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arXiv:1506.04468

Absolute irreversibility resolves the Gibbs paradox

Yûto Murashita, Masahito Ueda

Published Jun 15, 2015• 0 citations
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Abstract

The Gibbs paradox is often erroneously believed to have been resolved by quantum mechanics. In fact, it has been resolved by the assumption of extensivity only in non-interacting macroscopic thermodynamic systems as implied by Pauli. Yet, this resolution is far from satisfactory not only because real systems are interacting but also because it cannot apply to small thermodynamic systems. Here, we offer the resolution applicable to all of these systems based on a new concept of absolute irreversibility, where extensivity is shown to emerge automatically with increasing the system size. Quite surprisingly, our resolution also removes the fundamental flaw of fluctuation theorems in modern statistical physics.

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The Gibbs paradox is often erroneously believed to have been resolved by quantum mechanics. In fact, it has been resolved by the assumption of extensivity only in non-interacting macroscopic thermodynamic systems as implied by Pauli. Yet, this resolution is far from satisfactory not only because real systems are interacting but also because it cannot apply to small thermodynamic systems. Here, we offer the resolution applicable to all of these systems based on a new concept of absolute irreversibility, where extensivity is shown to emerge automatically with increasing the system size. Quite surprisingly, our resolution also removes the fundamental flaw of fluctuation theorems in modern statistical physics.
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 “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 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.
Two distinct puzzles, which are both known as Gibbs' paradox, have interested physicists since they were first identified in the 1870s. They each have significance for the foundations of statistical mechanics and have led to lively discussions with a wide variety of suggested resolutions. Most proposed resolutions had involved quantum mechanics, although the original puzzles were entirely classical and were posed before quantum mechanics was invented. In this paper, I show that contrary to what has often been suggested, quantum mechanics is not essential for resolving the paradoxes. I present a resolution of the paradoxes that does not depend on quantum mechanics and includes the case of colloidal solutions, for which quantum mechanics is not relevant.
The Gibbs paradox is a conventional paradox in classical statistical mechanics, typically resolved by invoking quantum indistinguishability through the 1/N! correction. In this letter, we present a resolution within classical ensemble theory, which relies solely on the equal probability principle and does not invoke the 1/N! correction. Our resolution can be naturally interpretated from a purely informational perspective, where the Gibbs entropy is explicitly regarded as the Shannon entropy, quantifying ignorance rather than disorder. From this informational perspective, we also clarify the connection between information and extractable work in the gas mixing processes. Our work opens a new avenue to reconsider the role of information in statistical mechanics.
The Gibbs paradox of the first kind (GP1) refers to the false increase in entropy which, in statistical mechanics, is calculated from the process of combining two gas systems S1 and S2 consisting of distinguishable particles. Presented in a somewhat modified form, the GP1 manifests as a contradiction to the second law of thermodynamics. Contrary to popular belief, this contradiction affects not only classical but also quantum statistical mechanics. This paper resolves the GP1 by considering two effects. (i) The uncertainty about which particles are located in S1 and which in S2 contributes to the entropies of S1 and S2. (ii) S1 and S2 are correlated by the fact that if a certain particle is located in one system, it cannot be located in the other. As a consequence, the entropy of the total system consisting of S1 and S2 is not the sum of the entropies of S1 and S2.