal explicitly with the difference between the gases. The proper approach employs quantum thermodynamics, which deals with finite quantum systems coupled to a large bath and a macroscopic work source. Within quantum thermodynamics, entropy generally loses its dominant place and the target of the paradox is naturally shifted to a decrease of the maximally available work before and after mixing (mixing ergotropy). In contrast to entropy this is an unambiguous quantity. For almost identical gases the mixing ergotropy continuously goes to zero, thus resolving the paradox. In this approach the concept of "difference between the gases" gets a clear operational meaning related to the possibilities of controlling the involved quantum states. Difficulties which prevent resolutions of the paradox in its entropic formulation do not arise here. The mixing ergotropy has several counterintuitive features. It can increase when less precise operations are allowed. In the quantum situation (in contrast to the classical one) the mixing ergotropy can also increase when decreasing the degree of mixing between the gases or when decreasing their distinguishability. These points go against a direct association of physical irreversibility with lack of information.

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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.

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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.

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

Chih‐Yuan Tseng

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.

arXiv:cond-mat/0109324•AIP conference proceedings•Jan 1, 2002•18 citations
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.
The issue of the Gibbs paradox is that when considering mixing of two gases within classical thermodynamics, the entropy of mixing appears to be a discontinuous function of the difference between the gases: it is finite for whatever small difference, but vanishes for identical gases. The resolution offered in the literature, with help of quantum mixing entropy, was later shown to be unsatisfactory precisely where it sought to resolve the paradox. Macroscopic thermodynamics, classical or quantum, is unsuitable for explaining the paradox, since it does not deal explicitly with the difference between the gases. The proper approach employs quantum thermodynamics, which deals with finite quantum systems coupled to a large bath and a macroscopic work source. Within quantum thermodynamics, entropy generally loses its dominant place and the target of the paradox is naturally shifted to a decrease of the maximally available work before and after mixing (mixing ergotropy). In contrast to entropy this is an unambiguous quantity. For almost identical gases the mixing ergotropy continuously goes to zero, thus resolving the paradox. In this approach the concept of "difference between the gases" gets a clear operational meaning related to the possibilities of controlling the involved quantum states. Difficulties which prevent resolutions of the paradox in its entropic formulation do not arise here. The mixing ergotropy has several counterintuitive features. It can increase when less precise operations are allowed. In the quantum situation (in contrast to the classical one) the mixing ergotropy can also increase when decreasing the degree of mixing between the gases or when decreasing their distinguishability. These points go against a direct association of physical irreversibility with lack of information.
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.
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.
It is often stated that the Gibbs ‘‘paradox’’ is resolved or at least alleviated in quantum mechanics because the value of the quantum mechanical entropy of mixing continuously approaches zero if the difference between the particles that are being mixed is made to vanish (i.e., if the value of the inproduct of the respective quantum states goes to one). In this article, the justification of that claim is discussed. It is pointed out that, strictly speaking, an entropy of mixing is not defined if the quantum states of the particles are no longer orthogonal. It seems, therefore, that from a fundamental point of view the Gibbs ‘‘paradox’’ is there no less in quantum mechanics than in classical statistical mechanics.
The classical Gibbs paradox concerns the entropy change upon mixing two gases. Whether an observer assigns an entropy increase to the process depends on their ability to distinguish the gases. A resolution is that an "ignorant" observer, who cannot distinguish the gases, has no way of extracting work by mixing them. Moving the thought experiment into the quantum realm, we reveal new and surprising behaviour: the ignorant observer can extract work from mixing different gases, even if the gases cannot be directly distinguished. Moreover, in the macroscopic limit, the quantum case diverges from the classical ideal gas: as much work can be extracted as if the gases were fully distinguishable. We show that the ignorant observer assigns more microstates to the system than found by naive counting in semiclassical statistical mechanics. This demonstrates the importance of accounting for the level of knowledge of an observer, and its implications for genuinely quantum modifications to thermodynamics.
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.