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arXiv:1811.03967 · International Journal of Quantum Information

The Gibbs paradox and the physical criteria for indistinguishability of identical particles

C. S. Unnikrishnan

Published Aug 18, 2016• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0219749916400372

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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.
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.
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.
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.
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.