Entropic bounds on currents in Langevin systems

Abstract

We derive a bound on generalized currents for Langevin systems in terms of the total entropy production in the system and its environment. For overdamped dynamics, any generalized current is bounded by the total rate of entropy production. We show that this entropic bound on the magnitude of generalized currents imposes power-efficiency tradeoff relations for ratchets in contact with a heat bath: Maximum efficiency-Carnot efficiency for a Smoluchowski-Feynman ratchet and unity for a flashing or rocking ratchet-can only be reached at vanishing power output. For underdamped dynamics, while there may be reversible currents that are not bounded by the entropy production rate, we show that the output power and heat absorption rate are irreversible currents and thus obey the same bound. As a consequence, a power-efficiency tradeoff relation holds not only for underdamped ratchets but also for periodically driven heat engines. For weak driving, the bound results in additional constraints on the Onsager matrix beyond those imposed by the second law. Finally, we discuss the connection between heat and entropy in a nonthermal situation where the friction and noise intensity are state dependent.

Theoretical perspectives on biological machines

“…General and elegant proofs for TUR have not only been given for the case of Markov jump processes on kinetic networks by employing the large deviation theory (Gingrich et al, 2016), but also can be deduced from the equality relation for the Fano factor of entropy production for over-damped Langevin processes (Pigolotti et al, 2017). More recently, Dechant and Sasa (Dechant and Sasa, 2018b) have generalized Eq.66 to underdamped processes in the following form…”

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

“…Note that there is no unique way to choose odd parameters for the time-reversal process. For example, one may regard a magnetic field as an odd parameter, so change the sign of a magnetic field in the time-reverse dynamics [21,47]. On the other hand, one may keep the sign of the magnetic field for the irreversibility [48][49][50][51][52], where the † dynamics is identical to the original time-forward dynamics.…”

“…The velocity v and work W mr are stochastic random variables. From the equation of motion (21) and the definition of work, one can write the stochastic differential equation for the velocity and work asv…”

Operationally Accessible Bounds on Fluctuations and Entropy Production in Periodically Driven Systems

“…For steady-state heat engines, the relation shows that an inevitable side-effect of reaching Carnot efficiency at finite power are diverging power fluctuations [29][30][31][32][33]. Refinements and generalizations of the TUR have been found for diffusive dynamics [34][35][36], for data allocated over a finite time [37][38][39][40], for ballistic transport [41], for underdamped Langevin dynamics with and without magnetic fields [42][43][44]. Rather than looking at the fluctuations of currents, it is also possible to constrain the fluctuations of time-symmetric quantities like residence times or activity [45][46][47] and the fluctuations of first passage times [48,49].…”