Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics
OOptimal diffusive search: nonequilibrium resettingversus equilibrium dynamics
Martin R. Evans (1) , Satya N. Majumdar (2) and Kirone Mallick (3) (1)
SUPA, School of Physics and Astronomy, University of Edinburgh, MayfieldRoad, Edinburgh EH9 3JZ, United Kingdom (2)
Univ. Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-01405, France (3)
Institut de Physique Th´eorique, CEA Saclay, Gif-Sur-Yvette F-91191, France
Abstract.
We study first-passage time problems for a diffusive particle withstochastic resetting with a finite rate r . The optimal search time is comparedquantitatively with that of an effective equilibrium Langevin process with the samestationary distribution. It is shown that the intermittent, nonequilibrium strategywith non-vanishing resetting rate is more efficient than the equilibrium dynamics. Ourresults are extended to multiparticle systems where a team of independent searchers,initially uniformly distributed with a given density, looks for a single immobile target.Both the average and the typical survival probability of the target are smaller in thecase of nonequilibrium dynamics. PACS numbers: 05.40.-a, 02.50.-r, 87.23.Ge a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics
1. Introduction
Stochastic search problems occur in many fields of science as well as in daily life. Thequest for an optimal strategy for locating a target whether inanimate (such as a bindingsite for a protein at the molecular level [1] or an element in a list) or living (such as aprey for a predator [2]) has been the source of a large number of different algorithmsthat combine observation, physical mechanisms and computation. Depending on thecontext, search strategies can be very different, leading to a variety of interestingmodels (see the special issue [3] devoted to this field of research). One robust class ofmodels called intermittent target search strategies combine phases of slow motion, thatallow target detection, and phases of fast motion, during which the searcher relocatesbut is not reactive (see [4] for a recent review). Such strategies have been observedat different scales: foraging animals, such as humming birds or bumblebees, displayintermittent search patterns [5, 6]. The E. Coli bacteria alternates ballistic moves (or‘runs’) with random changes of direction (‘tumbles’) in order to reach regions withhigh concentration of a chemo-attractant (chemotactic search) [7, 8, 9]. A proteinefficiently localizes a specific DNA sequence by alternating 1d sliding phases with free 3ddiffusion (‘relocation phases’): this mechanism of ‘facilitated diffusion’, first proposedby Adam and Delbr¨uck [1], enhances the association rate by two orders of magnitude ascompared to the diffusion limit and leads to numerical values that agree quantitativelywith experimental results [10, 11] (see [12] for a review).A simple model of diffusion with stochastic resetting, in which a Brownian particleis stochastically reset to its initial position with a constant rate r was defined andstudied in [13]. The stationary state of this process is a non-Gaussian distribution andviolates detailed-balance: a non-vanishing steady-state current is directed towards theresetting position. This process can be viewed as an elementary model of an intermittentstrategy in which the searcher, having explored its environment unsuccessfully for awhile, returns to its initial position and begins a new search. It was also shown in [13]that there exists an optimal resetting rate r ∗ that minimizes the average hitting time tothe target. Extensions to space depending rate, resetting to an random position with agiven distribution and to a spatial distribution of the target were considered in [14].The effect of resetting was previously studied in a stochastic multiplicative modelof population growth where stochastic resetting events of the population size was shownto lead to a stationary power-law distributed population size distribution [15]. Acontinuous-time random walk model in the presence of a drift and resetting has alsobeen studied recently [16]. Finally, in the context of search process, a related modelhas been studied by Gelenbe [17] where searchers are introduced stochastically intothe system: there is a single searcher present at a given time with a random lifetimeand when the searcher dies, a new searcher is introduced into the system at the initialstarting point.In the mathematics literature, mean first-passage time for a class of random walkswith stochastic restarting events has been studied recently from an algorithmic point of ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics
2. Diffusion with stochastic resetting: nonequilibrium dynamics
In this section, we recall the definition of the model of diffusion with stochastic resettingand briefly review some basic results, derived in [13].We consider a single particle on an infinite line starting at the initial position x at t = 0. The position of the particle at time t is updated in a small time interval dt bythe following stochastic rule [13]: x ( t + dt ) = x with prob . r dt = x ( t ) + η ( t ) dt with prob . (1 − r dt ) (1)where η ( t ) is a Gaussian white noise with mean (cid:104) η ( t ) (cid:105) = 0 and the two point correlator (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = 2 D δ ( t − t (cid:48) ). The dynamics thus consists of a stochastic mixture of resettingto the initial position with rate r (long range move) and ordinary diffusion (short rangemove) with diffusion constant D (see Fig. (1)). Resetting introduces a new length scale α − = (cid:112) D/r in the ordinary diffusion problem.As shown in [13], the probability density p ( x, t ) of the particle evolves via the ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics (cid:0)(cid:0)(cid:1)(cid:1) space0 x r time rr rresetting rate = r Figure 1.
Schematic space-time trajectory of a one dimensional Brownian motionthat starts at x and resets stochastically to its initial position x at rate r . following Fokker-Planck equation ∂ t p ( x, t ) = D ∂ x p ( x, t ) − r p ( x, t ) + rδ ( x − x ) , (2)starting from the initial condition p ( x,
0) = δ ( x − x ). We emphasize that the dynamicsviolates detailed balance manifestly: the current from a position x to x via the resettingmove is not compensated by a current from x to x .The Fokker-Planck equation (2) admits a stationary solution in the t → ∞ limit,given by p st ( x ) = α − α | x − x | ] where α = (cid:112) r/D . (3)Even though the stationary solution can be expressed as an effective Boltzmann weight: p st ( x ) ∝ exp [ − V eff ( x )] with the effective potential V eff ( x ) = α | x − x | , one should keepin mind that this is actually a nonequilibrium stationary state , and not an equilibriumstationary state. Indeed, in this stationary state detailed balance is violated by a nonzerocurrent in the configuration space. Had the stationary state been an equilibrium one,the current would have exactly vanished.As shown in [13, 14], the additional resetting parameter r allows us to tune thetarget search process to make it more efficient. Consider an immobile target at theorigin x = 0 and let the searcher undergo diffusion with resetting dynamics specified inEq. (1) starting from the initial position x >
0. What is the mean first-passage time T reset ( x , r ) to the origin knowing that the searcher starts at x ? This average time, T reset ( x , r ), that the searcher takes to find the target will be taken as a measure of theefficiency of the search process: the smaller the value of T reset ( x , r ) (for a fixed x ), thebetter is the search. ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics T reset ( x , r ), it is useful to consider the survival probability q reset ( x , t ) of the target given that the searcher starts at the initial position x andresets to x . The mean-hitting time is then given quite generally by [20, 21, 22], T reset ( x , r ) = (cid:82) ∞ t ( − ∂ t q reset ( x , t )) dt = (cid:82) ∞ q reset ( x , t ) dt .More generally, we define Q ( x, x , t ), the survival probability of the target giventhat the starting position of the searcher is x and its resetting position is x . The initialposition x of the searcher can be treated as a variable leading to a backward Fokker-Planck equation for Q ( x, x , t ) [13, 14]. Eventually, at the end of the calculation, one sets x = x and obtains q ( x , t ) = Q ( x = x , x , t ). The backward Fokker-Planck equationreads for x ≥ ∂ t Q ( x, x , t ) = D ∂ x Q ( x, x , t ) − r Q ( x, x , t ) + r Q ( x , x , t ) . (4)We require the initial condition Q ( x, x , t = 0) = 1 for all x > Q ( x = 0 , x , t ) = 0 and Q ( x → ∞ , x , t ) is finite. Equation (4) canbe solved explicitly in terms of the Laplace transform ˜ Q ( x, x , s ) = (cid:82) ∞ Q ( x, x , t ) e − st dt .In particular, setting x = x , we obtain˜ q reset ( x , s ) = ˜ Q ( x = x , x , s ) = 1 − exp (cid:16) − (cid:113) r + sD x (cid:17) s + r exp (cid:16) − (cid:113) r + sD x (cid:17) , (5)thus leading to the mean first-passage time [13] T reset ( x , r ) = (cid:90) ∞ q ( x , t ) dt = ˜ q ( x , s = 0) = 1 r [exp( α x ) − , (6)where we recall that α = (cid:112) r/D . In terms of the dimensionless parameter γ = α x = (cid:112) r/D x , we obtain T reset ( x , γ ) = (cid:20) e γ − γ (cid:21) x D . (7)We observe that T reset ( x , γ ) diverges in both the limits γ → γ → ∞ (for an infinite resetting rate the particle is localized at x ). The mean first-passage time has a unique minimum (see Fig. (2)) at γ = γ where γ is the solutionof ∂ γ T = 0, i.e., γ = 2(1 − e − γ ), giving γ = 1 . . . . . Hence, for fixed values of D and x , there is an optimal resetting rate r = r = γ D/x that makes the searchtime minimum and the search process most efficient. The corresponding optimal meanfirst-passage time is given by T optreset = (cid:20) e γ − γ (cid:21) x D = 1 . . . . x D . (8)
3. An Effective Equilibrium Dynamics
We have seen that the resetting with diffusion leads to a current-carrying nonequilibriumstationary state given in Eq. (3). This stationary distribution can be expressed as a ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics p st ( x ) = ( α /
2) exp [ − V eff ( x )] with an effective potential V eff ( x ) = α | x − x | , (9)where α was defined in Eq. (3).Let us now consider the following Langevin evolution of the particle position withtime t dxdt = − B ∂ x V eff ( x ) + η ( t ) (10)where B is the amplitude of the external force and η ( t ) is the same Gaussian whitenoise as in the previous case, i.e., with (cid:104) η ( t ) (cid:105) = 0 and (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = 2 D δ ( t − t (cid:48) ). Notethat we have chosen the same diffusion constant D as in the resetting case to reflect thefact that without the reset (in the former case) or without the external potential (inthe Langevin case), this equation describes ordinary diffusion with the same diffusionconstant D in both cases. The corresponding Fokker-Planck equation for the probabilitydensity P ( x, t ) of the particle is ∂ t P ( x, t ) = D ∂ x P ( x, t ) + ∂ x [ B ( ∂ x V eff ( x )) P ( x, t )] . (11)Eq. (11) can be expressed as a continuity equation, ∂ t P = − ∂ x J where the currentdensity J ( x, t ) = − D∂ x P − B ( ∂ x V eff ( x )) P . The system will then reach a stationary stateat long times and if we set the current in the stationary state to be 0, we arrive at theequilibrium Gibbs-Boltzmann solution which reads, P eq ( x ) = N exp [ − ( B/D ) V eff ( x )],where N is the normalization constant. By choosing B = D and V eff ( x ) = α | x − x | , wecan engineer a zero current equilibrium state which has the same weight as the currentcarrying nonequilibrium stationary state in the resetting case, i.e., P eq ( x ) = p st ( x ) = α − α | x − x | ] where α = (cid:112) r/D . (12)The following natural question then arises. Consider the target search problem,where we have an immobile target at the origin. In the previous section, the searcherwas performing normal diffusion with stochastic resetting to its initial position x . Now,suppose that the searcher undergoes instead the Langevin dynamics as in Eq. (10)with the choice B = D and V eff ( x ) = α | x − x | that guarantees that both dynamicslead to the same steady state. One is then tempted to compare the efficiency of thenonequilibrium search process with diffusion and reset to the Langevin search processwhere the searcher’s position evolves via Eq. (10). Which process is the more efficient?To address this question, we need to compute the mean first-passage time T lange ( x , r ) to the origin of the Langevin process in Eq. (10). With the choice B = D and V eff ( x ) = α | x − x | where α = (cid:112) r/D , the Langevin equation reads dxdt = − α D sgn( x − x ) + η ( t ) (13)where sgn( z ) denotes the sign of z . We define Q lange ( x, x , t ), the probability that thesearcher, starting at the initial position x ≥ ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics t . Equivalently, Q lange ( x, x , t ) is the survivalprobability of the target up to time t under the Langevin dynamics of the searcher.Treating the initial position x as a variable, Q lange ( x, x , t ) satisfies a backward Fokker-Planck equation [20, 21] ∂ t Q lange ( x, x , t ) = D ∂ x Q lange ( x, x , t ) − α D sgn( x − x ) ∂ x Q lange ( x, x , t ) (14)which holds for all x ≥ Q lange ( x, x , t = 0) = 1 for all x > Q lange ( x = 0 , x , t ) = 0 for all t (absorbing boundaryat the origin) and (ii) Q lange ( x → ∞ , x , t ) = 1.To solve Eq. (14), we consider the Laplace transform ˜ Q lange ( x, x , s ) = (cid:82) ∞ Q lange ( x, x , t ) e − st dt , which, taking into account the initial condition, satisfies − s ˜ Q lange ( x, x , s ) = D d ˜ Q lange dx − α D sgn( x − x ) d ˜ Q lange dx . (15)Making a shift ˜ Q lange ( x, x , s ) = 1 /s + ˜ F ( x, x , s ), one finds a homogeneous differentialequation for ˜ F in x ≥ D d ˜ Fdx − α D sgn( x − x ) d ˜ Fdx − s ˜ F = 0 (16)with the boundary conditions: (i) ˜ F ( x = 0 , x , s ) = − /s and (ii) ˜ F ( x → ∞ , x , s ) = 0.Taking the boundary condition (ii) into account, we obtain˜ F ( x, x , s ) = A exp[ − µ ( x − x )] for x > x (17)˜ F ( x, x , s ) = B exp [ − µ ( x − x )] + B exp [ µ ( x − x )] for x < x , (18)where µ = ( (cid:113) α + 4 s/D + α ) / , and µ = ( (cid:113) α + 4 s/D − α ) / . (19)From the boundary condition (i) at x = 0 and using that ˜ F ( x, x , s ) and its firstderivative ∂ x ˜ F ( x, x , s ) are continuous at x = x , the three unknown constants A , B and B are determined B = 2 µ µ + µ A ; B = µ − µ µ + µ A ; and A = − s ( µ + µ )[2 µ e µ x + ( µ − µ ) e − µ x ] . (20)Finally, inserting x = x leads to˜ q lange ( x , s ) = ˜ Q lange ( x = x , x , s ) = 1 s (cid:20) − ( µ + µ )[2 µ e µ x + ( µ − µ ) e − µ x ] (cid:21) . (21)The mean first-passage time is then given by T lange ( x , r ) = (cid:90) ∞ t ( − ∂ t q lange ( x , t )) dt = (cid:90) ∞ q lange ( x , t ) dt = ˜ q lange ( x , s = 0) . ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics s → T lange ( x , r ) = 1 α D [2 ( e α x − − α x ] = (cid:20) e γ − − γγ (cid:21) x D (22)where γ = α x = (cid:112) r/D x is the same dimensionless parameter as defined above (seeEq. (7)). We can now compare the result in Eq. (22) with that of the resetting case,Eq. (7). Using the fact that e γ − ≥ γ , we see that T lange ( x , r ) ≥ T reset ( x , r ) for fixedvalues of x and D .The minimum value of T lange ( x , r ) is obtained (see Fig. (2)) for γ = γ where γ is given by the solution of ∂ γ T lange ( x , r ) = 0, i.e., it is the positive root of the equation2( γ − e γ + γ +4 = 0, leading to γ = 1 . . . . . Thus the optimal mean first-passagetime with Langevin dynamics of the searcher is given by T optlange = (cid:20) e γ − − γ γ (cid:21) x D = 2 . . . . x D . (23)Comparing with the corresponding result in Eq. (8) for the reset dynamics, we concludethat T optreset T optlange = 1 . . . . . . . . = 0 . · · · ≤ . (24)This shows that the search process via the nonequilibrium diffusion combined with resetmechanism is significantly more efficient than the equilibrium Langevin dynamics of thesearcher although the stationary distributions induced by both dynamics are the same:by this measure the nonequilibrium strategy beats the equilibrium dynamics.
4. Multiparticle problem: Nonequilibrium vs. Langevin dynamics
In the multiparticle version of the search process, we have a single immobile target atthe origin and a team of searchers which are initially uniformly distributed on the linewith uniform density ρ (see [23] for a colorful example). The searchers are independentof each other and the position of each searcher evolves stochastically (identical in lawfor all searchers) starting at its own initial position. This stochastic process, for themoment, is general. When any of the searchers finds the target, the search process isterminated. Let P s ( t ) denote the survival probability of the target up to time t , i.e, theprobability that the target has not been found up to t by any of the searchers. To set theproblem, we consider N searchers initially distributed uniformly in a box [ − L/ , L/ L and will eventually take the limit N → ∞ , L → ∞ , but keeping the density ρ = N/L fixed. Let x i denote the initial position of the i -th searcher. Thus, x i is arandom variable uniformly distributed in the box [ − L/ , L/ { x i } , P s ( t ) = N (cid:89) i =1 q ( x i , t ) (25) ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics γ T Figure 2.
Mean first-passage time T plotted versus the dimensionless parameter γ (setting x = 1 and D = 1) for the nonequilibrium reset dynamics (shown by thesolid (black) line) (see Eq. (7)) and the equilibrium Langevin dynamics (dashed (red)line) (see Eq. (22)). Evidently, the optimal (minimum) T is higher in the equilibriumLangevin case. where q ( x i , t ) is the survival probability of the target due to a single searcher startingat x i . For both (i) diffusion with resetting dynamics and (ii) Langevin dynamics inan effective potential V eff ( x ) = α | x − x | , the Laplace transforms of q ( x i , t ) are givenrespectively in Eqs. (5) and (21). The explicit dependence of P s ( t ) on x i ’s has beensuppressed in Eq. (25) for notational convenience.We first compute the average survival probability (averaged over the initial positionsof the searchers) following Ref. [13]. Taking the average of Eq. (25) we obtain (cid:104) P s ( t ) (cid:105) = N (cid:89) i =1 [1 − (cid:104) (1 − q ( x i , t ) (cid:105) ]= N (cid:89) i =1 (cid:34) − L (cid:90) L/ − L/ (1 − q ( x i , t )) dx i (cid:35) → exp (cid:20) − ρ (cid:90) ∞−∞ [1 − q ( x, t )] dx (cid:21) (26)where in the last step we have exponentiated for large L and then taken thethermodynamic limit N → ∞ , L → ∞ but keeping the ratio ρ = N/L fixed. Usingfurther the symmetry q ( x, t ) = q ( − x, t ), one finally obtains the general expression ofthe average survival probability in the multiparticle system (cid:104) P s ( t ) (cid:105) = exp (cid:20) − ρ (cid:90) ∞ [1 − q ( x, t )] dx (cid:21) . (27) ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics q ( x, t ) of the target at the origin for a singlesearcher starting at x , one has an exact formula for the average survival probability inthe multiparticle problem. The quantity M ( t ) = (cid:82) ∞ [1 − q ( x, t )] dx has the interpretationas the expected maximum up to time t of the single particle process (starting at theorigin) and the general formula in Eq. (27) and its discrete-time analogue have been usedrecently to compute exactly the average survival probability for a number of processesincluding continuous-time random walks as well as discrete-time L´evy flights [24].Similarly, the typical survival probability can be estimated [13] by first takinglogarithm of Eq. (25), followed by averaging over the initial positions and finallyreexponentiating P typ s ( t ) = exp (cid:34) N (cid:88) i =1 (cid:104) ln[ q ( x i , t )] (cid:105) (cid:35) → exp (cid:20) ρ (cid:90) ∞ ln[ q ( x, t )] dx (cid:21) . (28)Thus, if we know the survival probability q ( x, t ) in the single searcher case, we can usethe two exact formulae in Eqs. (27) and (28) to estimate respectively the average andthe typical survival probability of the target in the multiparticle case. Here, we analyze the asymptotic large t behavior of (cid:104) P s ( t ) (cid:105) in Eq. (27) and compare theexpression obtained for the resetting dynamics with the one for the Langevin dynamics.Let us denote M ( t ) = (cid:90) ∞ [1 − q ( x, t )] dx so that (cid:104) P s ( t ) (cid:105) = exp [ − ρ M ( t )] (29)In terms of the Laplace transform of M ( t ), ˜ M ( s ) = (cid:82) ∞ M ( t ) e − s t dt , Eq. (29) reads˜ M ( s ) = (cid:90) ∞ (cid:20) s − ˜ q ( x, s ) (cid:21) dx (30)We now consider the two cases (i) diffusion with reset and (ii) Langevin dynamicsseparately. Substituting ˜ q reset ( x, s ) from Eq. (5)in Eq. (30) and performing the integration over x , one obtains exactly [13]˜ M ( s ) = (cid:112) D ( r + s ) r s ln (cid:18) r + ss (cid:19) . (31)The large t behavior of M ( t ) will be derived from small s behavior of ˜ M ( s ):As s → , ˜ M ( s ) ≈ (cid:114) Dr (cid:20) − ln ss + ln( r ) s + . . . (cid:21) . (32) ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics M ( t )for large t is given by M ( t ) ≈ (cid:114) Dr ln t + (cid:114) Dr ln( r ) + . . . (33)Thus, the average survival probability from Eq. (29) decays algebraically for large t [13] (cid:104) P reset s ( t ) (cid:105) ≈ a t − θ where θ = 2 ρ (cid:112) D/r and a = exp [ − θ ln r ] = r − θ . (34) Here, we use the expression (21) of˜ q lange ( x, s ) in Eq. (30), perform the integration over x exactly and get˜ M ( s ) = (cid:20) µ + µ µ µ s (cid:21) F (cid:18) , µ µ + µ , µ + µ µ + µ , µ − µ µ (cid:19) (35)where µ , are defined in Eq. (19), α in Eq. (3) and the hypergeometric function F ( a, b, c, z ) is given by F ( a, b, c, z ) = 1 + abc z + a ( a + 1) b ( b + 1) c ( c + 1) z
2! + a ( a + 1)( a + 2) b ( b + 1)( b + 2) c ( c + 1)( c + 2) z
3! + . . . (36)Expanding µ , for small s , keeping terms up to O ( s ) and using α D = r , one gets˜ M ( s ) ≈ α D s (cid:20) sr + · · · (cid:21) F (cid:20) , − sr , − sr , − r s −
12 + s r (cid:21) (37)To make further progress, we use the following identity F (1 , , , − z ) = ln(1 + z ) /z which follows from the definition Eq. (36). Expanding for small s , we find the followingleading order behavior of ˜ M ( s ) from Eq. (37)˜ M ( s ) = − α s ln( s ) + 1 α s ln( r/
2) + . . . (38)where . . . correspond to lower order terms that vanish as s →
0. This indicates that forlarge t , M ( t ) ≈ α ln( t ) + 1 α ln( r/
2) + . . . (39)Hence, from Eq. (29), we obtain the average survival probability (cid:104) P lange s ( t ) (cid:105) ≈ a t − θ where θ = θ = 2 ρ (cid:112) D/r and a = e − θ ln( r/ = ( r/ − θ (40)Thus the power law exponent θ characterizing the algebraic decay of the averagesurvival probability in the Langevin case is identical to the nonequilibrium case, thoughthe amplitude a = 2 θ a is larger than a . Hence, for large t , the average survivalprobability in the Langevin case is greater than that of the nonequilibrium case (cid:104) P reset s ( t ) (cid:105) ≤ (cid:104) P lange s ( t ) (cid:105) . (41)Thus, we conclude that on average the target is found faster in the nonequilibrium casethan in the equilibrium one. ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics We now analyze the asymptotic large t behavior of the typical survival probability,defined in Eq. (28). We define W ( t ) = (cid:90) ∞ ln[ q ( x, t )] dx so that P typ s ( t ) = exp [2 ρ W ( t )] . (42)We recall that the Laplace transform of q ( x, t ) is denoted by ˜ q ( x, s ) and its expressionsare given in Eqs. (5) and (21) respectively for (i) the diffusion with reset dynamics and(ii) the equilibrium Langevin dynamics in the effective potential V eff ( x ) = α | x − x | .The two cases will be considered separately. To analyze the asymptotic large t behavior of W ( t ), we need to know how q ( x, t ) behaves for large t . The Laplace transformof q ( x, t ), given in Eq. (5), has a pole at s = s (for fixed r and fixed x ) which satisfies s + r exp (cid:34) − (cid:114) r + s D x (cid:35) = 0 . (43)Clearly s = s ( x ) depends implicitly on x and one must have s ( x ) <
0. Then, toleading order for large t , it follows from Laplace inversion that q ( x, t ) ∼ exp [ s ( x ) t ] = exp [ −| s ( x ) | t ] . (44)Consequently, from Eq. (42), we find to leading order for large tW ( t ) ≈ − (cid:20)(cid:90) ∞ | s ( x ) | dx (cid:21) t . (45)Hence, the typical survival probability decays exponentially for large t as P typ s ( t ) ∼ exp [ − ρ κ t ] where κ = (cid:90) ∞ | s ( x ) | dx . (46)To compute s ( x ), it is useful to first define s ( x ) = − r (1 − u ), so that Eq. (43)reads, in terms of u and the dimensionless length z = α xu − (cid:2) −√ u z (cid:3) = 0 . (47)Clearly, as z → ∞ , u ( z ) → z → u ( z ) →
0. Hence κ = (cid:90) ∞ | s ( x ) | dx = rα (cid:90) ∞ [1 − u ( z )] dz . (48)The idea then is to transform the integral over z to an integral over u . We then use dz = du/ | u (cid:48) ( z ) | where u (cid:48) ( z ) = du ( z ) /dz . The derivative can be easily computed fromEq. (47). Expressing Eq. (47) as z = − ln(1 − u ) / √ u and taking derivative, we get dzdu = 12 u / ln(1 − u ) + 1 √ u (1 − u ) . (49) ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics α = (cid:112) r/D , κ = √ r D (cid:90) ∞ [1 − u ( z )] dz = √ r D (cid:90) (1 − u ) (cid:20) u / ln(1 − u ) + 1 √ u (1 − u ) (cid:21) du . (50)The integral in Eq. (50) can be done explicitly to give [13] κ = 4 (1 − ln 2) √ r D . (51)Hence finally we find P typ s ( t ) ∼ exp (cid:104) − − ln 2) ρ √ rD t (cid:105) = exp (cid:104) − (2 . . . . ) ρ √ rD t (cid:105) . (52)As explained in [13], the fact that the average and the typical survival probabilitieshave different behaviours in the large time limit is a consequence of the memory of theinitial conditions in the diffusion process with resetting. In this case, we proceed in exactly thesame way as the nonequilibrium case, except that to evaluate W ( t ) = (cid:82) ∞ ln[ q ( x, t )] dx in Eq. (42), we need the expression (21) of the Laplace transform of q lange ( x, t ). Thefunction ˜ q lange ( x, s ) has a pole at s = s ( x ) which is a root of the equation (with fixed r and fixed x )2 µ ( s ) exp[ µ ( s ) x ] + ( µ ( s ) − µ ( s )) exp[ − µ ( s ) x ] = 0 , (53)where we emphasize that µ , ( s ) defined in Eq. (19) depend on s . Because µ > µ ,one must have µ ( s ) < s ( x ) < q ( x, t ) ∼ exp[ −| s ( x ) | t ] for large t and W ( t ) ∼ − (cid:2)(cid:82) ∞ | s ( x ) | dx (cid:3) t . Consequently P typ s ( t ) = exp [2 ρ W ( t )] ∼ exp [ − ρ κ t ] (54)where κ = (cid:90) ∞ | s ( x ) | dx . (55)To compute κ , we reorganize Eq. (53) slightly (using the explicit expressions of µ and µ ) and express it in terms of the dimensionless length z = α x as (cid:112) s /r − (cid:104) − (cid:112) s /r z (cid:105) = 0 . (56)where we have used α D = r . Let us further define v = 1 + 4 s /r in terms of which Eq.(56) reads √ v − (cid:2) −√ v z (cid:3) = 0 . (57)Hence, from Eq. (55) we get κ = √ r D (cid:90) ∞ [1 − v ( z )] dz (58) ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics v ( z ) is the solution of Eq. (57). Now the solution of Eq. (57) has two branches: v = 0, 0 ≤ z < v ( z ) > z ≥
1. Thus (58) becomes κ = √ r D (cid:20) (cid:90) ∞ [1 − v ( z )] dz (cid:21) (59)To compute the integral in (59), we use the same trick as above. (cid:90) ∞ [1 − v ( z )] dz = (cid:90) [1 − v ] dz ( v ) dv dv . (60)where z ( v ) = − ln(1 − √ v ) / √ v , which follows from (57). Here, the integral (60) is easilyevaluated by integration by parts (cid:90) (1 − v ) dz ( v ) dv dv = [(1 − v ) z ( v )] v =0 + (cid:90) z ( v ) dv = − − (cid:90) ln(1 − v / ) v / dv = 1 . (61)Therefore from (59) we have κ = √ r D . (62)Hence finally we obtain P typ s ( t ) ∼ exp (cid:104) − ρ √ rD t (cid:105) . (63)Comparing this result to Eq. (52), we find that for the multiparticle case as well,the typical survival probability of the target in the equilibrium Langevin case is largerthan the nonequilibrium dynamics of diffusion with resetting P typ s ( t ) (cid:12)(cid:12) reset ≤ P typ s ( t ) (cid:12)(cid:12) lange . (64)This means that the target is found faster in the nonequilibrium case than theequilibrium Langevin case. In other words, the target search process by multiplesearchers, as in the case of a single searcher, is more efficient when the searcher undergoesnonequilibrium diffusion and reset dynamics, rather than the equilibrium Langevindynamics.
5. Concluding Remarks
Equilibrium thermodynamics teaches us that dissipation is reduced when a systemremains close to an equilibrium state and that the transformations that affect it arequasistatic and reversible. Such a statement of local optimum can not be taken asa paradigm: it can be advantageous in some circumstances to be driven away fromequilibrium, by creating non-vanishing stationary currents that break detailed balanceand time-reversal invariance.Optimal search problems provide us with concrete examples in which nonequi-librium can defeat equilibrium. In the present work, we have undertaken a systematic ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics [1] G. Adam and M. Delbr¨uck, Reduction of dimensionality in biological diffusion processes, in
Structural Chemistry and Molecular Biology, A. Rich and N. Davidson Eds. (W.H. Freemanand Company, San Francisco; London, 1968).[2] W. J. Bell,
Searching behaviour: the behavioural ecology of finding resources , (Chapman and Hall,London 1991).[3] G. Oshanin, K. Lindenberg, H. S. Wio, and S. F. Burlatsky, Efficient search by optimizedintermittent random walks,
J. Phys. A: Math. Theor. , 434008 (2009).[4] O. B´enichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod.Phys. , 81 (2011).[5] Bartumeus F and Catalan J, Optimal search behavior and classic foraging theory J. Phys. A:Math. Theor. J. Phys. A: Math. Theor. Phys. Rev. Lett. ,238101 (2008).[8] M. Sheinman and Y. Kafri, Effects of intersegmental transfers of target location by proteins,
Phys.Biol. , 016003 (2009).[9] J. Tailleur and M. E. Cates, Statistical Mechanics of Interacting Run-and-Tumble Bacteria, Phys.Rev. Lett. , 218103 (2008).[10] A. D. Riggs, S. Bourgeois and M. Cohn, The lac repressor-operator interaction. 3. Kinetic studies,
J. Mol. Biol. , 401 (1970).[11] O. G. Berg, R. B. Winter and P. H. Von Hippel, Diffusion-driven mechanisms of proteintranslocation on nucleic acids. 1. Models and theory, Biochemistry , 6929 (1981).[12] L. Mirny, M. Slutsky, Z. Wunderlich, A. Tafvizi, J. Leith and A. Kosmrlj, How a protein searchesfor its site on DNA: the mechanism of facilitated diffusion, J. Phys. A: Math. Theor. , 434013(2009).[13] M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. , 160601(2011).[14] M. R. Evans and S. N. Majumdar, Diffusion with optimal resetting,
J. Phys. A: Math. Theor. ,435001 (2011). ptimal diffusive search: nonequilibrium resetting versus equilibrium dynamics [15] S. C. Manrubia and D. H. Zanette, Stochastic multiplicative processes with reset events, Phys.Rev. E , 4945 (1999).[16] M. Montero and J. Villarroel, Monotonous continuous-time random walks with drift and stochasticreset events, Preprint arXiv: 1206.4570[17] E. Gelenbe, Search in unknown environments, Phys. Rev. E , 061112 (2010).[18] S. Janson and Y. Peres, Hitting times for random walks with restarts, SIAM J. Discrete Math. , 537 (2012)[19] M. Tachiya, Theory of diffusion-controlled reactions: formulation of the bulk reaction rate in termsof the pair probability, Radiat. Phys. Chem. , 167 (1983)[20] S. Karlin and H. E. Taylor, A First Course in Stochastic Processes (Academic Press, San Diego,1975).[21] S. Redner,
A guide to First-Passage Processes (Cambridge University Press, Cambridge 2001).[22] P. L. Krapivsky, S. Redner and E. Ben-Naim,
A Kinetic View of Statistical Physics (CambridgeUniversity Press, Cambridge 2010).[23] P. L. Krapivsky and S. Redner, Kinetics of a diffusion capture process: lamb besieged by a prideof lions,
J. Phys. A: Math. Gen. , 5347 (1996).[24] J. Franke and S. N. Majumdar, Survival probability of an immobile target surrounded by mobiletraps, J. Stat. Mech.