Abstract
A system that violates detailed balance evolves asymptotically into a non-equilibrium steady state (NESS) with non-vanishing currents. Analogously, when detailed balance holds at any instant of time but the system is driven through time-periodic variations of external parameters, it evolves toward a time-periodic state, which can also support non-vanishing currents. In both cases the maintenance of currents throughout the system incurs a cost in terms of entropy production. Here we compare these two scenarios for one dimensional diffusive systems with periodic boundary condition, a framework commonly used to model biological and artificial molecular machines. We first show that the entropy production rate in a periodically driven system is necessarily greater than that in a stationary system without detailed balance, when both are described by the same (time-averaged) current and probability distribution. Next, we show how to construct both a NESS and a periodic driving that support a given time averaged probability distribution and current. Lastly, we show that although the entropy production rate of a periodically driven system is higher than that of an equivalent steady state, the difference between the two entropy production rates can be tuned to be arbitrarily small.
Highlights
A system that violates detailed balance evolves asymptotically into a nonequilibrium steady state with non-vanishing currents
It is natural and potentially fruitful to ask: are Stochastic Pumps (SP) and nonequilibrium steady states (NESS) essentially equivalent in terms of currents, probabilities and entropy production? In other words – can any current, probability distribution and entropy production achievable using one type of driving can be achieved with the other type as well? In terms of potential applications, this question can be stated as follows: can an artificial molecular motor driven by periodic changes in the environment exactly mimic a biological molecular motor driven by consuming fuel? For finite-state systems, this question has been recently addressed in [20], where it was shown that SP and NESS are equivalent – both systems can in principle have the same time-averaged probabilities, currents and entropy production rates
In this work we discussed similarities and differences between two types of driving that maintain a diffusive system on a ring out of equilibrium: periodic variations of a potential along the ring, and static driving by breaking the detailed balance condition
Summary
We aim to compare two types of driving in diffusive systems: the first is performed by the breakage of detailed balance in a time-independent system, and the second concerns the time-periodic variations of parameters of a detailed balanced system. Eq(1) sets the basic model for a diffusive system driven by periodic variations of external parameters, commonly referred to as a stochastic pump or as a thermal ratchet [30, 31] In this model, the time dependence of the driving is encoded in the temporal variations of the potential U (x, t). As we have just shown, in contrast with discrete state models, for diffusive systems in NESS the current and probability distribution uniquely define the entropy production, Eq(10). We establish that if a given NESS and SP support the same probability and current (after time-averaging in the case of the SP), the entropy production in the SP is no less than that in the NESS. This implies a lower bound, Eq 13, for the time-averaged entropy production of a SP
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