Péclet number governs transition to acceleratory restart in drift-diffusion
PP´eclet number governs transition to acceleratory restart in drift-diffusion
Somrita Ray , Debasish Mondal , and Shlomi Reuveni ∗ School of Chemistry, The Center for Physics and Chemistry of Living Systems,The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science,& The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 6997801, Israel Department of Chemistry, Indian Institute of Technology Tirupati, Tirupati 517506, India (Dated: May 14, 2019)First-passage processes can be divided in two classes: those that are accelerated by the introduction of restartand those that display an opposite response. In physical systems, a transition between the two classes may occuras governing parameters are varied to cross a universal tipping point. However, a fully tractable model system toteach us how this transition unfolds is still lacking. To bridge this gap, we quantify the effect of stochastic restarton the first-passage time of a drift-diffusion process to an absorbing boundary. There, we find that the transitionis governed by the P´eclet number ( Pe ) — the ratio between the rates of advective and diffusive transport. When Pe > ≤ Pe < ∼ / Pe . Such speedupoccurs when the process is restarted at an optimal rate r (cid:63) (cid:39) r (cid:63) ( − Pe ) , where r (cid:63) stands for the optimal restartrate in the pure-diffusion limit. The transition considered herein stands at the core of restart phenomena andis relevant to a large variety of processes that are driven to completion in the presence of noise. Each of theseprocesses has unique characteristics, but our analysis reveals that the restart transition resembles other phasetransitions — some of its central features are completely generic. I. INTRODUCTION
Restart—a situation in which a dynamical process stopsand starts anew—is a naturally occurring phenomenon as wellas an integral part of many man-made systems. Randomcatastrophes can drastically reduce the population of a livingspecies and restart its growth [1]. Computer algorithms are of-ten restarted to shorten their total run times [2], and financialcrises may result in a crash of the stock market thus resettingasset prices to past levels [3]. Search processes may also un-dergo restart e.g., when bad weather forces teams to return tobase [4], and when hunger or fatigue brings foraging animalsback to their shelters [5]. At the microscopic level, restart isan integral part of the renowned Michaelis-Menten reactionscheme and is thus crucial to the understanding of enzymaticand heterogeneous catalysis [6–9]. For these reasons and oth-ers, restart has recently attracted considerable attention.Diffusion with stochastic resetting played a central role inshaping our understanding of restart phenomena [10–20]. Inthis model, one considers a Brownian particle that is returnedto its initial position at random time epochs. This simple setupalready gives rise to non-equilibrium steady states, but themodel can also be considered in the presence of an absorb-ing boundary to provide a quintessential example of a sys-tem where restart acts to facilitate the completion of a first-passage process [21, 22]. This counter-intuitive phenomenonis not unique to diffusion [4, 23]. And yet, the fact that it canbe demonstrated so vividly with a process that is known tophysicists since the days of Einstein, Smoluchowski, and Per-rin has really brought it to center stage and welcomed the ad-vent of many important applications and generalizations [24–47]. In fact, some universal properties of first-passage underrestart [4, 23] were first demonstrated using diffusion. How-ever, since restart always acts to expedite the first-passagetime (FPT) of a free Brownian particle to a marked target,
Position V0 TimeAbsorbing boundary L FIG. 1. An illustration of drift-diffusion under stochastic restart. simple diffusion fails to display a transition that is genericallyobserved in many other systems.Restart can either hinder or facilitate the completion of astochastic process. Knowing the distribution of the randomcompletion-time, one can tell—in advance—which of the twowill happen [4, 6, 7, 23]; but in real-world systems the an-swer will typically depend on environmental conditions. In-deed, changes in governing parameters such as temperature,viscosity, and concentration, can directly affect the progres-sion of physical processes and chemical reactions, thus al-tering their stochastic completion-time distributions. Whensubject to such changes, restart may invert its role: hinderingthe completion of a process it previously facilitated and viceversa. This transition has already been predicted to have uni-versal features [4, 23], as well as dramatic real-life implica-tions [6–9], but an analytically tractable model system whereit could be understood in full has so far been missing.In many important situations diffusion occurs in the pres-ence of a bias. This could happen, e.g., when a particle is dif-fusing in a flow-field or under the influence of some potential[48]. Sometimes, the particle under consideration is not even a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y real but rather a symbolic entity that stands, e.g., for a reactioncoordinate, the price of a stock, or the height of a fluctuatinginterface. This marked generality allows drift-diffusion to de-scribe anything from charge transport [49] to biological evo-lution [50] and from polymer translocation [51, 52] to the dy-namics of neuron firing [53]. Moreover, like many other first-passage processes [4] drift-diffusion may also become subjectto restart (Fig. 1); and it then provides a non-trivial example ofa system where stochastic restart could either hinder or facil-itate the first-passage of a particle to an absorbing boundary.A transition between the two types of behavior was shown tooccur at a universal tipping point [54], but how this transitionunfolds is still unknown. Here, we address this question andshow that the answer to it reflects not only on the toy modelillustrated in Fig. 1, but also on a large variety of processesthat are driven to completion in the presence of noise.In this paper, we provide a comprehensive analysis ofdrift-diffusion under stochastic restart. We start in Sec. IIwhere we utilize the Fokker-Planck approach to derive adetailed spatio-temporal description of the process illustratedin Fig. 1. A result for the FPT to the absorbing boundaryfollows and this is further corroborated using a general ap-proach to first-passage under restart [4, 23]. Both approachesare well-suited to treat generalizations of our problem, butaside from deductive reasons the latter is given to show thatgeneralizations are not limited to processes described bythe Fokker-Planck equation. In Sec. III, we demonstratethe restart transition in drift-diffusion and explain why it isreasonable to expect that it will be governed by the P´ecletnumber ( Pe ) — the ratio between the rates of advective anddiffusive transport. In Sec. IV, we show that the P´ecletnumber decides the effect of restart in our problem, and that italso sets the optimal restart rate, i.e., the rate which minimizesthe mean FPT of the particle to the absorbing boundary. Thespeedup conferred by optimal restart is discussed in Sec. V,and it is shown that it too can be given in terms of Pe . In Sec.VI, we go on to show that the restart transition has a universalfingerprint on relative stochastic fluctuations of the FPT. InSec. VII, we continue to discuss the generic nature of ourresults. The analogy with thermodynamic phase-transitionsis then pointed out and discussed side-by-side with otherbroad implications of our findings and possible future ex-tensions of our work. Final conclusions are given in Sec. VIII. II. FIRST-PASSAGE TIME OF DRIFT-DIFFUSION UNDERSTOCHASTIC RESTARTA. Fokker-Planck approach
Consider a particle undergoing drift-diffusion in one di-mension and further assume that the particle has a constantdiffusion coefficient D and drift velocity V . This situationcould e.g., arise for diffusion in a viscous medium under a lin-ear potential U ( x ) = − U x , where U is an arbitrary constant. Letting ζ stand for the drag coefficient, the drift velocity inthis case will be V = U / ζ , and the diffusion constant willbe given by the Einstein-Smoluchowski relation D = ( β ζ ) − with β standing for the thermodynamic beta.To study the effect of resetting, we assume that the particleis further subjected to stochastic restart with a constant rate r ,which means that it is returned to its initial position, x = / r . The process ends when the particle hitsan absorbing boundary located at x = L . The master equationfor p ( x , t ) , the conditional probability density of finding theparticle at position x at time t provided its initial position was x =
0, then reads ∂ p ( x , t ) ∂ t = − V ∂ p ( x , t ) ∂ x + D ∂ p ( x , t ) ∂ x − rp ( x , t ) + r δ ( x ) Q ( t ) , (1)where δ ( x ) is a Dirac delta function and Q ( t ) : = (cid:82) L − ∞ p ( x , t ) dx denotes the survival probability, i.e., the probability that theparticle is not absorbed at the boundary by time t . We notethat when the particle hits the boundary it is immediately ab-sorbed, which sets the boundary condition to p ( L , t ) =
0. Inthe absence of restart ( r = x and a gain of probability at x =
0, where the particle restarts its motion. The last twoterms on the right hand side of Eq. (1) account for this ad-ditional probability flow.Calculating the survival probability, Q ( t ) , is our principalobjective here as this function holds all the information on thedistribution of the FPT of the particle to the boundary, but inwhat follows we will also solve for p ( x , t ) . To do this, we takethe Laplace transform of Eq. (1) and rearrange to obtain ∂ ˜ p ( x , s ) ∂ x − (cid:18) VD (cid:19) ∂ ˜ p ( x , s ) ∂ x − (cid:18) s + rD (cid:19) ˜ p ( x , s )= − − r ˜ Q ( s ) D δ ( x ) , (2)where ˜ p ( x , s ) : = (cid:82) ∞ e − st p ( x , t ) dt and ˜ Q ( s ) : = (cid:82) ∞ e − st Q ( t ) dt denote the Laplace transforms of p ( x , t ) and Q ( t ) , respec-tively. The above differential equation can be solved exactly[see Appendix A] to give˜ p ( x , s ) = + r ˜ Q ( s ) D ( α − − α + ) (cid:104) e α + ( x − L )+ α − L − e α − x (cid:105) if 0 ≤ x ≤ L + r ˜ Q ( s ) D ( α − − α + ) (cid:104) e α + ( x − L )+ α − L − e α + x (cid:105) if − ∞ < x ≤ , (3)where α ± = D (cid:104) V ± (cid:112) V + D ( s + r ) (cid:105) .The result in Eq. (3) gives p ( x , t ) in terms of Q ( t ) inLaplace space. To get the Laplace transform of Q ( t ) in a self-consistent manner from the above, we observe that ˜ Q ( s ) : = (cid:82) L − ∞ ˜ p ( x , s ) dx . Integrating Eq. (3) over the spatial coordinateand rearranging we find˜ Q ( s ) = − exp (cid:104) L D (cid:16) V − (cid:112) V + D ( s + r ) (cid:17)(cid:105) s + r exp (cid:104) L D (cid:16) V − (cid:112) V + D ( s + r ) (cid:17)(cid:105) . (4)Note that the conventional approach to first-passage usuallystarts with the backward Fokker-Planck description of the sys-tem [55]. This leads directly to the survival probability, by-passing the calculation of the spatio-temporal probability dis-tribution function. In contrast, by taking the forward Fokker-Planck approach, we can plug Eq. (4) back into Eq. (3) andobtain an explicit expression for the propagator, ˜ p ( x , s ) , inLaplace space. The forward Fokker-Planck approach thushelped us extract additional information regarding the posi-tion of the particle and this may be useful in future studies.Eq. (4) allows us to compute the FPT of the particle to theabsorbing boundary. Letting T r denote this FPT, we recallthat the probability density function of this random variableis given by − dQ ( t ) / dt [55]. This allows us to calculate anymoment of T r from ˜ Q ( s ) following the relation (cid:104) T nr (cid:105) = ( − ) n − n (cid:20) d n − ˜ Q ( s ) ds n − (cid:21) s = . (5)The above relation will be useful in calculating the standarddeviation, σ ( T r ) = (cid:113) (cid:104) T r (cid:105) − (cid:104) T r (cid:105) in Sec. VI. Here we focuson the the first moment, i.e., the mean FPT, which is given by (cid:104) T r (cid:105) = (cid:2) ˜ Q ( s ) (cid:3) s = . Thus from Eq. (4) we obtain (cid:104) T r (cid:105) = exp (cid:104) L D (cid:16) √ V + Dr − V (cid:17)(cid:105) − r . (6)Eq. (6) relates the mean FPT of a drift-diffusion process underrestart with the fundamental physical parameters that governthis problem, viz. V , D , L and r . In the next subsection, wederive the mean and the distribution of T r following a generalapproach to first-passage under restart. B. General approach
A general theory of first-passage under restart was devel-oped in [4, 23]. This theory asserts that one can write the FPTdistribution of a process that is restarted at a rate r in termsof the FPT distribution of the process without restart. Morespecifically, letting T denote the FPT of some generic processand T r its FPT under restart, we define ˜ T ( s ) : = (cid:104) exp ( − sT ) (cid:105) and ˜ T r ( s ) : = (cid:104) exp ( − sT r ) (cid:105) to be the Laplace transforms ofthese random variables. The following relation then holds [23]˜ T r ( s ) = ˜ T ( s + r ) ss + r + rs + r ˜ T ( s + r ) . (7)The mean FPT of a generic process under stochastic restartfollows directly from Eq. (7) and is given by [23] (cid:104) T r (cid:105) = r (cid:20) − ˜ T ( r ) ˜ T ( r ) (cid:21) . (8) Eqs. (7) and (8) are completely general. To demonstrate this,we will now use them to analyze drift-diffusion under stochas-tic restart.The first-passage time T of a drift-diffusion process to anabsorbing boundary is known to be governed by the followingprobability density function [56] f T ( t ) = L √ π Dt exp (cid:20) − ( L − V t ) Dt (cid:21) , (9)where once again we have L , D , and V standing respectivelyfor the initial distance from the boundary, the diffusion coef-ficient, and the drift velocity. The Laplace transform of T isalso known exactly and is given by˜ T ( s ) = exp (cid:20) L D (cid:16) V − (cid:112) V + Ds (cid:17)(cid:21) . (10)Plugging in Eq. (10) into Eq. (8), one readily recovers Eq. (6)for the mean FPT, (cid:104) T r (cid:105) . Plugging in Eq. (10) into Eq. (7), wefind ˜ T r ( s ) = s + rr + s exp (cid:16) L D (cid:104)(cid:112) V + D ( s + r ) − V (cid:105)(cid:17) . (11)To show that this result is equivalent to the one in Eq. (4) werecall that Q ( t ) = (cid:82) ∞ t f T r ( t ) dt , where f T r ( t ) is the FPT dis-tribution of drift-diffusion under restart; hence ˜ Q ( s ) = [ − ˜ T r ( s )] / s . Equivalence of Eqs. (4) and (11) is then evident. III. RESTART TRANSITION
Having obtained an explicit expression for the mean FPTof a drift-diffusion process under stochastic restart [Eq. (6)],we now turn to explore how this depends on physical param-eters. In Fig. 2, we plot the mean FPT, (cid:104) T r (cid:105) , vs. the restartrate, r for different values of the drift velocity V . For V = r , obtaining a minimum at an optimalrestart rate r = r (cid:63) , as was first observed by Evans and Majum-dar in [10]. A similar behavior is also observed when the driftvelocity is positive, but sufficiently small. However, at somepoint, as V increases, a transition occurs, and (cid:104) T r (cid:105) becomesmonotonically increasing with r . This transition can also beidentified by examining the optimal restart rate. This is maxi-mal for pure diffusion, and gradually decreases as V increasesuntil it vanishes at some critical drift velocity. When V < (cid:104) T r (cid:105) is non-monotonic [Fig. 2] and the optimal restart rate in-creases with the magnitude of the (negative) drift velocity. Inwhat follows, we focus on the V > V < V >
0) lim-its. For pure diffusion [10], substituting V = (cid:104) T r (cid:105) = (cid:104) exp (cid:16)(cid:112) rL / D (cid:17) − (cid:105) / r . Then it is easy to see V = V = V = V = V = V = - 〈 T r 〉 FIG. 2. The mean FPT from Eq. (6) vs. the restart rate for differentvalues of the drift velocity V . Here, D = L =
1. Circles indicateminima. that (cid:104) T r (cid:105) diverges for the very high and very low values of r , but is finite otherwise. Thus one expects the mean FPT tohave a minimum with respect to the restart rate. In the otherlimit, when V → ∞ , one can approximate the mean FPT as (cid:104) T r (cid:105) (cid:39) [ exp ( rL / V ) − ] / r . Taylor expanding the exponentialterm, we find that the mean FPT attains the value L / V at r = r from that point onward.Recapitulating, we see that for high values of r , (cid:104) T r (cid:105) divergesin all cases, because frequent resetting makes it harder for theparticle to hit the absorbing boundary. In contrast, the behav-ior at low values of r is sensitive to the drift velocity. Thisanalysis suggests that the mean FPT should generally showa non-monotonic variation at the diffusion-controlled regimeand a monotonic increase at the drift-controlled regime.A standard way to compare drift and diffusion is by use ofthe P´eclet number which is defined as the ratio between therates of advective and diffusive transport Pe : = LV / D . (12)The P´eclet number naturally appears in classic first-passagetime problems that involve drift-diffusion [21]. In the absenceof drift, the above definition makes it clear that Pe =
0. Asthe drift velocity increases, or as the diffusion coefficient de-creases, the P´eclet number increases. In other words, low val-ues of Pe correspond to a diffusion-controlled regime, whereashigh values of Pe correspond to a drift-controlled regime. Inthis way, the P´eclet number beautifully captures the interplaybetween drift and diffusion, and it thus makes sense to tryand characterize the transition seen in Fig. 2 in terms of thisdimensionless quantity. Before moving on, we note that for V < IV. OPTIMAL RESTART
The theory of first-passage under restart asserts that theintroduction of stochastic restart will result in a decreaseof the mean FPT whenever the ratio between the standarddeviation and mean of the FPT distribution—in the absenceof restart—is larger than unity, and vice versa [4, 6, 7, 23]. Inour case, the mean and standard deviation of the underlyingfirst-passage process can be calculated directly from Eq. (9)to give: (cid:104) T (cid:105) = L / V and σ ( T ) = (cid:112) DL / V . Their ratio, thecoefficient of variation ( CV ), is then given by CV = / √ Pe .It is thus clear that Pe = r (cid:63) and show that it too can be expressedin terms of Pe .We start by defining a reduced variable z : = L D (cid:104)(cid:112) V + Dr − V (cid:105) , (13)and rewriting the restart rate r in terms of z to give r = (cid:18) VL (cid:19) z + (cid:18) DL (cid:19) z . (14)Substituting Eq. (14) into Eq. (6), we express the mean FPTin terms of z as (cid:104) T r (cid:105) = (cid:18) L LV + Dz (cid:19) (cid:20) exp ( z ) − z (cid:21) . (15)To find the optimal restart rate, we now look for a solutionto d (cid:104) T r (cid:105) dr = LV + Dz (cid:20) d (cid:104) T r (cid:105) dz (cid:21) = . (16)Substituting Eq. (15) into Eq. (16), we obtain the followingtranscendental equation [( z − ) exp ( z ) + ] Pe = (cid:104)(cid:16) − z (cid:17) exp ( z ) − (cid:105) z . (17)We now solve Eq. (17) and calculate the optimal restart ratein terms of its solutions. In Fig. 3, we simultaneously plotthe left hand side, F ( z , Pe ) : = [( z − ) exp ( z ) + ] Pe (coloredlines), and the right hand side, F ( z ) : = (cid:2)(cid:0) − z (cid:1) exp ( z ) − (cid:3) z (black line), of Eq. (17). The solutions, denoted as z (cid:63) , are the z values for which F ( z , Pe ) and F ( z ) intersect. We observethat for Pe <
1, Eq. (17) has one non-trivial positive solution z (cid:63) >
0. In the pure-diffusion limit, Pe =
0, and z (cid:63) attains itsmaximal value z (cid:63) (cid:39) .
59. It then gradually decreases as Pe increases until it vanishes for Pe ≥
1. Following Eq. (14),and the solution of Eq. (17), the optimal restart rate for a drift-diffusion process can then be written as r (cid:63) r (cid:63) = Pe z (cid:63) + z (cid:63) z (cid:63) , (18) Pe = Pe = Pe = Pe = Pe = Pe = FIG. 3. Plots of the left-hand side (colored lines) and right-hand side(black line) of Eq. (17) vs. the reduced variable z from Eq. (13). Col-ored circles denote solutions to Eq. (17) for different P´eclet numbers. where r (cid:63) = Dz (cid:63) / L stands for the optimal restart rate inthe pure diffusion limit [10]. Recalling that z (cid:63) is uniquelydetermined by the P´eclet number, we see from Eq. (18) thatthe same can be said about the scaled optimal restart rate: r (cid:63) / r (cid:63) .In Fig. 4, we present the scaled optimal restart rate vs.the P´eclet number. We observe that optimal restart rates arestrictly positive for Pe < Pe ≥ Pe =
1. Finally, we graph-ically observe that for Pe ≤ r (cid:63) / r (cid:63) (cid:39) ( − Pe ) . (19)Eq. (19) provides a simple and effective way to approximatethe optimal restart rate for Pe ≤
1. In Sec. VII, we utilizethis expression in order to explore how temperature affectsthe restart transition.
V. MAXIMAL SPEEDUP
The mean FPT of a drift-diffusion process with a constantdrift velocity V , diffusion coefficient D , and initial distance L from an absorbing boundary, can be readily calculated fromthe FPT distribution in Eq. (9) to give (cid:104) T (cid:105) = L / V , and notethat this result does not depend on the diffusion coefficient.When the same drift-diffusion process is subjected to optimalrestart, its mean FPT can be expressed as (cid:104) T r (cid:63) (cid:105) = (cid:40) L ( LV + Dz (cid:63) ) (cid:104) exp ( z (cid:63) ) − z (cid:63) (cid:105) if 0 ≤ Pe < LV if Pe ≥ , (20) r * / r * FIG. 4. The scaled optimal restart rate from Eq. (18) vs. the P´ecletnumber from Eq. (12) (solid black line). A transition occurs at Pe = Pe values from Fig. 3.The linear approximation from Eq. (19) is marked as a dot-dashedblack line. where we substituted z in Eq. (15) by z (cid:63) and recalled that z (cid:63) = Pe ≥
1. The speedup optimal restart confers on the meanFPT of a drift-diffusion process then reads (cid:104) T (cid:105)(cid:104) T r (cid:63) (cid:105) = (cid:40) z (cid:63) + Pe z (cid:63) Pe ( exp ( z (cid:63) ) − ) if 0 ≤ Pe <
11 if Pe ≥ . (21)Equation (21) shows that the maximal speedup, just like thescaled optimal restart rate, can be uniquely characterized bythe P´eclet number.When Pe (cid:28) z (cid:63) (cid:39) z (cid:63) , suggesting a simple scaling law forthe maximal speedup (cid:104) T (cid:105) / (cid:104) T r (cid:63) (cid:105) ∼ / Pe . (22)To understand the importance of Eq. (22), consider drift-diffusion without restart. There, the mean FPT to the bound-ary is given by (cid:104) T (cid:105) = L / V which asserts that doubling the driftvelocity will half the mean time to completion. In contrast,Eq. (22) asserts that, when Pe is sufficiently small, optimalrestart will expedite completion many folds more. This pointis further discussed in Sec. VII.In Fig. 5, we plot (cid:104) T (cid:105) / (cid:104) T r (cid:63) (cid:105) as a function of the P´eclet num-ber to show that the maximal speedup diverges for a pure dif-fusion process ( Pe = Pe →
1. This figure also shows that when Pe (cid:28) Pe ≥ VI. FLUCTUATIONS
Having fully characterized the transition on the level of themean FPT, we now turn to look at stochastic fluctuations.Eq. (5) allows us to calculate all the moments of the FPT. In CV ( T r * ) 〈 T 〉 / 〈 T r * 〉 FIG. 5. Main: The maximal speedup conferred by optimal restart[Eq. (21)] vs. the P´eclet number from Eq. (12). The scaling lawfrom Eq. (22) is clearly visible. Inset: The coefficient of variationunder optimal restart from Eq. (25) vs. the P´eclet number. In boththe inset and main panel a clear transition is seen at Pe = particular, we find that the standard deviation of the FPT isgiven by σ ( T r ) = (cid:114) exp ( z ) − ( z ) Lr √ Dr + V − r , (23)where z is defined in Eq. (13). The coefficient of variation, CV ( T r ) = σ ( T r ) / (cid:104) T r (cid:105) , is a dimensionless measure of stochas-tic fluctuations in the FPT. Combining Eqs. (6) and (23), weobtain CV ( T r ) = (cid:113) exp ( z ) − z exp ( z ) Pe + z / Pe + z − ( z ) − . (24)Eq. (24) holds for any restart rate r . To demonstrate a tran-sition, we once again focus on the optimal restart rate r (cid:63) .Recalling that r (cid:63) > Pe < z exp ( z )( Pe + z / ) / ( Pe + z ) = exp ( z ) −
1, and substitute back into Eq. (24) to obtain CV ( T r (cid:63) ) =
1. For Pe ≥
1, however, r (cid:63) =
0, i.e., the processis nothing but the underlying drift-diffusion process itself. Inthis regime, CV ( T r (cid:63) ) = CV ( T ) = / √ Pe as we discussed ear-lier in Sec. IV. Summing up, we have CV ( T r (cid:63) ) = (cid:40) ≤ Pe < √ Pe if Pe ≥ . (25)The transition in Eq. (25) is graphically illustrated in the insetof Fig. 5. Importantly, it should be noted that this transition isa concrete manifestation of a more general result: CV ( T r (cid:63) ) = r (cid:63) > CV ( T r (cid:63) ) = CV ( T ) . This quantity then depends on thedetails of the underlying first-passage process and is clearlynon-universal. ∞ β τ * FIG. 6. The mean optimal time between restart events, τ (cid:63) = / r (cid:63) ,diverges at a critical temperature β − c as described by Eq. (26). Thisgeneric feature of the restart transition is akin to that observed inother phase transitions. Plot was made using Eq. (18) with parame-ters chosen such that β c = VII. DISCUSSION
A constant drift velocity emerges when a particle diffusesunder a linear potential, e.g., the potential considered herein, U ( x ) = − U x ; and note that when U > x = L ). More complicated potentials can also be considered, butnote that if the potential U ( x ) has a single minimum at the ab-sorbing boundary a restart transition is expected in the genericcase. This transition will occur, e.g., as the diffusion con-stant is ramped up from an initial value of zero or rampeddown from a very high value. Recalling that D = ( β ζ ) − , wenote that this could be achieved by heating or cooling the sys-tem, thus suggesting a critical transition temperature β − c . Forthe model analyzed in this paper, we find that β − c = LV ζ / τ (cid:63) = / r (cid:63) ∝ (cid:18) β c β − (cid:19) − , (26)which governs optimal restart in the limit β → β c (Fig. 6).The analogy with thermodynamic phase transitions is strik-ing and raises the question whether the critical exponent inEq. (26) is universal. Some evidence in support of this con-jecture have recently been given in [57], but a general andformal proof still awaits to be found.Our analysis reveals another generic feature of restartedprocesses that are driven to completion in the presence ofnoise. To illustrate this, we plot the mean FPT, (cid:104) T (cid:105) = L / V ,of a drift diffusion process in the absence of restart vs. thedrift velocity (Fig. 7, dashed black line). The mean FPT un-der optimal restart is also plotted for different values of thediffusion coefficient (Fig. 7, colored lines). The drift veloc-ity serves here as a proxy for the drive strength, and one cantrivially observe that increasing the drive always results in alower mean FPT. Moreover, if the process is driven hard—soas to overwhelm noise—restart will not carry any benefit and D = D = D = D = D = L / 〈 T r * 〉 〈 T r * 〉 FIG. 7. Main: The optimal mean FPT from Eq. (20) vs. the driftvelocity V for different values of the diffusion coefficient D (coloredlines). The mean FPT in the absence of any restart is also indicated(dashed black line). In all plots, L =
1. Inset: The optimal meanFPT for pure diffusion vs. L / D . Circles correspond to asymptotic, V →
0, values of the mean FPT in the main panel. should be avoided. However, in reality, one’s ability to drive aprocess towards a desired outcome is typically limited due topractical constraints, and restart may then come to the rescue.Fig. 7 shows that restart can effectively surrogate for drivein the high-noise/low-drive limit. Moreover, in this limit op-timal restart renders the mean FPT insensitive to the drivestrength. Thus, when the application of drive is costly, compli-cated, or hard (in comparison to restart) it is best not to drivethe system at all and restart it instead. This is clearly visiblefrom the plots which show results for simple drift-diffusion,but it is important for us to emphasize that the behaviour seenhere is totally generic. Namely, it does not depend on thedetails of the potential U ( x ) that drives the particle. Indeed,when noise is very strong one can forget about the potentialaltogether and treat the process as if it were purely diffusive.Optimal mean FPTs then follow the familiar ∼ L / D scal-ing (Fig. 7, inset) which governs pure diffusion with resetting[10]. VIII. CONCLUSIONS AND OUTLOOK
In this paper, we presented a thorough analysis of drift-diffusion under stochastic restart. We calculated the first-passage time of a particle to an absorbing boundary by twodifferent methods: first, using a Fokker-Planck description ofthe system and then using a general approach to first-passageunder restart. Closed-form formulæ were obtained for theLaplace transform of the propagator governing this processand for the distribution of the FPT to the absorbing bound-ary. Exact expressions for the mean FPT and the restart ratethat minimizes it were also obtained. For positive drift veloc-ities, this optimal restart rate was found to transition from avalue of zero attained at the drift-controlled regime to a valueof r (cid:63) (cid:39) r (cid:63) ( − Pe ) attained at the diffusion-controlled regime.The speedup conferred by optimal restart also undergoes atransition. In the diffusion-controlled regime optimal restart Unfolded ProteinFolded ProteinUnfolding via Chaperone Misfolded
Protein
FIG. 8. A schematic of protein folding envisioned as diffusion in arugged potential landscape. Chaperones save misfolded proteins byrestarting the folding process, but the connection with the problemof first-passage under restart has so far been overlooked. can reduce the mean FPT by a factor of ∼ / Pe , which canbe huge for small Pe . However, in the drift-controlled regimethe optimal restart rate is zero, i.e., restart cannot acceleratethe process at all. Another hallmark of the transition is ob-served when examining relative fluctuations of the FPT underoptimal restart. These deviate from a universal value of unityas one transitions from the diffusion-controlled regime to thedrift-controlled regime.Taken together, our results provide a comprehensive quan-tification of the effect stochastic restart has on drift-diffusion.It also suggests drift-diffusion as a concrete model systemwhere first-passage under restart can be experimentally real-ized to demonstrate the restart transition. Drift-diffusion canbe realized experimentally, e.g., with a colloidal particle ina flow chamber. Such setup was recently used as part of anexperimental demonstration of an information machine [58].There, a barrier made of light was used to prevent the particlefrom being carried away by the flow, but light can also be usedto trap and manipulate the particle. Indeed, optical tweezerscan be used to return the particle to its initial position when-ever it fails to hit the boundary within a given time windowthat could, in principle, be random. Experimental realizationof this setup could thus be used to demonstrate some of theinteresting physics that was predicted above, and in particularserve to show that in the diffusion-controlled regime optimalrestart sets the relative standard deviation of the FPT to unity[23]. This would be the first experimental verification of thisuniversal principle.Finally, we note that the approach taken herein can also beapplied to diffusion in higher dimensions and in complicatedpotential landscapes. Strong motivation to do so comes e.g.,from the problem of protein folding (Fig. 8). There, the setof unfolded states are high in energy, and the natively foldedstate is depicted as the global minimum of the energy func-tion. Meta-stable states, in which the protein is only partlyfolded, also exist and are represented by local minima thatare scattered across the energy landscape. A problem thenarises: some of these local minima are deep enough to formlong-lived meta-stable states which pose a grave danger toliving cells as partially-folded proteins are not only dysfunc-tional but also tend to aggregate into harmful clusters. Enterchaperones—a group of molecular machines whose job in thecell is to prevent, or reverse, protein misfolding. In particular,by using energy some chaperones act as unfolders that restartthe folding process, thus giving misfolds another chance tofind the native state.Situations of the type described above clarify why onewould want to study how restart affects the FPT—to amarked target—of a particle (reaction coordinate) diffusingin a rugged potential landscape. The general theory of first-passage under restart then asserts that the presence of a restart-ing agent, e.g., a molecular chaperone, would be favorablewhenever stochastic fluctuations in the FPT are large by the CV measure [4, 6, 7, 23]. For example, in protein foldinglarge fluctuations in the FPT to the native state can arise whene.g., most folding attempts successfully come to completionafter some typical time τ but a small fraction requires a time τ (cid:29) τ as a result of prolonged trapping in meta-stable states.Restarting ongoing folding processes at a rate τ − (cid:28) r (cid:28) τ − is then expected to be highly beneficial; but the effect of restartcould be easily inverted as changes in pH or temperature al-ter the ratio between τ and τ and thus the CV of the fold-ing process. To study this, and many other similar problems,one can apply the same methodology that was utilized in Sec.II.B. First, define the first-passage time problem in the ab-sence of resetting and solve it (or get the solution from theliterature in case it is known). Then, substitute this solutioninto Eqs. (7) and (8) to get the solution to the problem in thepresence of resetting. As parameters are varied in the system,a transition in the effect of restart will occur when the CV ofthe restart-free first-passage time crosses the value of unity(this transition point is universal). Here, we analysed Eq. (8)and showed that the P´eclet number governs this transition fordrift-diffusion. Equation (8) can also be analyzed for morecomplicated situations, but the challenge is then to identify aP´eclet like quantity that governs the restart transition near andfar from the critical point. This challenge will be addressedelsewhere. ACKNOWLEDGMENTS
S. Ray acknowledges support from the Raymond and Bev-erly Sackler Center for Computational Molecular and Mate-rials Science, Tel Aviv University. S. Reuveni acknowledgessupport from the Azrieli Foundation. D. Mondal acknowl-edges the Ratner Center for Single Molecule Science for sup-porting his visit to the School of Chemistry at Tel Aviv Univer-sity. All authors thank Yair Shokef, Arnab Pal, Yuval Scher,Sarah Kostinski and Ofek Lauber for reading and commentingon early versions of this manuscript.
Appendix A: Derivation of Eq. (3)
Eq. (2) in the main text is a second-order, linear, non-homogeneous differential equation. It has general spatialcoordinate-dependent solution˜ p + ( x , s ) = A ( s ) e α + x + B ( s ) e α − x if 0 ≤ x ≤ L ˜ p − ( x , s ) = A ( s ) e α + x + B ( s ) e α − x if − ∞ < x ≤ . (A.1)Here α ± = D (cid:104) V ± (cid:112) V + D ( s + r ) (cid:105) are the two roots ofthe characteristic equation associated with the homogeneousdifferential equation ∂ ˜ p ( x , s ) ∂ x − (cid:18) VD (cid:19) ∂ ˜ p ( x , s ) ∂ x − (cid:18) s + rD (cid:19) ˜ p ( x , s ) = . (A.2)Since V is real and D , s and r all are real and positive, it is evi-dent from the expressions of α ± that α + > α − <
0. Inorder to find a specific solution to Eq. (2), we need to calculate A ( s ) , A ( s ) , B ( s ) and B ( s ) explicitly. We can accomplishthat in the following way.To prevent ˜ p − ( x , s ) from diverging at x → − ∞ , we set B ( s ) =
0. The other boundary condition, ˜ p + ( L , s ) =
0, gives A ( s ) e α + L + B ( s ) e α − L = . (A.3)In addition, ˜ p − ( x , s ) should be continuous at x =
0, i.e.,˜ p + ( , s ) = ˜ p − ( , s ) , leading to A ( s ) + B ( s ) = A ( s ) . (A.4)The fourth and final condition can be obtained by integrat-ing Eq. (2) in the main text over the narrow spatial interval[ − ε , ε ]. Doing so, we get (cid:20) ∂ ˜ p ( x , s ) ∂ x (cid:21) + ε − ε − (cid:18) VD (cid:19) [ ˜ p ( x , s )] + ε − ε − (cid:18) s + rD (cid:19) (cid:90) + ε − ε ˜ p ( x , s ) dx = − − r ˜ Q ( s ) D , (A.5)where we have utilized the identity (cid:82) + ε − ε δ ( x ) dx =
1. Due tothe continuity of ˜ p ( x , s ) at x =
0, lim ε → [ ˜ p ( x , s )] + ε − ε =
0. Inaddition, lim ε → (cid:82) + ε − ε ˜ p ( x , s ) dx =
0. Thus in the limit of ε → ∂∂ x [ ˜ p + ( x , s ) − ˜ p − ( x , s )] x = = − − r ˜ Q ( s ) D , (A.6)i.e., a finite discontinuity in the first derivative. Plugging inEq. (A.1) into Eq. (A.6) we obtain α + A ( s ) + α − B ( s ) = α + A ( s ) − (cid:20) + r ˜ Q ( s ) D (cid:21) . (A.7)Solving Eqs.(A.3), (A.4), and (A.7), we get A ( s ) = e ( α − − α + ) L (cid:20) + r ˜ Q ( s ) D ( α − − α + ) (cid:21) A ( s ) = (cid:16) e ( α − − α + ) L − (cid:17) (cid:20) + r ˜ Q ( s ) D ( α − − α + ) (cid:21) B ( s ) = − (cid:20) + r ˜ Q ( s ) D ( α − − α + ) (cid:21) . (A.8) Pe =- Pe =- Pe = Pe =- Pe =- Pe =- - - - - -
20 z
FIG. 9. Left hand side (colored lines) and right hand side (blackline) of Eq. (17) vs. the reduced variable z from Eq. (13) for nega-tive P´eclet numbers. Colored circles denote solutions to Eq. (17) fordifferent Pe . - - - - r * / r * FIG. 10. The scaled optimal restart rate from Eq. (18) vs. the P´ecletnumber from Eq. (12) (solid black line). Colored circles correspondto Pe values from Fig. 9. Plugging in Eq. (A.8) into Eq. (A.1), we obtain Eq. (3) in themain text.
Appendix B: Optimal restart rate for V < Analysis in the main text was focused on positive drift ve-locities ( V > V < V <
0, the drive is di-rected away from the absorbing boundary and the particle hasa non-zero probability to escape to infinity, which means thatthe mean FPT necessarily diverges. Introducing restart ren-ders the mean completion time finite [Eq. (6)], thereby lead-ing to infinite speedup. Therefore, the effect of restart in the V < Pe ∈ ( − ∞ , ) . In order to calculate the optimal restart ratein this case, we first solve Eq. (17) with Pe < z (cid:63) , correspond to z valuesat which the two curves intersect. The optimal restart ratesare calculated from these z (cid:63) values following Eq. (18). Weplot the scaled optimal restart rates in Fig. 10 to observe thatit monotonically increases with the magnitude of the negativeP´eclet number. ∗ Author for correspondence: [email protected][1] Visco, P., Allen, R. J., Majumdar, S. N., and Evans, M. R., 2010.Switching and growth for microbial populations in catastrophicresponsive environments. Biophys. J. 98, p.1099.[2] Luby, M., Sinclair, A., and Zuckerman, D., 1993. Optimalspeedup of Las Vegas algorithms, Inform. Process. Lett. 47,p.173.[3] Sornette, D., 2003. Critical market crashes. Phys. Rep. 378,p.198.[4] Pal, A. and Reuveni, S., 2017. First passage under restart. Phys-ical review letters, 118(3), p.030603.[5] Viswanathan, G. M., da Luz, M. G. E., Raposo, E. P., and H.E., 2011. The Physics of Foraging: An Introduction to RandomSearches and Biological Encounters, Cambridge U. Press, NewYork.[6] Reuveni, S., Urbakh, M. and Klafter, J., 2014. Role of substrateunbinding in MichaelisMenten enzymatic reactions. Proceed-ings of the National Academy of Sciences, p.201318122.[7] Rotbart, T., Reuveni, S. and Urbakh, M., 2015. Michaelis-Menten reaction scheme as a unified approach towards the op-timal restart problem. Physical Review E, 92(6), p.060101.[8] Berezhkovskii, A.M., Szabo, A., Rotbart, T., Urbakh, M. andKolomeisky, A.B., 2016. Dependence of the Enzymatic Veloc-ity on the Substrate Dissociation Rate. The Journal of PhysicalChemistry B, 121(15), pp.3437-3442.[9] Robin, T., Reuveni, S. and Urbakh, M., 2018. Single-moleculetheory of enzymatic inhibition. Nature communications, 9(1),p.779.[10] Evans, M.R. and Majumdar, S.N., 2011. Diffusion withstochastic resetting. Physical review letters, 106(16), p.160601.[11] Evans, M.R. and Majumdar, S.N., 2011. Diffusion with optimalresetting. Journal of Physics A: Mathematical and Theoretical,44(43), p.435001.[12] Evans, M.R., Majumdar, S.N. and Mallick, K., 2013. Optimaldiffusive search: nonequilibrium resetting versus equilibriumdynamics. Journal of Physics A: Mathematical and Theoretical,46(18), p.185001.[13] Evans, M.R. and Majumdar, S.N., 2014. Diffusion with reset-ting in arbitrary spatial dimension. Journal of Physics A: Math-ematical and Theoretical, 47(28), p.285001.[14] Christou, C. and Schadschneider, A., 2015. Diffusion with re-setting in bounded domains. Journal of Physics A: Mathemati-cal and Theoretical, 48(28), p.285003.[15] Pal, A., 2015. Diffusion in a potential landscape with stochasticresetting. Physical Review E, 91(1), p.012113.[16] Pal, A., Kundu, A. and Evans, M.R., 2016. Diffusion undertime-dependent resetting. Journal of Physics A: Mathematicaland Theoretical, 49(22), p.225001.[17] Nagar, A. and Gupta, S., 2016. Diffusion with stochastic reset-0