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Mixing indistinguishable systems leads to a quantum Gibbs paradox

Yadin, Benjamin; Morris, Benjamin; Adesso, Gerardo

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Benjamin Yadin

Benjamin Morris


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. Despite its phenomenological beginnings, thermodynamics has been inextricably linked throughout the past century with the abstract concept of information. Such connections have proven essential for solving paradoxes in a variety of thought experiments, notably including Maxwell's demon [1] and Loschmidt's paradox [2]. This integration between classical thermodynamics and information is also one of the main motivating factors in extending the theory to the quantum realm, where information held by the observer plays a similarly fundamental role [3]. In this work, we study the transition from classical to quantum thermodynamics in the context of the Gibbs paradox [4-6]. This thought experiment considers two gases on either side of a box, separated by a partition and with equal volume and pressure on each side. If the gases are identical, then the box is already in thermal equilibrium, and nothing changes after removal of the partition. If the gases are distinct, then they mix and expand to fill the volume independently, approaching thermal equilibrium with a corresponding entropy increase. The (supposed) paradox can be summarised as follows: what if the gases differ in some unobservable or negligible way-should we ascribe an entropy increase to the mixing process or not? This question sits uncomfortably with the view that thermodynamical entropy is an objective physical quantity. Various resolutions have been described, from phenomeno-logical thermodynamics to statistical mechanics perspectives, and continue to be analysed [6-8]. A crucial insight by Jaynes [9] assuages our discomfort at the observer-dependent nature of the entropy change. For an informed observer, who sees the difference between the gases, the entropy increase has physical significance in terms of the work extractable through the mixing process-in principle, they can build a device that couples to the two gases separately (for example, through a semi-permeable membrane) and thus let each gas do work on an external weight independently. An ignorant observer, who * has no access to the distinguishing degree of freedom, has no device in their laboratory that can exploit the difference between the gases, and so cannot extract work. For Jaynes, there is no paradox as long as one considers the abilities of the experimenter-a viewpoint central to the present work. We study the Gibbs mixing process for quantum gases of identical bosons or fermions. This is motivated by recognising that the laws of thermodynamics must be modified to account for quantum effects such as coherence [10], which can lead to enhanced performance of thermal machines [11-13]. The thermodynamical implications of identical quantum particles have received renewed interest for applications such as Szi-lard engines [14, 15], thermodynamical cycles [16, 17] and energy transfer from boson bunching [18]. Moreover, the particular quantum properties of identical particles, including en-tanglement, can be valuable resources in quantum information processing tasks [19-21] We consider a toy model of an ideal gas with non-interacting quantum particles, distinguishing the two gases by a spin-like degree of freedom. We describe the mixing processes that can be performed by both informed and ignorant observers, taking into account their different levels of control, from which we can calculate the corresponding entropy changes and thus work extractable by each observer. For the informed observer, we recover the same results as obtained by classical statistical mechanics arguments. However, for the ignorant observer, there is a marked divergence from the classical case. Counter-intuitively, the ignorant observer can typically extract more work from distinguishable gases-even though they appear indistinguishable-than from truly identical gases. In the continuum and large particle number limit which classically recovers the ideal gas, this divergence is maximal: the ignorant observer can extract as much work from apparently indistinguishable gases as the informed observer. Our analysis hinges on the symmetry properties of quantum states under permutations of particles. For the ignorant observer, these properties lead to non-trivial restrictions on the possible work extraction processes. Viewed another way, the microstates of the system described by the ignorant observer are highly non-classical entangled states. This implies a fundamentally different way of counting microstates, and therefore

Journal Article Type Article
Acceptance Date Jan 27, 2021
Online Publication Date Mar 5, 2021
Publication Date Mar 5, 2021
Deposit Date Feb 2, 2021
Publicly Available Date Mar 5, 2021
Journal Nature Communications
Electronic ISSN 2041-1723
Publisher Nature Publishing Group
Peer Reviewed Peer Reviewed
Volume 12
Issue 1
Article Number 1471
Keywords General Biochemistry, Genetics and Molecular Biology; General Physics and Astronomy; General Chemistry
Public URL
Publisher URL
Additional Information Yadin, B., Morris, B. & Adesso, G. Mixing indistinguishable systems leads to a quantum Gibbs paradox. Nat Commun 12, 1471 (2021).


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