Efficiency at maximum power: An analytically solvable model for stochastic heat engines

We study a class of cyclic Brownian heat engines in the framework of finite-time thermodynamics. For infinitely long cycle times, the engine works at the Carnot efficiency limit producing, however, zero power. For the efficiency at maximum power, we find a universal expression, different from the endoreversible Curzon-Ahlborn efficiency. Our results are illustrated with a simple one-dimensional engine working in and with a time-dependent harmonic potential. Introduction. – According to Carnot, the second law leads to a maximal efficiency ηC ≡ 1 − Tc/Th for heat engines working between two heat baths at temperatures Th > Tc. However, this efficiency can only be achieved in the quasistatic limit, where transitions occur infinitesimally slowly and hence the power output vanishes. For real heat engines, it is more meaningful to calculate the efficiency at the maximal power output of the machine. The Curzon-Ahlborn efficiency at maximum power ηCA ≡ 1 − √ Tc/Th was originally derived for a Carnot-engine with finite thermal resistance coupling to the thermal reservoirs [1]. This behaviour is even recovered if path optimization techniques are used to maximize the work output with respect to the driving scheme, see [2] and references therein. Recent extensions [3,4] of this apparent universal law to an infinitesimal series of coupled heat engines operating in a linear response regime (i.e. between heat baths with small temperature difference) have again risen the question whether the Curzon-Ahlborn result is quite generally a bound on the efficiency of heat engines working at maximum power output. In contrast to the macroscopic heat engines considered in conventional endoreversible thermodynamics, thermal fluctuations play a crucial role in most biologically relevant systems and dynamics cannot be described on a deterministic (macroscopic) level. In this regime, models for thermodynamic machines must incorporate fluctuation effects and thus allow also for backward steps even in a directed motion [5,6]. Brownian motors, in contrast to genuine heat engines, are mostly driven by time-dependent potentials or chemical potential differences. In the last two decades, a variety of aspects of such Brownian motors has been studied, including dynamics [6,7], stochastic energetics [8–11] and efficiencies [12–14], both on a continuous (Langevin equation) and a discrete (Master equation) state space. Since biological motors are typically driven far out of equilibrium, linear response thermodynamics is not appropriate to describe these systems. The seminal work of Sekimoto [8, 9] opened the door for a thermodynamic description of Langevin systems driven far out of equilibrium. Thermodynamic quantities such as work, heat, internal energy, and entropy can even be defined on a single stochastic trajectory [9, 15, 16], yielding the respective ensemble quantities after averaging. Brownian heat engines have been investigated with a model system using a spatial temperature profile and a ratchet potential [17–19]. This thermal ratchet model has also been extended to a Brownian particle in the underdamped regime [20]. Efficiencies of such ratchet heat engines [8, 21–23] and related Brownian heat engines [24] have been calculated. Recent studies on the efficiency at maximum power, however, have either been restricted to the linear response regime [23] or use a questionable definition of work (which allows for the efficiency to exceed the Carnot bound) [22]. For a ratchet heat engine on a discrete state space, a violation of the Curzon-Ahlborn bound has been found [25]. Although widely used as a model system for a stochastic heat engine, ratchet heat engines are hard to realize experimentally because of the necessary temperature gra-