Work and entropy of mixing in isolated quantum systems

TL;DR

This paper investigates the entropy of mixing in isolated quantum systems with non-equilibrium states, revealing how measurement resolution influences entropy and work extraction, and resolving the Gibbs paradox within a quantum framework.

Contribution

It introduces an observer-dependent framework for entropy and work in isolated quantum systems, extending the thermodynamics of mixing beyond equilibrium and classical cases.

Findings

Entropy of mixing is a special case of observational entropy.

Discontinuous entropy increase does not enhance work extraction if particles are indistinguishable.

Derived a Landauer-like bound on energy differences based on accessible information.

Abstract

The mixing of two different gases is one of the most common natural phenomena, with applications ranging from CO$_2$ 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.