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arXiv:2006.12482 · Nature Communications

Mixing indistinguishable systems leads to a quantum Gibbs paradox

Benjamin Yadin, Benjamin Morris, Gerardo Adesso

Published Mar 5, 2021• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41467-021-21620-7

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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 mixing of two different gases is one of the most common natural phenomena, with applications ranging from CO2_22​ capture to water purification. Traditionally, mixing is analyzed in the context of local thermal equilibrium, where systems exchange energy with a heat bath. Here, we study mixing in an isolated system with potentially non-equilibrium initial states, characterized solely by macroscopic observables. We identify the entropy of mixing as a special case of observational entropy within an observer-dependent framework, where both entropy and extractable work depend on the resolution of measurement. This approach naturally resolves the Gibbs mixing paradox in quantum systems: while an observer experiences a discontinuous increase in entropy upon learning of the existence of two particle types, this knowledge does not provide an advantage in work extraction if the types of particles remain operationally indistinguishable in their measurements. Finally, we derive a Landauer-like bound on the difference in energy extracted by two observers, where an "observational temperature" emerges, determined by the accessible information. These results provide a foundation for rigorously determining the energy required to unmix in non-equilibrium settings and extend beyond quantum systems, offering insights into the thermodynamics of isolated classical gases.
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.
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.
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.
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.