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Ergotropic rearrangement of phase space density

Michele Campisi

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 $ρ=f(H_0)$ is asymptotically passive, where $H_0$ is the system Hamiltonian and $f$ a generic function.

Ergotropic rearrangement of phase space density

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 is asymptotically passive, where is the system Hamiltonian and a generic function.

Paper Structure

This paper contains 5 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quick review of classical ergotropy
  3. Ergotropic rearrangement
  4. Fate of ergotropy in the thermodynamic limit
  5. Conclusions

Figures (1)

  • Figure 1: The energy shell $S$, and its ergotropic rearrangement $\breve S$. The sets are displayed in the momenta space $(p_1,p_2)$, for the case of only two degrees of freedom. The set $S$ is bounded by the hypersurfaces of energy $E$ and $E-\varepsilon$. The set $\breve S$ is bounded by the hypersurface of energy $E'$, Eq. (\ref{["eq:E'"]}). $\rho$ and $\breve \rho$ are evenly distributed on $S$ and $\breve S$, respectively.