Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature Variations

TL;DR

This paper develops a comprehensive thermodynamic framework for small systems driven by periodic temperature changes, deriving new bounds on power and efficiency, and illustrating these concepts with a colloidal heat engine example.

Contribution

It introduces a general approach to analyze thermodynamics of micro- and nano-systems under periodic driving, including new bounds on power and efficiency.

Findings

Derived a general expression for entropy production without linear response assumptions.

Established a symmetry in kinetic coefficients related to microscopic reversibility.

Proved a new efficiency-dependent power bound that vanishes at Carnot efficiency.

Abstract

We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This set-up covers, in particular, periodic micro and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck type dynamics, we show that these coefficients can be expressed as a sum of an adiabatic contribution and one reminiscent of a Green-Kubo expression that contains deviations from adiabaticity. Furthermore, we show that the generalized kinetic coefficients fulfill an Onsager-Casimir type symmetry tracing back to microscopic reversibility. This symmetry allows for non-identical off-diagonal coefficients if the driving protocols are not symmetric under time-reversal. We then derive a novel constraint on the kinetic coefficients that is sharper than the second law and provides an efficiency-dependent bound on power. As one consequence, we can prove that the power vanishes at least linearly when approaching Carnot efficiency. We illustrate our general framework by explicitly working out the paradigmatic case of a Brownian heat engine realized by a colloidal particle in a time-dependent harmonic trap subject to a periodic temperature profile. This case study reveals inter alia that our new general bound on power is asymptotically tight.