Mixing indistinguishable systems leads to a quantum Gibbs paradox

Benjamin Yadin,Benjamin Morris,Gerardo Adesso

doi:10.1038/s41467-021-21620-7

Benjamin Yadin, Benjamin Morris + Show 1 more

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https://doi.org/10.1038/s41467-021-21620-7

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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.

Highlights

The classical Gibbs paradox concerns the entropy change upon mixing two gases
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 and let each gas do work on an external weight independently
A study of Gibbs mixing for identical quantum bosons or fermions 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[11,12,13]

Summary

Introduction

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. 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. 1234567890():,; 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 work is concerned with the transition from classical to quantum thermodynamics in the context of the Gibbs paradox[4,5,6] 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. The particular quantum properties of identical particles, including entanglement, can be valuable resources in quantum information processing tasks[19,20,21]

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