Ergotropic rearrangement of phase space density

TL;DR

This paper generalizes the concept of ergotropy for classical systems by introducing ergotropic rearrangement, enabling analysis of energy extractability without restrictions on phase space density shape.

Contribution

It provides a new, general expression for ergotropy applicable to arbitrary phase space densities, extending previous results limited to continuous, plateau-free densities.

Findings

Ergotropy can be expressed via phase space rearrangement as a function problem.

Any density of the form ρ=f(H₀) becomes asymptotically passive in the thermodynamic limit.

The approach generalizes symmetric decreasing rearrangement to classical phase space densities.

Abstract

The explicit expression of ergotropy (a.k.a. available energy) of a classical system is known for the case when the system phase space density is continuous and with no plateaus. Here we provide the general expression of ergotropy that applies without those limitations. It easily follows upon casting the ergotropy problem as a function rearrangement problem. This leads to the notion of "ergotropic rearangement" which generalises that of "symmetric decreasing rearrangement" (an advanced topic of measure theory). We apply it to investigate the fate of classical ergotropy in the thermodynamic limit, and find that any density of the form $\rho=f(H_0)$ is asymptotically passive, where $H_0$ is the system Hamiltonian and $f$ a generic function.