Work Fluctuations and Jarzynski Equality in Stochastic Resetting

Work ﬂuctuations and Jarzynski equality in stochastic resetting

Deepak Gupta,1 Carlos A. Plata,1 and Arnab Pal2, 3, 4, ∗ 1Dipartimento di Fisica ‘G. Galilei’, INFN, Universitádi Padova, Via Marzolo 8, 35131 Padova, Italy 2School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel 3Center for the Physics and Chemistry of Living Systems. Tel Aviv University, 6997801, Tel Aviv, Israel 4The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel (Dated: February 4, 2020) 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. Finally, we have pointed out possible realistic implementations for resetting in experiments using the so-called engineered swift equilibration.

Introduction.— Stochastic thermodynamics is a cor- of thermodynamics were interpreted by identifying the nerstone in non-equilibrium statistical physics [1–5]. Mi- contributions to the total entropy production [53], and croscopic systems satisfy stochastic laws of motion gov- furthermore it was shown to satisfy a universal integral erned by force fields and thermal fluctuations which arise fluctuation relation [54]. While these first studies focused due to the surrounding. The subject then teaches us that exclusively on the entropy production, efforts are yet to thermodynamic observables such as work, heat, entropy be made to understand other response functions. More- production etc. measured along the stochastic trajecto- over, not much is known about the distribution of these ries taken from ensembles of such dynamics will fluctuate observables. In particular, one important observable is too. Understanding the distribution and the statistical the work function which is produced due to external per- properties of these fluctuations is of great interest since turbations to the system. Work statistics encodes im- they hold a treasure trove of information about micro- portant features of an out-of-equilibrium thermodynamic scopic systems and how they respond to external pertur- process but its computation is usually quite daunting. bations. Indeed there has been a myriad of studies to un- Here, we set out to characterize work fluctuations in a derstand e.g., non-equilibrium dynamics of biopolymers stochastic system which is subjected to resetting. Our de- [6,7], colloidal particles [8–14], efficiency of molecular tailed analysis to this account then reveals emergence of bio-motors [15, 16] and microscopic engines [17], heat robust universal pattern in work-fluctuations: firstly re- conduction [18, 19], electronic transport in quantum sys- setting renders work-fluctuations Gaussian independent tems [20], trapped-ion systems [21] and many more [22]. of the nature of the external perturbation that produces Although we observe such diverse small systems with no it. Secondly, work fluctuations are found to obey the JE apparent similarity, it is remarkable to find that there under certain conditions which we identify through this exist some universal relations which are shared in com- comprehensive study. mon. One of the most celebrated ones is perhaps the General theory.— For the sake of generality, we put Jarzynski equality (JE) that relates the non-equilibrium forward our results in the paradigmatic framework of a fluctuations of the work to the equilibrium free energy one-dimensional overdamped Brownian particle in a po- difference [23–25]. Universalities of such kind have al- tential U(x, λ(t)), which is modulated externally through arXiv:1909.08512v2 [cond-mat.stat-mech] 2 Feb 2020 ways been considered as an important feature in physical the protocol λ(t). Motion of such a particle is governed sciences and in this paper we seek out for thermodynamic by the Langevin equation of the form invariant principles in stochastic resetting systems [26]. √ x˙(t) = −γ−1∂ U(x, λ(t)) + 2Dη(t), (1) Dynamics with stochastic reset has drawn a lot of at- x tention recently because of its rich non-equilibrium prop- where γ and D are the friction and diffusion coefficients erties [26–41] and its broad applicability in first passage respectively that satisfy the fluctuation-dissipation rela- processes [42–52]. Nevertheless, thermodynamical per- tion, i.e., Dγ = kBT , with kB being the Boltzmann con- spective of resetting systems has been largely overlooked stant and T is the temperature of the medium. We as- so far. It was only recently when first and second laws sume hη(t)i = 0 and hη(t)η(t0)i = δ(t − t0). Moreover, let 2 us consider that the position of the particle at t = 0 is distributed according to the probability density function (PDF) pini(x0). At random times taken from an exponential distribution f(t) = re−rt, the particle in motion is stopped and teleported to the initial configuration. The external modulation of the potential performs work on the system which can be defined as [23]

Z t 0 0 0 1 0 ∂U[x(t ), λ(t )] dλ(t ) W = dt 0 , (2) kBT 0 ∂λ dt measured in units of kBT . In what follows, we will set kBT = D = 1 without any loss of generality. Since the reset process is instantaneous, we will assume that no work was done during this course (a finite FIG. 1: Schematic of a Brownian particle confined in a har- resetting-work will be discussed later). In order to quan- monic trap U(x, λ(t)) = κ(t)[x − y(t)]2/2, where λ(t) = tify the work fluctuations, it is convenient to first define {y(t), κ(t)} represents the set of time-dependent protocols the moment generating function (MGF) namely which are independently regulated. λ = y(t) and λ = κ(t) Z ∞ indicate the center of the trap (panel A) and the stiffness −kW (panel B) respectively. The resetting mechanism acts both on Hr(k, t) ≡ dW e Pr(W, t) (3) −∞ the particle and the protocols as mentioned in the text. Here, we show the modulation of the protocol when it is renewed where Pr(W, t) is the PDF of the work at time t, averaged after each resetting (panel C) or remains unaltered (panel D). over the initial distribution pini(x0) and the underlying dynamics with stochastic resetting. To delve deeper, we make use of the renewal structure in resetting dynamics of W recursively from the knowledge of the moments of to construct a relation that connects the MGF for r > 0 the process without resetting. to that of r = 0 for any initial and subsequent resetting Universal work fluctuations.—The infinite set of positions moments given by Eq. (5) contains the same information as that of the full distributions Pr(W, t). However, ˜ H0(k, s + r) physical intuition tells us that not all the moments con- H˜r(k, s) = , (4) 1 − rH˜0(k, s + r) tribute significantly at long time. To see this, we consider a trajectory of time length t with multiple possible ˜ R ∞ −st where Hr(k, s) = 0 dt e Hr(k, t) and the subscript resetting events. The total work done along this long 0 indicates the observables with r = 0. We have added a trajectory can then be decomposed into the sum of the proof of Eq. (4) in [55]; but it is imperative to stress the partial works produced in each time interval between the following points here. Note that Eq. (4) holds for any resetting events. However, these intervals are statistically initial condition and naturally adheres to a fixed initial independent since the entire configuration of the system condition which was derived in [56, 57], but in the ab- (comprising the particle and the trap) is renewed after sence of any protocol λ(t). In the presence of protocol, each resetting event, and hence there are no correlations one needs to be meticulous since the structure of this between the intervals. Therefore, for a long enough ob- equation relies on the fact that λ(t) is also renewed after servation time t one would expect on an average ∼ rt each resetting. As we will see later, Eq. (4) does not hold number of resetting events and the total work W (t) can when the protocol is unaffected under resetting [55]. then be written as W ≈ W1 +W2 +W3 +···+W[rt]. Since The MGF, given by Eq. (4), can be inverted to obtain the intervals are disjoint, the Wi-s are also independent the full work statistics at a given time. Nonetheless, we and identically distributed. Moreover, if Wi-s are regular will show that it suffices to know the first and second (with finite mean and variance), one would expect that moment to predict the universal behavior of the work the distribution of W , according to the central limit the- fluctuations in the large time limit. To this end, we first orem, would converge to a Gaussian irrespective of the note that the n-th moment of W in Laplace space can nature of the potential and choice of the external protocol R ∞ −st n ∂n ˜ be written as dte hW (t)ir = n Hr(k, s) , 0 ∂(−k) k→0 1 (W − µ )2 which satisfies a recursive-renewal structure [55] P (W, t) = exp − t , (6) r p 2 2σ2 (t) 2πσW (t) W " n # s + r X n W˜ n(s) = W˜ n(s + r) + r W˜ n−l(s) W˜ l(s + r) , (5) 2 r s 0 l r 0 where the mean µt ≡ hW ir and the variance σW (t) ≡ l=1 2 2 hW ir − hW ir are computed from Eq. (5). In Fig.2, ˜ n R ∞ −st n where we have defined Wr (s) ≡ 0 dt e hW (t)ir. we demonstrate Eq. (6) in the set up of a 1D Brow- Eq. (5) gives a simple recipe to compute all the moments nian particle confined in a harmonic trap U(x, λ(t)) = 3

κ(t)[x − y(t)]2/2, where κ(t) and y(t) represent the stiﬀ- ness and center of the trap respectively (leaving details

of the simulation in [55]). z z Jarzynski equality—reset protocol. The JE relates the finite time work fluctuations to the equilibrium free energy and here we ask whether such relations hold generically in resetting systems. We consider the same set-up as before and assume that each resetting act re- z z news both the particle and the protocol. We further assume that the initial condition is taken from an equilib- FIG. 2: Numerical computation of the PDF of the rescaled work z = (W − µt)/σW (t) performed on a Brownian particle rium distribution pini(x0) ∝ exp [−U(x0, λ(0))], which is in a harmonic trap for the linear modulation of the trap center an essential prerequisite for the JE. Employing Eq. (4) i.e., y(t) = ut (panel a) and the stiffness i.e., κ(t) = κ0 + vt and substituting k = 1 there, we find a renewal expres- (panel b) respectively. Simulations are performed for fixed sion for the average of the exponentiated work which con- initial condition (FIC): pini(x0) = δ(x0) (circle markers) nects to the same with r = 0 in the Laplace space [55] and random initial condition (RIC): pini(x0) = peq(x0) ∝ exp [−U(x0, λ(0))] (square markers) respectively for each of −W the above cases. Parameters for panel (a): κ0 = 1.5, u = 0.2 −W Lt→s+r[he i0] for FIC and κ0 = 0.5, u = 0.5 for RIC respectively where Lt→s he ir = −W , (7) 1 − rLt→s+r[he i0] r = 0.5 and t = 10 are set identical for both of these cases. Similarly, parameters for panel (b): κ0 = 0.5, v = 0.002, y = where L is the Laplace transform operator. Several 0, r = 5, and t = 500 for both FIC and RIC. Numerical simu- comments are in order now. The exponential average lations are corroborated with the theoretical prediction (solid 2 √ on the RHS is along the trajectory without resetting line in both cases) given by P (z) = e−z /2/ 2π, and we see −W and therefore must satisfy the JE i.e., he [0,t] i0 = an excellent Gaussian collapse. −[F0(λ(t))−F0(λ(0))] e , where F0(λ(t)) is the free energy of the underlying system (i.e., when the dynamics is not in- terrupted by resetting) corresponding to the value of λ evaluated at time t. However, it is evident that substi- protocol at a time keeping the others fixed. The horizon- tuting this in Eq. (7) will not essentially lead to e−∆F0(t) tal lines shown in the panel correspond to the theoretical −∆F0(t) (where ∆F0(t) = F0(λ(t)) − F0(λ(0)) is the free-energy prediction of e which takes the values 1.0, 2.0 and difference) along the entire trajectory of length t in the ∼ 1.65 respectively for each of the modulations. The ex- presence of resetting i.e., JE will not be obeyed gener- act computation has been reserved to [55]. It is evident ically for any arbitrary protocol. Nonetheless, we iden- from Fig.3a that the JE holds for modulations (i) and tify the protocols which will indeed satisfy this condition. (iii), but not for modulation (ii). This happens when the modulation of the protocol ren- Jarzynski equality is invariant under non-reset ders a linear change in the free energy i.e., ∆F0(t) = αt. protocol.—The discussion so far focused on the case The trivial scenario i.e., ∆F0(t) = 0 is true under any when we reset both the protocol and the particle. In the external perturbation which is of the following form: following, we relax this condition and assume that only U(x, y(t)) = U(x − y(t)). This could happen, e.g., when the particle is reset while the protocol keeps evolving in we move the center of the trap y(t) according to some spe- time. Moreover, we consider that after each resetting cific schedule. On the other hand, the nontrivial linear event, position of the particle is drawn from the equi- change in ∆F0(6= 0) occurs when e.g., the stiffness κ(t) is librium distribution pini(x0) ∝ exp [−U(x0, λ(ti))] cor- varied exponentially as a function of time. Utilizing this responding to λ measured at the times ti of resetting. −W −∆F0(t) condition in Eq. (7), we obtain he ir = e which In this way, the particle is effectively equilibrated after holds along the entire trajectory with multiple resetting each resetting event which is essential for the JE to hold. events [55]. This construction correlates the intervals between reset- We now briefly summarize the numerical setups which ting events: since the initial configuration of a given in- are used to verify these findings. We have simulated terval depends on the time spent in the previous one and an overdamped Brownian particle in a harmonic trap hence renewal structure of Eq. (4) is lost [55]. However, U(x, λ(t)) = κ(t)[x − y(t)]2/2 in the presence of reset- notice that (i) the particle is prepared at the equilibrium −W ting (r = 0.5), and measured e till time t = 5. In state pini after each resetting event, and (ii) consequently, Fig.3a, we have shown the convergence of the statistical the equality is satisfied in any interval between two reset- −W average he ir as a function of realizations NR for the ting events. Taking these two facts into account, one can following protocol modulations (i) moving the center of show that the equality holds along the entire trajectory the trap with y(t) = 0.2t, (ii) changing stiffness with a independent of the nature of the protocol [55] −2 power law κ(t) = κ0(1 + 0.2t) , and (iii) an exponential −0.2t −W −∆F0(t) law κ(t) = κ0e . As before, we have regulated one he ir = e . (8) 4

the resetting mechanism minimum.

(ii) (ii) As a representative case, we will consider the Brown- ian particle diffusing in a harmonic trap whose stiffness (iii) (iii) is timely modulated, that is, U(x, t) = κ(t)x2/2. At time zero, the particle is prepared in equilibrium with a (i) (i) zero-mean Gaussian and standard deviation σ(0) ≡ σ0 = p 2 2 hx (0)i with κ(0) = κ0 which satisfies σ0 = kBT/κ0 from the equipartition theorem. Due to the nature of the potential, position density remains to be zero-mean −x2/2σ2(t) Gaussian at all times i.e., p (x, t) = e√ , where FIG. 3: Numerical verification of the JE: we have demon- 0 2 −W 2πσ (t) strated convergence of he ir as a function of the number of σ(t) satisfies the following evolution equation: σ(t)σ ˙ (t) = realizations N . We have used three different types of proto- R κ(t) 2 col modulations as mentioned in the main text. The analytical − γ σ (t)+D [55]. Upon first restart at time tr, the par- values of e−∆F0 , shown by the horizontal lines (dotted for (i) ticle returns from a position which is distributed accord- moving trap, dashed and solid for the (ii) power law and (iii) ing to a Gaussian distribution with σ(tr) (corresponding exponential stiffness respectively in both panels), are plotted to κ(tr)). The goal is then to design an optimal pro- against numerical points for he−W i (shown by the triangles, r tocol κ(t)(tr < t < tr + ∆r) which drives the system squares, and circles respectively). Panel (a): Reset protocol. from σ(t ) to σ in time ∆ within the most efficient en- JE is seen to hold for protocols (i) and (iii) but not (ii). Pa- r 0 r ergy consumption budget. The average work performed rameters: κ0 = 1.5 and κ0 = 0.5 respectively for the center and stiffness modulation. Panel (b): Non-reset protocol. JE during this interval (with reset-protocol) over many such holds for any protocols. Parameters: κ0 = 1 and κ0 = 0.35 trajectories is given by respectively for the center and stiffness modulation. In all the Z tr +∆r simulations, we have set r = 0.5 and t = 5. 1 2 irr hWrpi = dt σ (t)κ ˙ (t) = W1 + ∆Fr + W ,(9) 2 tr

1 2 2 where W1 = 2 [σ0κ(0) − σ (tr)κ(tr)], and ∆Fr = 2 We numerically check Eq. (8) in Fig.3b and show that 1 σ0 − kBT ln 2 is the difference of free energy be- 2 σ (tr ) indeed JE is invariant under non-reset protocol modula- tween the equilibrium states characterized by the ini- tions. tial and final variance [55]. Finally, the third term Discussion.— Up to this point, we have strictly as- tr +∆r W irr[σ ˙ (t)] = γ R dt σ˙ 2(t) can be identified as the sumed that resetting is an instantaneous process and thus tr irreversible work of the process which is always positive. neglected any possible contributions coming from it to Note that W and ∆F are determined given the initial the total work done. However, in real world taking one 1 r and target states leaving the dependence of the specific particle from location A to B will require work and this protocol and ∆ only in W irr. The optimal profile σ (t) contribution must be taken into account. Another es- r opt that minimizes W irr[σ ˙ (t)] can be immediately obtained sential aspect of the paper is the imposed equilibrated using variational calculus and this reads [55] condition upon each resetting which may appear arti- ficial from an experimental point of view. To fill up σ0 − σ(tr) σopt(t) = σ(tr) + (t − tr) . (10) these conceptual gaps, in this section, we put forward ∆r an explicit proposal for practical implementation of resetting which accounts for both these issues. To pro- Substituting this into the evolution equation for σ(t), one ceed further, recall that the unhindered process, charac- immediately finds the corresponding profile for the optimal protocol κ (t). Interestingly, κ (t) develops fi- terized by p0(x, t), satisfies the Fokker-Planck equation opt opt h 1 0 i 2 nite discontinuities at the threshold times i.e., κ(tr) 6= ∂tp0(x, t) = ∂x γ U (x, λ(t))p0(x, t) + D∂xp0(x, t). Let lim κopt(t) and κ(tr + ∆r) = κ(0) 6= lim κopt(t), us now focus on the first resetting event which occurs at t→tr t→tr +∆r as was also found in other studies [60–62]. Implement- a time tr. Then the job of this engineered restart mech- ing the protocol, we get the optimal irreversible work anism would be to take the current distribution p0(x, tr) 2 W irr = γ (σ0−σ(tr )) , which is exactly proportional to in the resetting time tr and make a transformation to opt ∆r −1 reach an arbitrary target distribution pf (x) (which in our ∆r [55]. This ensues that a perfect instantaneous reset- case is the equilibrium density) in a fixed time ∆r > 0. ting (i.e., the ∆r → 0 limit) is not physically viable (if a There have been recent developments to design proto- work indeed is accounted for the entire resetting mech- cols, namely Engineered Swift Equilibration (ESE) [58– anism) since the energetic cost for each resetting jump 61], that shortcuts the relaxation times between two tar- will be divergent. A similar analysis follows also for the get distributions whose properties can be controlled in non-reset protocol [55]. time. We will now show how to choose the optimal pro- Conclusions and outlook.— In summary, this let- tocol that renders the average irreversible work during ter discusses statistical properties of work fluctuations in 5 a stochastic resetting system. We find that the introduc- theorems and molecular machines. 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