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We investigate a kinetic heat engine model constituted by particles enclosedin a box where one side acts as a thermostat and the opposite side is a pistonexerting a given pressure. Pressure and temperature are varied in a cyclicalprotocol of period $\tau$ : their relative excursions, $\delta$ and $\epsilon$respectively, constitute the thermodynamic forces dragging the systemout-of-equilibrium. The analysis of the entropy production of the system allowsto define the conjugated fluxes, which are proportional to the extracted workand the consumed heat. In the limit of small $\delta$ and $\epsilon$ the fluxesare linear in the forces through a $\tau$-dependent Onsager matrix whoseoff-diagonal elements satisfy a reciprocal relation. The dynamics of the pistoncan be approximated, through a coarse-graining procedure, by a Klein-Kramersequation which - in the linear regime - yields analytic expressions for theOnsager coefficients and the entropy production. A study of the efficiency atmaximum power shows that the Curzon-Ahlborn formula is always an upper limitwhich 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 directlyrelated.

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