Unified theory of classical and quantum ergotropy
Michele Campisi
TL;DR
The paper develops a unified theory of ergotropy for classical and quantum systems under Liouville dynamics, showing that classical ergotropy is the classical limit of quantum ergotropy for classically ergodic dynamics. By extending Gardner’s ground-state framework to general Hamiltonian dynamics, it derives a closed-form classical ergotropy $\mathcal{E}_c = \int d\Phi [P_0(\Phi) - P_1(\Phi)] E_0(\Phi)$, with $P_1(\Phi)=\Sigma^{-1}(\Omega_0(H_0))$, and identifies the passive companion $\rho_1$ via $\rho_1(\mathbf z) = \Sigma^{-1}(\Omega_0(H_0(\mathbf z)))$. It shows that the quantum-to-classical decomposition survives the transition, i.e., $\mathcal{E}_c = \mathcal{E}_c^c + \mathcal{E}_c^i$, implying coherences do not necessarily signal quantumness. It also proposes a quench-adiabat (QA) protocol that can extract ergotropy in the classical regime by suddenly quenching to $H_1 = f(\rho_0)$ and adiabatically returning to $H_0$, providing a practical route to energy extraction in Liouville systems. Overall, the framework unifies disparate results across plasma physics and quantum thermodynamics and lays groundwork for cross-boundary methods and the systematic study of genuine quantum signatures in non-equilibrium energy exchange.
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.