Bounds on current fluctuations in periodically driven systems

Abstract

Small nonequilibrium systems in contact with a heat bath can be analyzed with the framework of stochastic thermodynamics. In such systems, fluctuations, which are not negligible, follow universal relations such as the fluctuation theorem. More recently, it has been found that, for nonequilibrium stationary states, the full spectrum of fluctuations of any thermodynamic current is bounded by the average rate of entropy production and the average current. However, this bound does not apply to periodically driven systems, such as heat engines driven by periodic variation of the temperature and artificial molecular pumps driven by an external protocol. We obtain a universal bound on current fluctuations for periodically driven systems. This bound is a generalization of the known bound for stationary states. In general, the average rate that bounds fluctuations in periodically driven systems is different from the rate of entropy production. We also obtain a local bound on fluctuations that leads to a trade-off relation between speed and precision in periodically driven systems, which constitutes a generalization to periodically driven systems of the so called thermodynamic uncertainty relation. From a technical perspective, our results are obtained with the use of a recently developed theory for 2.5 large deviations for Markov jump processes with time-periodic transition rates.

Effect of a magnetic field on the thermodynamic uncertainty relation

“…Since its first discovery [1], the thermodynamic uncertainty relation has been proven for Markov jump dynamics on a discrete set of states [2,4,5] and Langevin dynamics [5][6][7][8][9] for continuous variables. Generalizations to finite time statistics [10][11][12] and to periodically driven systems [4,13,14] have also been reported.…”

“…The scatter plot shows that the uncertainty product is always greater than the bound depicted as solid red line. The bound takes on 1 for κ0B0 < 0 and otherwise corresponds to Eq (13)…”

Thermodynamic uncertainty relation in quantum thermoelectric junctions

“…(1)] implies that a more precise output requires higher entropy production (cost). Given its fundamental and conceptual importance, the TUR has been refined [8,9] and generalized to finite times [10][11][12], discrete time and periodic dynamics [13][14][15][16], multidimensional systems [17], and bounds on counting observables and first-passage times [18,19], with applications to biochemical motors [20], heat engines [17,[21][22][23] and a variety of nonequilibrium problems [24][25][26]. Specifically, for an engine operating in a nonequilibrium steady state, the TUR translates into a trade-off relation between the output power, power fluctuations, and the engine's efficiency: According to the bound, power fluctuations diverge when operating an engine at finite power while approaching the Carnot efficiency [21].…”

Thermodynamic Cost, Speed, Fluctuations, and Error Reduction of Biological Copy Machines

“…This form of TUR holds for most of biological processes that can be represented either by stochastic jump processes on a kinetic network or by overdamped Langevin dynamics [23][24][25][26], though extensions to more general conditions, which adjust the lower bound of the original relation, have also been discussed in recent years [14,[27][28][29][30][31][32][33][34]. Briefly,…”