Unified theory of classical and quantum ergotropy

TL;DR

This paper develops a unified analytical framework for ergotropy, the maximum extractable energy, applicable to both classical and quantum systems, revealing classical limits and quantum signatures across scales.

Contribution

It provides the first general analytical expression for classical ergotropy and demonstrates its emergence as the classical limit of quantum ergotropy in ergodic systems.

Findings

Derived a universal formula for classical ergotropy applicable to all system sizes and interactions.

Showed quantum ergotropy decomposition persists in the classical regime, indicating non-quantum origins of coherence.

Solved the longstanding open problem of ergotropy extraction in classical systems.

Abstract

Quantifying the ergotropy (a.k.a. available energy), namely the maximal amount of energy that can be extracted from a thermally isolated system, is a central problem in quantum thermodynamics. Notably, the same problem has been long studied for classical systems as well, e.g. in plasma physics and astrophysics, where the basic principles for its solution are known for the case of collisionless fluids. Here we provide the general analytical expression of ergotropy of classical systems valid regardless of their size and the type of interparticle interactions, and show that it emerges as the classical limit of the quantum expression of ergotropy, for quantum systems that are classically ergodic. We thus establish a unified theory of classical and quantum ergotropy, whose applicability ranges from atomic to galactic scale. Such unified theory is indispensable for studying the genuine quantum signatures of ergotropy: We show that the celebrated decomposition of quantum ergotropy into coherent ant inchoherent parts survives in the classical regime, indicating that coherences do not necessarily reveal quantumness. The unified theory also allows to port tools and methods across the classical-quantum boundary to unlock the solution of standing problems. We apply this to swiftly solve the open problem of ergotropy extraction in the classical regime.