Work Fluctuations and Jarzynski Equality in Stochastic Resetting

Abstract

We consider the paradigm of an overdamped Brownian particle in a potential well, which is modulated through an external protocol, in the presence of stochastic resetting. Thus, in addition to the short range diffusive motion, the particle also experiences intermittent long jumps which reset the particle back at a preferred location. Due to the modulation of the trap, work is done on the system and we investigate the statistical properties of the work fluctuations. We find that the distribution function of the work typically, in asymptotic times, converges to a universal Gaussian form for any protocol as long as that is also renewed after each resetting event. When observed for a finite time, we show that the system does not generically obey the Jarzynski equality which connects the finite time work fluctuations to the difference in free energy, albeit a restricted set of protocols which we identify herein. In stark contrast, the Jarzynski equality is always fulfilled when the protocols continue to evolve without being reset. We present a set of exactly solvable models, demonstrate the validation of our theory and carry out numerical simulations to illustrate these findings.Introduction.-Stochastic thermodynamics is a cornerstone in non-equilibrium statistical physics [1][2][3][4][5]. Microscopic systems satisfy stochastic laws of motion governed by force fields and thermal fluctuations which arise due to the surrounding. The subject then teaches us that thermodynamic observables such as work, heat, entropy production etc. measured along the stochastic trajectories taken from ensembles of such dynamics will fluctuate too. Understanding the distribution and the statistical properties of these fluctuations is of great interest since they hold a treasure trove of information about microscopic systems and how they respond to external perturbation. Indeed there has been a myriad of studies to understand e.g., non-equilibrium dynamics of biopolymers [6,7], colloidal particles [8][9][10][11][12][13], efficiency of molecular bio-motors [14,15] and microscopic engines [16], heat conduction [17,18], electronic transport in quantum systems [19], trapped-ion systems [20] and many more [21]. Despite there exists a long catalogue of such diverse small systems with no apparent similarity, it is quite remarkable to find universal relations which are obeyed regardless. One of the most celebrated ones is perhaps the Jarzynski equality (JE) that relates the non-equilibrium fluctuations of the work to the equilibrium free energy difference [22][23][24]. Universalities of such kind have always been considered as a holy grail in physical sciences and in this paper we seek out for thermodynamic invariant principles in stochastic resetting systems [25].Dynamics with stochastic reset has drawn a lot of attention recently because of its rich non-equilibrium properties [25][26][27][28][29][30][31][32][33][34][35][36][37] and its broad applicability in first passage processes [38][39][40][41][42][43][44][45][46][47]. Nevertheless, thermodyna...

Stochastic resetting and applications

“…They showed how the Hatano-Sasa integral fluctuation theorem [154] which pertains to nonequilibrium steady states is also valid for systems with resetting. Further integral theorems have been considered in [153,156].…”

Entropy production in systems with unidirectional transitions

“…Stochastic resetting is a mechanism in which the system undergoes a stochastic dynamics in the state space as well as stochastically resets to a prescribed location with a given transition rate (i.e., a unidirectional process) [16,25,26,[49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64]. These resetting transitions involve jumps of the system into given locations and can be called the controlled transitions (i.e., these can be tuned from external sources).…”

Optimal mean first-passage time for a Brownian searcher subjected to resetting: Experimental and theoretical results

“…In standard models of resetting one usually assumes instantaneous resetting [5], which is, however, impossible to achieve experimentally. There have been recent theoretical studies that incorporate a "refraction" period before the particle's position is reset [20,[31][32][33]. In our experiment, we have an analog of this refraction period: After the period T , we switch on the optimal harmonic trap and we let the particle relax back to its equilibrium thermal distribution.…”

“…In our experiment, we have an analog of this refraction period: After the period T , we switch on the optimal harmonic trap and we let the particle relax back to its equilibrium thermal distribution. This relaxation can, in principle, be made arbitrarily fast using, e.g., the recently developed "engineered swift equilibration" (ESE) technique [33][34][35][36][37]. In our experiment we determine the characteristic relaxation time inside the trap τ c = /k.…”