Optimisation of an active heat engine

TL;DR

This paper develops an exact mapping for active heat engines using a time-dependent effective temperature, enabling optimization of their power and efficiency, and clarifies the importance of a consistent temperature definition for thermodynamic efficiency.

Contribution

It introduces a novel exact mapping between passive and active models, defining an effective temperature for active engines, and derives explicit formulas for optimal cycle parameters and efficiency.

Findings

Effective temperature $T_{eff}(t)$ differs from known active temperatures.

Optimal cycle parameters are derived for maximum power and efficiency.

In the quasi-static limit, the efficiency formula simplifies to a known thermodynamic expression.

Abstract

Optimisation of heat engines at the micro-scale has applications in biological and artificial nano-technology, and stimulates theoretical research in non-equilibrium statistical physics. Here we consider non-interacting overdamped particles confined by an external harmonic potential, in contact either with a thermal reservoir or with a stochastic self-propulsion force (active Ornstein-Uhlenbeck model). A cyclical machine is produced by periodic variation of the parameters of the potential and of the noise. An exact mapping between the passive and the active model allows us to define the effective temperature $T_{eff}(t)$ which is meaningful for the thermodynamic performance of the engine. We show that $T_{eff}(t)$ is different from all other known active temperatures, typically used in static situations. The mapping allows us to optimise the active engine, whatever are the values of the persistence time or self-propulsion velocity. In particular - through linear irreversible thermodynamics (small amplitude of the cycle) - we give explicit formula for the optimal cycle period and phase delay (between the two modulated parameters, stiffness and temperature) achieving maximum power with Curzon-Ahlborn efficiency. In the quasi-static limit, the formula for $T_{eff}(t)$ simplifies and coincides with a recently proposed temperature for stochastic thermodynamics, bearing a compact expression for the maximum efficiency. A point - overlooked in recent literature - is made about the difficulty in defining efficiency without a consistent definition of effective temperature.