Linear and non-linear thermodynamics of a kinetic heat engine with fast transformations

TL;DR

This paper models a kinetic heat engine with cyclical pressure and temperature variations, analyzing its thermodynamic behavior, linear response, and efficiency limits, including the Curzon-Ahlborn bound, in both linear and non-linear regimes.

Contribution

It introduces a detailed kinetic model of a heat engine with explicit analysis of linear and non-linear thermodynamics, including Onsager relations and efficiency bounds.

Findings

Onsager matrix is τ-dependent with reciprocal off-diagonal elements.

Efficiency at maximum power approaches Curzon-Ahlborn limit outside linear regime.

Analytic expressions for Onsager coefficients derived from Klein-Kramers dynamics.

Abstract

We investigate a kinetic heat engine model constituted by particles enclosed in a box where one side acts as a thermostat and the opposite side is a piston exerting a given pressure. Pressure and temperature are varied in a cyclical protocol of period $\tau$ : their relative excursions, $\delta$ and $\epsilon$ respectively, constitute the thermodynamic forces dragging the system out-of-equilibrium. The analysis of the entropy production of the system allows to define the conjugated fluxes, which are proportional to the extracted work and the consumed heat. In the limit of small $\delta$ and $\epsilon$ the fluxes are linear in the forces through a $\tau$-dependent Onsager matrix whose off-diagonal elements satisfy a reciprocal relation. The dynamics of the piston can be approximated, through a coarse-graining procedure, by a Klein-Kramers equation which - in the linear regime - yields analytic expressions for the Onsager coefficients and the entropy production. A study of the efficiency at maximum power shows that the Curzon-Ahlborn formula is always an upper limit which is approached at increasing values of the thermodynamic forces, i.e. outside of the linear regime. In all our analysis the adiabatic limit $\tau \to \infty$ and the the small force limit $\delta,\epsilon \to 0$ are not directly related.