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Several recent theories address the efficiency of a macroscopic thermodynamic motor at maximum power and question the so-called Curzon-Ahlborn (CA) efficiency. Considering the entropy exchanges and productions in an n-sources motor, we study the maximization of its power and show that the controversies are partly due to some imprecision in the maximization variables. When power is maximized with respect to the system temperatures, these temperatures are proportional to the square root of the corresponding source temperatures, which leads to the CA formula for a bi-thermal motor. On the other hand, when power is maximized with respect to the transitions durations, the Carnot efficiency of a bi-thermal motor admits the CA efficiency as a lower bound, which is attained if the duration of the adiabatic transitions can be neglected. Additionally, we compute the energetic efficiency, or sustainable efficiency, which can be defined for n sources, and we show that it has no other universal upper bound than 1, but that in certain situations, favorable for power production, it does not exceed 1/2.
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Molecular motors transduce chemical energy obtained from hydrolizing ATP into mechanical work exerted against an external force. We calculate their efficiency at maximum power output for two simple generic models and show that the qualitative behaviour depends crucially on the position of the transition state. Specifically, we find a transition state near the initial state (sometimes characterized as a power stroke) to be most favorable with respect to both high power output and high efficiency at maximum power. In this regime, driving the motor further out of equilibrium by applying higher chemical potential differences can even, counter-intuitively, increase the efficiency.
We investigate the efficiency of systems of molecular motors operating at maximum power. We consider two models of kinesin motors on a microtubule: for both the simplified and the detailed model, we find that the many-body exclusion effect enhances the efficiency at maximum power of the many-motor system, with respect to the single motor case. Remarkably, we find that this effect occurs in a limited region of the system parameters, compatible with the biologically relevant range.
We study the efficiency at maximum power, $eta^*$, of engines performing finite-time Carnot cycles between a hot and a cold reservoir at temperatures $T_h$ and $T_c$, respectively. For engines reaching Carnot efficiency $eta_C=1-T_c/T_h$ in the reversible limit (long cycle time, zero dissipation), we find in the limit of low dissipation that $eta^*$ is bounded from above by $eta_C/(2-eta_C)$ and from below by $eta_C/2$. These bounds are reached when the ratio of the dissipation during the cold and hot isothermal phases tend respectively to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency $eta_{CA}=1-sqrt{T_c/T_h}$ is recovered.
Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is theoretically studied. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the side wall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call stochastic mean field model.
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.
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