Diffusion with resetting in arbitrary spatial dimension
DDiffusion with resetting in arbitrary spatialdimension
Martin R. Evans (1) and Satya N. Majumdar (2) (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
Abstract.
We consider diffusion in arbitrary spatial dimension d with the additionof a resetting process wherein the diffusive particle stochastically resets to a fixedposition at a constant rate r . We compute the nonequilibrium stationary state whichexhibits non-Gaussian behaviour. We then consider the presence of an absorbingtarget centred at the origin and compute the survival probability and mean timeto absorption of the diffusive particle by the target. The mean absorption time isfinite and has a minimum value at an optimal resetting rate r . Finally we consider theproblem of a finite density of diffusive particles, each resetting to its own initial position.While the typical survival probability of the target at the origin decays exponentiallywith time regardless of spatial dimension, the average survival probability decaysasymptotically as exp( − A (ln t ) d ) where A is a constant. We explain these findingsusing an interpretation as a renewal process and arguments invoking extreme valuestatistics. a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r esetting in arbitrary dimension
1. Introduction
Diffusion is a fundamental dynamical process dating back to Einstein and the propertiesof the diffusion equation have been much studied. In this work we consider the problemof diffusion with a stochastic resetting process whereby the diffusive particle is reset toa fixed position (which we usually take to coincide with the particle’s initial position)at random times and the diffusive process begins anew. It turns out [1] that resettingfundamentally affects the properties of diffusion.Resetting holds the system away from any equilibrium state by constantly returningthe system to the initial condition. It is thus a simple way of generating a nonequilibriumstationary state. In such states probability currents are non-zero and detailed balancedoes not hold—for resetting systems a non-vanishing steady-state current is directedtowards the resetting position. The nature and properties of nonequilibrium stationarystate are questions of fundamental importance within statistical physics [2]. Thus,the resetting paradigm provides a convenient framework within which to study suchnonequilibrium properties.Resetting was first considered in a stochastic multiplicative model of populationgrowth where stochastic resetting events of the population size was shown to lead to astationary distribution in which the population size has a power-law distribution [3].A continuous-time random walk model in the presence of a constant drift and resettinghas also been studied and the stationary distribution established [4]. Related models ofpopulation growth include those involving catastrophic events in which the populationis stochastically reduced and reset to some value lower value [5].The resetting paradigm has a natural realisation in the context of search processes[6, 7]. The optimal stochastic search appears as a classic problem in areas as diverse ascomputer science [8] (e.g. searching for an element in an array) through biochemistry [9](e.g. a protein searching a binding site) to macrobiology [10] (e.g. a predator seekingits prey) . It is also a teasing question in everyday life—how best does one search forlost keys? One general class of search strategies are termed intermittent and combineperiods of slow, local motion, termed foraging, in which the target may be detectedwith periods of fast motion, termed relocation, during which the searcher relocates tonew territory (see [7] for a recent review). A diffusive process mimics the foraging phaseand a resetting process mimics the relocation. Thus diffusion with stochastic resettingprovides a simple realisation of an intermittent search process. Another related searchprocess introduced in [11] is one in which an individual has a random lifetime and whenthe searcher dies, a new searcher is introduced into the system at the initial startingpoint. In the mathematical literature, the mean hitting time for a class of random walksin which the walker may choose to restart the walk has been studied recently from analgorithmic point of view [12].In this paper we consider diffusion with stochastic resetting as a fundamentalnonequilibrium process in which the statistics of first passage properties may becomputed exactly. A simple model of diffusion with stochastic resetting, in which a esetting in arbitrary dimension r wasdefined and studied in [1] with the focus on one spatial dimension. Amongst the resultsobtained, it was shown in [1] that there exists an optimal resetting rate r ∗ that minimizesthe average hitting time to the target. Extensions to space-dependent resetting rate,resetting to a random position with a given distribution and to a spatial distribution ofthe target were considered in [13]. How the average absorption time is increased whenthe searcher is only partially absorbed by the target, corresponding to an imperfectsearcher, has been studied in [14]. Finally a comparison between the statistics of firstpassage times for diffusion with stochastic resetting and for an equilibrium dynamicsthat generates the same stationary state has been made [15].In the present work we extend these studies to diffusion in arbitrary spatialdimension d . In Section 2 we define the model and solve the forward master equationfor the probability distribution of the diffusive particle both in the stationary state andrelaxing to the stationary state. In Section 3 we compute the survival probability of afixed absorbing target (or trap) in the presence of a diffusing particle with resetting. Wealso present a simple interpretation of the result in terms of the extreme value statisticsof a renewal process. In Section 4 we compute the optimal mean first absorption time inarbitrary dimension. In Section 5 we consider many diffusive particles in the presenceof a single trap and study the average and typical behaviours of the survival probabilityof the trap. The system furnishes one of the few models for which all these propertiescan be computed exactly in arbitrary dimensions. Finally we conclude in Section 6 witha summary and outlook.
2. Diffusive resetting problem in arbitrary spatial dimension
First let us define diffusion with resetting in arbitrary spatial dimension d . We considera single particle (or searcher) in R d with initial position (cid:126)x at t = 0 and resetting toposition (cid:126)X r . We stress here that the initial position (cid:126)x and resetting position (cid:126)X r are ingeneral distinct, although at the end of some calculations it is convenient to set themto be equal. Our notation for the resetting position thus differs from that of [1].The position (cid:126)x ( t ) of the particle at time t is updated by the following stochasticrule [1]: in a small time interval d t each component x i of the position vector (cid:126)x ( t ) becomes x i ( t + dt ) = ( X r ) i with probability r dt = x i ( t ) + ξ i ( t ) dt with probability (1 − r dt ) (1)where ξ i ( t ) is a Gaussian white noise with mean (cid:104) ξ i ( t ) (cid:105) = 0 and the two-point correlator (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = 2 D δ ij δ ( t − t (cid:48) ). The dynamics thus consists of a stochastic mixture ofresetting to the initial position with rate r (long range move) and ordinary diffusion(local move) with diffusion constant D (see Fig. (1)). esetting in arbitrary dimension (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) X x r Figure 1.
Illustration in d = 2 of the diffusion with resetting process: the particlestarts at initial position (cid:126)x and resets to position (cid:126)X r with rate r . The probability density for the particle to be at position (cid:126)x at time t , having startedfrom position (cid:126)x at time t = 0 with resetting to position (cid:126)X r , should, in principle, bewritten as p ( (cid:126)x, t | (cid:126)x ; (cid:126)X r ). However in the following, in order to lighten the notation, weshall suppress the initial and resetting positions and abbreviate to p ( (cid:126)x, t ).The forward master equation for the probability density for diffusion with resettingrate r to point (cid:126)X r reads ∂p ( (cid:126)x, t ) ∂t = D ∇ p ( (cid:126)x, t ) − rp ( (cid:126)x, t ) + rδ d ( (cid:126)x − (cid:126)X r ) , (2)with initial condition p ( (cid:126)x,
0) = δ d ( (cid:126)x − (cid:126)x ). The first term on the right hand side (r.h.s)of Equation (2) expresses the diffusive spread of probability; the second term expressesthe loss of probability from (cid:126)x due to resetting to (cid:126)X r ; the final term corresponds to thegain of probability at (cid:126)X r due to resetting from all other positions.One can write down the solution to (2) in a simple and intuitive way as follows.We first note that the (initial value) diffusive Green function in the absence of resetting( r = 0), which we denote G ( (cid:126)x, t | (cid:126)x ), satisfies ∂G ( (cid:126)x, t | (cid:126)x ) ∂t = D ∇ G ( (cid:126)x, t | (cid:126)x ) , (3)with initial condition G ( (cid:126)x, t = 0 | (cid:126)x ) = δ d ( (cid:126)x − (cid:126)x ), and is given by G ( (cid:126)x, t | (cid:126)x ) = 1(4 πDt ) d/ exp (cid:34) − | (cid:126)x − (cid:126)x | Dt (cid:35) . (4) esetting in arbitrary dimension p ( (cid:126)x, t ) is a sum over two contributions: one which comes fromtrajectories where no resetting events have occurred in time t and a second contributionwhich comes from summing over trajectories where the last resetting event occurred attime t − τ . The probability of no resetting events having occurred up to time t is e − rt and the probability of the last resetting event having occurred at t − τ (and no resettingevents since) is r e − rτ . Thus the full time-dependent solution can be written down as p ( (cid:126)x, t ) = e − rt G ( (cid:126)x, t | (cid:126)x ) + r (cid:90) t d τ e − rτ G ( (cid:126)x, τ | (cid:126)X r ) . (5)The stationary state is attained when t (cid:29) /r where (5) tends to the stationarydistribution p ∗ ( (cid:126)x ) = r (cid:90) ∞ d τ e − rτ G ( (cid:126)x, τ | (cid:126)X r ) . (6)The relaxation to the steady state may be obtained from p ( (cid:126)x, t ) = p ∗ ( (cid:126)x ) + e − rt G ( (cid:126)x, t | (cid:126)x ) − r (cid:90) ∞ t d τ e − rτ G ( (cid:126)x, τ | (cid:126)X r ) . (7)In order to evaluate the integral in (6) we use the identity (Equation 3.471.9 of [16]) (cid:90) ∞ d t t ν − e − βt − γt = 2 (cid:32) βγ (cid:33) ν/ K ν (2 (cid:113) βγ ) , (8)where K ν is the modified Bessel function of the second kind (also known as Macdonaldfunction) of order ν . The relevant case of this identity is ν = 1 − d/ , (9)and one obtains from (4) and (6), after setting (cid:126)x = (cid:126)X r p ∗ ( (cid:126)x ) = (cid:32) α π (cid:33) − ν ( α | (cid:126)x − (cid:126)X r | ) ν K ν ( α | (cid:126)x − (cid:126)X r | ) , (10)where α = (cid:18) rD (cid:19) / . (11)Expression (10), for the stationary distribution of diffusion in the presence of resettingin arbitrary dimension, is the central result of this section. Note that (10) tends tozero as | (cid:126)x | → ±∞ and has a cusp at (cid:126)x = (cid:126)X r (see Figure 2). Also note that (13)is a nonequilibrium stationary state by which it is meant that there is circulation ofprobability (even in a one-dimensional geometry). This is because resetting implies asource of probability at (cid:126)X r while probability is lost from all other values of (cid:126)x (cid:54) = (cid:126)X r . d = 1 and d = 3 : In the case d = 1 ( ν = 1 /
2) we use the identity K / ( y ) = (cid:32) π y (cid:33) / e − y (12)to find that the stationary distribution (10) reduces to p ∗ ( x ) = α − α | x − X r | ) , (13) esetting in arbitrary dimension −5 −3 −1 1 3 5 x p * ( x ) d=3d=2d=1 Figure 2.
The stationary probability densities p ∗ ( x = | (cid:126)x | ) given by Equation 10 forthe case α = 1 and (cid:126)X r = 0: d = 1 full lines; d = 2 dotted lines; d = 3 dashed lines. thus recovering the result of [1], [13].Also in the case d = 3 ( ν = − / K − / ( y ) = K / ( y ) tofind a simple form for the stationary distribution p ∗ ( (cid:126)x ) = α π | (cid:126)x − (cid:126)X r | exp( − α | (cid:126)x − (cid:126)X r | ) . (14)
3. Survival probability in the presence of a trap at the origin
We now consider the presence of a trap at the origin, i.e., an absorbing d -dimensionalsphere of radius a centred at (cid:126)x = 0 which absorbs the particle. The particle startsat the initial position | (cid:126)x | > a and undergoes diffusion with diffusion constant D andstochastic resetting to (cid:126)X r with a constant rate r . When it reaches the surface of thetarget sphere, the particle is absorbed (see Figure 3).Once we set (cid:126)x = (cid:126)X r , there are only three length scales in the system. Theparameter α defined in (11) is an inverse length scale which corresponds to the typicaldistance diffused by the particle between resets. The other two length scales are R r = | (cid:126)X r | , the distance from the origin to resetting position, and a the radius of esetting in arbitrary dimension (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) x X a r O Figure 3.
Illustration in d = 2 of the diffusion of a particle with initial position (cid:126)x and resetting to (cid:126)X r , in the presence of an absorbing trap of radius a with centre atthe origin O . the trap. These three length scales can be combined to define dimensionless reducedvariables γ = α R r (15) (cid:15) = aR r . (16)The dimensionless quantity γ measures the ratio of the distance R r of the reset pointfrom the target at the origin to the diffusion length α − ; (cid:15) is simply the ratio of theradius of the absorbing sphere to the distance R r of the reset point to the target at theorigin. To study absorption problems it is most convenient to consider the backward masterequation for the survival probability [17]. In this formulation the initial position isconsidered as a variable. In the presence of resetting it then is important to distinguishthe initial position which is a variable (denoted henceforth by (cid:126)x ) from the resettingposition (cid:126)X r which is fixed. At the end of the calculation we will set (cid:126)x = (cid:126)X r .We are interested in the survival probability of a diffusive particle at time t , having started at (cid:126)x at t = 0 with resetting to (cid:126)X r , which in principle should be denoted Q ( (cid:126)x , t ; (cid:126)X r ). However, to lighten the notation we will suppress the resetting positionand write Q ( (cid:126)x , t ). In the same way we will write its Laplace transform, defined below(20), as q ( (cid:126)x , s ). esetting in arbitrary dimension Q ( (cid:126)x , t ) is constructed by considering eventsin the first infinitesimal time interval d t : the components ( x ) i of the initial positionbecome ( x ) i → ( X r ) i with probability r d t → ( x ) i + ξ i (0)d t with probability 1 − r d t (17)where ξ i (0) is the i th component of the initial noise. Then Q ( (cid:126)x , t + d t ) = (1 − r d t ) (cid:104) Q ( (cid:126)x + (cid:126)η (0)d t, t ) (cid:105) + r d tQ ( (cid:126)X r , t ) , (18)where (cid:104)·(cid:105) indicates averaging over the initial noise. Performing the average and takingthe d t → | (cid:126)x | > a , ∂Q ( (cid:126)x , t ) ∂t = D ∇ (cid:126)x Q ( (cid:126)x , t ) − rQ ( (cid:126)x , t ) + rQ ( (cid:126)X r , t ) , (19)with boundary and initial conditions Q ( | (cid:126)x | = a, t ) = 0 and Q ( (cid:126)x ,
0) = 1 for | (cid:126)x | > a .The Laplace transform q ( (cid:126)x , s ) = (cid:90) ∞ d t e − st Q ( (cid:126)x , t ) (20)satisfies D ∇ (cid:126)x q ( (cid:126)x , s ) − ( r + s ) q ( (cid:126)x , s ) = − − rq ( (cid:126)X r , s ) . (21)The solution to the homogeneous equation D ∇ (cid:126)x q ( (cid:126)x , s ) − ( r + s ) q ( (cid:126)x , s ) = 0 , (22)which is radially symmetric about the origin and does not diverge as | (cid:126)x | → ∞ , is q hom ( (cid:126)x , s ) = ( α | (cid:126)x | ) ν K ν ( α | (cid:126)x | ) (23)where now α ( s ) = (cid:18) r + sD (cid:19) / , (24)and as before, ν = 1 − d/
2. Note that α (0) = α given in (11). Hereafter, we write α ≡ α ( s ) for brevity.In order to determine the solution to (21) we write q ( (cid:126)x , s ) = B | (cid:126)x | ν K ν ( α | (cid:126)x | ) + C (25)where C is independent of | (cid:126)x | . Then the absorbing boundary condition implies that C = − Ba ν K ν ( αa ) and substituting (25) back into (21) determines B = − sa ν K ν ( αa ) + r | (cid:126)X r | ν K ν ( α | (cid:126)X r | ) . Therefore q ( (cid:126)x , s ) = a ν K ν ( αa ) − | (cid:126)x | ν K ν ( α | (cid:126)x | ) r | (cid:126)X r | ν K ν ( α | (cid:126)X r | ) + sa ν K ν ( αa ) . (26) esetting in arbitrary dimension (cid:126)x equal to the resetting position (cid:126)X r , andusing isotropy, we have q ( (cid:126)X r , s ) = q ( R r , s ) = a ν K ν ( αa ) − R νr K ν ( αR r ) rR νr K ν ( αR r ) + sa ν K ν ( αa ) , (27)where R r = | (cid:126)X r | . Equation (27) is the main result of this section and is an exactexpression for the Laplace transform of the survival probability in arbitrary dimension ν = 1 − d/ Q ( R r , t ) ≡ Q ( | (cid:126)x | = R r , t ; | (cid:126)X r | = R r ) . (28)However, it appears difficult to invert the Laplace transform (27) explicitly for allparameters. Nevertheless there are two limits where one may extract the forms of Q ( R r , t ): (i) the natural scaling regime r → t → ∞ with scaling variable y = rt fixed;(ii) for fixed r , the large t limit. In the scaling regime, we consider the limit r → s → λ = sr . (29)The scaling variable is y = rt . (30)In this limit we obtain the scaling distribution Q ( R r , t ) → F ( R r , y ) (31)and (27) becomes q ( R r , s ) → f ( R r , λ ) = (cid:90) ∞ d y e − λy F ( R r , y ) r (32)i.e. we obtain a simple scaling form of the Laplace transform f ( R r , λ ). We may theninvert the Laplace transform in (32) with respect to λ to obtain the scaling form F ( R r , y )for the survival probability.To proceed, we use the small argument expansion K ν ( x ) (cid:39) − ln( x/ − γ E if ν = 0 (33) (cid:39) Γ( ν )2 (cid:18) x (cid:19) ν + Γ( − ν )2 (cid:18) x (cid:19) − ν + . . . if 0 < ν < , (34) (cid:39) Γ( ν )2 (cid:18) x (cid:19) ν − Γ( ν − (cid:18) x (cid:19) ν − + . . . if ν > , (35)where γ E = 0 . . . . is the Euler constant. esetting in arbitrary dimension ν > ν = 0, ν < d < d = 2, d > ν = 1 − d/ i) > ν > < d < q ( R r , s ) (cid:39) Γ( − ν )Γ( ν ) r ν − (1 + λ ) ν − (4 D ) ν (cid:104) a ν − R νr (cid:105) . (36)One may then invert (36) using (32) to obtain the final result in the scaling limit Q ( R r , t ) = F ( R r , rt ) (37)where the scaling function is given by F ( R r , y ) = 1 ν Γ( ν ) (cid:18) γ (cid:19) ν (1 − (cid:15) ν )e − y y − ν , (38)and, as usual, γ = R r ( r/D ) / and (cid:15) = a/R r .In particular, the d = 1 ( ν = 1 /
2) case is F ( R r , y ) = γ √ π e − y y / . (39)Note that in the limit y → F ( R r , y ) (cid:39) R r √ Dπ (cid:32) ry (cid:33) / = R r √ Dπt , (40)which recovers the usual 1 d diffusive survival probability at large t . Also note that for y large the survival probability decays exponentially. ii) ν = 0 ( d = 2): In the case d = 2, which corresponds to ν = 0, the leading orderexpansion of (27) in the limit r → s → q ( R r , s ) (cid:39) (cid:15) ( s + r ) ln r . (41)Inverting the Laplace transform yields Q ( R r , t ) (cid:39) (cid:15) ln r e − rt . (42)So in the case d = 2 a scaling form, which is solely a function of y and (cid:15) , does not emerge. iii) ν < d > . In the case d >
2, which corresponds to ν <
0, we can use thesymmetry K − ν ( x ) = K ν ( x ). Then it turns out that only the leading order term in thesmall x expansions (34,35) matters. Substituting this in (27) gives in the scaling limit q ( R r , s ) (cid:39) r − (cid:15) − ν ( λ + (cid:15) − ν ) . (43)Inverting the Laplace transform yields F ( R r , y ) = (1 − (cid:15) − ν )e − y(cid:15) − ν . (44)Thus for d > y . esetting in arbitrary dimension The long-time asymptotics of the survival probability for fixed r may be deduced fromthe inversion of (27) Q ( R r , t ) = (cid:90) C d s e st πi a ν K ν ( αa ) − R νr K ν ( αR r ) rR νr K ν ( αR r ) + sa ν K ν ( αa ) , (45)where C is the Bromwich contour to the right of any singularities in the complex s plane. We may rewrite (45) as Q ( R r , t ) = (cid:90) C d s πi e st r (cid:34) r + g ( s ) s − g ( s ) (cid:35) (46)where g ( s ) = − r(cid:15) ν K ν ( α ( s ) R r ) K ν ( α ( s ) a ) , (47)with (cid:15) = a/R r and α ( s ) = (cid:115) r + sD . The analytic structure in the complex s plane ofthe integrand of (46) is a branch point s = − r coming from α ( s ), and a simple pole at s = s which satisfies s = g ( s ) . (48)In the long-time limit the dominant contribution to (46) will come from the pole at s which yields by the residue theorem Q ( R r , t ) (cid:39) e s t r [ r + s ]1 − g (cid:48) ( s ) . (49)Making the substitution s = r ( u −
1) 0 < u < , (50)equation (48) becomes a transcendental equation for u u = 1 − (cid:15) ν K ν ( γu / ) K ν ( γ(cid:15)u / ) , (51)where γ = α R r . Thus the asymptotic behaviour of the survival probability isln Q ( R r , t ) ∼ r ( u − t (52)where u satisfies (51).In the limit γ (cid:29) γ(cid:15) (cid:29)
1) we can use the asymptotic behaviour K ν ( y ) (cid:39) (cid:32) π y (cid:33) / e − y [1 + O (1 /y )] for large y (53)to obtain the behaviour of u u (cid:39) − (cid:15) / − ν e − γ (1 − (cid:15) ) . (54)Then (49) takes the form Q ( R r , t ) ∼ exp (cid:16) − rt(cid:15) / − ν e − γ (1 − (cid:15) ) (cid:17) . (55)This expression has the form of a Gumbel distribution, the origin of which we shall nowdiscuss. esetting in arbitrary dimension In order to betterunderstand the result (55) we consider the diffusion with resetting process as a processwhich is renewed each time the particle is reset—the renewed process has no memory ofits past history. We shall consider the long time regime where the number of resets toposition (cid:126)X r in time t is the mean number of resets N = rt plus fluctuations of O ( t / ).In order for the target to survive up to time t it must survive through each of theresets. Then we can write the asymptotic behaviour as Q ( R r , t ) ∼ (cid:16) (cid:104) P diff S ( τ ) (cid:105) τ (cid:17) rt , (56)where the P diff S is the survival probability of a diffusive particle up to time τ and (cid:104)·(cid:105) τ denotes an average over the duration of the reset (time until next resetting event). Asresetting is a Poisson process we have (cid:104) P diff S ( τ ) (cid:105) τ = (cid:90) ∞ d τ re − rτ P diff S ( τ ) = rq diff ( R r , r ) , (57)where now q diff ( R r , r ) is the Laplace transform, with Laplace variable s replaced by r ,of the survival probability of a diffusive particle with an absorbing target at the origin.The expression for q diff ( R r , r ) is easy to construct using, for example, a backward masterequation approach and is well-known in the literature (see e.g. [18]): q diff ( R r , r ) = 1 r (cid:34) − (cid:18) R r a (cid:19) ν K ν ( R r ( r/D ) / ) K ν ( a ( r/D ) / ) (cid:35) (58)= 1 r (cid:34) − (cid:15) − ν K ν ( γ ) K ν ( γ(cid:15) ) (cid:35) . (59)We then have Q ( R r , t ) ∼ exp (cid:40) rt ln (cid:34) − (cid:15) − ν K ν ( γ ) K ν ( γ(cid:15) ) (cid:35)(cid:41) (60) (cid:39) exp (cid:104) − rt(cid:15) / − ν e − γ (1 − (cid:15) ) (cid:105) , (61)where we have used the asymptotic behaviour (53) for large γ and (cid:15)γ , thus recovering(55).One may relate this result to the Gumbel distribution for the extremum ofindependent random variables [19] as follows. As we have seen above, for the targetto survive it must survive through N resets. Thus the leftmost point the searcher hasreached in the N resets must be less than R r , the distance to the target. In otherwords, the maximum of N independent random variables must be less than R r . Thusthe Gumbel distribution, which is the scaling form for the cumulative distribution of themaximum of N independent random variables with distribution decaying faster than apower law, naturally emerges. The computation of the survival probability can easily be generalised to the case ofa resetting distribution P ( (cid:126)z ) [13] where (cid:126)z is a randomly chosen resetting position. esetting in arbitrary dimension x i ( t + dt ) = z i with probability r P ( (cid:126)z ) dt = x i ( t ) + ξ i ( t ) dt with probability (1 − r dt ) . (62)Also, if we choose the distance of the initial position from the origin, R = | (cid:126)x | tobe drawn from the same distribution P ( R ) we find that the Laplace transform of thesurvival probability averaged over the resetting distribution becomes q ( s ) = a ν K ν ( αa ) − (cid:82) d R R ν K ν ( αR ) P ( R ) sa ν K ν ( αa ) + r (cid:82) d R R ν K ν ( αR ) P ( R ) . (63)Equation (63) gives the survival probability for a diffusive particle with a trap at theorigin for abitrary resetting distribution, thus generalising the result (27) for a fixedresetting position. Equation (63) may be used to study the optimal choices of resettingdistribution to optimize for example the mean time to absorption [13].
4. Mean First-passage time to absorption by sphere of radius a in arbitrarydimensions We now consider the mean time to absorption in the presence of an absorbing trap inthe case of diffusion with resetting to point (cid:126)X r . The mean time to absorption by the trap at the origin, with coincident initial andresetting positions, can be computed from the survival probability Q ( R r , t ) T ( R r ) = − (cid:90) ∞ d t t ∂Q ( R r , t ) ∂t = q ( R r , s = 0) . (64)(The integration by parts requires Q ( R r , t ) to vanish faster than 1 /t for large t .) Thus,the mean time to absorption is obtained by setting s = 0 in (27) T ( R r ) = 1 r (cid:34)(cid:18) aR r (cid:19) ν K ν ( α a ) K ν ( α R r ) − (cid:35) . (65)Note that for d ≤
2, (65) has a finite limit as a →
0, but for d >
2, (65) diverges as a → K ν ( z ) = K − ν ( z ) . (66)Then using the dimensionless quantities (15), (16) one may rewrite (65) as T = 1 γ (cid:34) (cid:15) ν K | ν | ( (cid:15) γ ) K | ν | ( γ ) − (cid:35) R r D , (67)where, as usual, γ = α R r = (cid:113) r/DR r and we have suppressed the R r dependence of T ( R r ) for convenience. The first point to note is that T is finite for 0 < r < ∞ . It esetting in arbitrary dimension r → T ∼ r − / in d = 1 , T ∼ / ( r | ln r | ) in d = 2 as can bechecked using the small argument expansion of the modified Bessel functions (33–35).It also diverges exponentially in r as r → ∞ which can be easily checked using theasymptotic behaviour (53). The r → r → ∞ limit merelyexpresses the fact that as the reset rate increase the diffusing particle has less timebetween resets to reach the origin.Now, for fixed (cid:15) and fixed R r /D , we wish to see how T in Eq. (67) behaves as afunction of the reduced variable γ = (cid:113) r/D R r . Since T diverges in the limits γ → γ → ∞ , there is a unique minimum of T at the optimal value γ ∗ where d T / d γ = 0. Thisoptimal γ ∗ is evidently a complicated function of (cid:15) and in general, is hard to determineexplicitly. However, we are interested in the limit (cid:15) = a/R r → (cid:15) limit Taking (cid:15) → K ν ( z ) in Eqs.(33) and (34), gives for ν = 1 − d/ (cid:54) = 0 T → γ (cid:34) (cid:15) ν −| ν | Γ( | ν | ) 2 | ν |− γ −| ν | K | ν | ( γ ) − (cid:35) R r D . (68) d < : Consider first the case d <
2. In this case, ν = 1 − d/ | ν | = 1 − d/ (cid:15) dependence of T in Eq. (68) just drops out in the (cid:15) → d <
2, one does not need to have a finitesize of the target and there is well defined point target limit. Hence, for d <
2, we findthe exact result T d< = 1 γ − d/ Γ (cid:32) − d (cid:33) γ d/ − K − d ( γ ) − R r D . (69)In particular, for d = 1, using (12) we recover the 1-d result [1] T d =1 = 1 γ [e γ − R r D , (70)which has a unique minimum at γ ∗ = γ = 1 . . . . where γ is the unique positiveroot of γ − − e − γ ) = 0. For general d <
2, one can similarly minimize Eq. (69)and obtain the optimal γ ∗ . d > : In the opposite case d >
2, the (cid:15) dependence of T does not dropout of Eq. (68), indicating that for d >
2, one needs a finite target size as otherwisea point particle will never meet a point target. In this case, to leading order in (cid:15) , Eq.(68) gives T d> → γ − − d/ K d − ( γ ) Γ (cid:32) d − (cid:33) d/ − (cid:15) − d R r D . (71) esetting in arbitrary dimension T as a function of γ . The optimal value γ ∗ is obtained by settingd T d γ = 0 and γ = γ ∗ . Taking the derivative of Eq. (71) with respect to γ and using theproperty of the modified Bessel function [16], yK (cid:48) ν ( y ) + νK ν ( y ) = − yK ν − ( y ) , (72)we obtain γ ∗ as a root of the equation2 K d − ( γ ∗ ) − γ ∗ K d − ( γ ∗ ) = 0 . (73)A particular simplification occurs for d = 3 where one can use K − / ( y ) = K / ( y )to obtain explicitly γ ∗ = 2. Hence, using K / ( y ) = (cid:113) π/ y e − y , the optimal meanfirst-passage time for d = 3 is given explicitly by T ∗ d =3 = e R r D (cid:15) . (74)In the marginal dimension d = 2, one has ν = 0 and hence from Eq. (67) we obtain T d =2 = 1 γ (cid:34) K ( (cid:15) γ ) K ( γ ) − (cid:35) R r D . (75)Taking again the (cid:15) → (cid:15) , T d =2 → − ln( (cid:15) ) γ K ( γ ) R r D . (76)Once again, one can minimize T as a function of γ . The optimal γ ∗ is obtained bysolving d T d γ = 0 and is given by the root of the equation2 K ( γ ∗ ) − γ ∗ K ( γ ∗ ) = 0 (77)which gives γ ∗ ( d = 2) = 1 . . . . . Substituting this result in Eq. (76), we find thatthe exact optimal mean first-passage time in d = 2 T ∗ d =2 → . . . . [ − ln( (cid:15) )] R r D . (78)
5. Many Searchers
We now consider the problem of many independent searchers (diffusive particles) andthe survival probability of a stationary target at the origin defined as in Section 4.1.Specifically, we consider N diffusive particles labelled µ = 1 , . . . , N , each of which isreset to its own resetting position (cid:126)X µ with rate r . We also take the initial position ofeach searcher to be identical to its resetting position.The survival probability of the target is given by P s ( t ) = N (cid:89) µ =1 Q ( (cid:126)X µ , t ) (79) esetting in arbitrary dimension Q ( (cid:126)X µ , t ) is the survival probability in the single searcher problem considered inSection 3. Note that in the absence of resetting ( r = 0) this survival probability P s ( t )has been studied in all dimensions [20–23].We consider the N resetting positions to be distributed uniformly with density ρ outside of the target volume and consequently, P s ( t ) is a random variable. Its averageis simply P av s ( t ) = (cid:104) P s ( t ) (cid:105) (cid:126)x where (cid:104)·(cid:105) (cid:126)X denotes averages over (cid:126)X µ ’s. However, as we shallsee, for a typical resetting configuration P s ( t ) is not captured by the average. This isbecause the average may be dominated by rare distributions of the resetting positionsof the searchers for which the survival probability is much larger than is typical. Onecan think of the resetting positions as initial conditions which are remembered for alltime by the system through the resetting dynamics.The typical P s ( t ) can be extracted by first averaging over the logarithm of P s ( t )followed by exponentiating: P typ s ( t ) = exp [ (cid:104) ln P s ( t ) (cid:105) (cid:126)X ]. One can draw an analogyto a disordered system with P s ( t ) playing the role of partition function Z and X µ ’scorresponding to disorder variables. Thus the average and typical behaviour correspondrespectively to the annealed average (where one averages the partition function Z ) andthe quenched average (where one averages the free energy ln Z ) in disordered systems.To compute the average behaviour of (79) (the annealed case) we may write P av s ( t ) = (cid:104) (cid:104) Q ( (cid:126)X, t ) (cid:105) (cid:126)X (cid:105) N (80)= exp (cid:110) N ln (cid:104) − (cid:104) − Q ( (cid:126)X, t ) (cid:105) (cid:126)X (cid:105)(cid:111) (81)where (cid:104)·(cid:105) (cid:126)X denotes an average over the resetting position (cid:126)X . We begin by considering (cid:126)X to be distributed uniformly over a volume V comprising a sphere of radius L with thetarget volume, which is a sphere of radius a , removed. Noting that Q ( (cid:126)X, t ) = Q ( R, t ),where R = | (cid:126)X | , we obtain (cid:104) − Q ( (cid:126)X, t ) (cid:105) (cid:126)X = 1 − V (cid:90) R>a d R Γ d R d − (1 − Q ( R, t )) , (82)where Γ d = 2 π d/ Γ( d/
2) (83)is the surface area of a d -dimensional unit sphere. Letting N, L → ∞ but keeping thedensity of walkers ρ = N/V fixed, we obtain P av s ( t ) → exp (cid:20) − ρ Γ d (cid:90) ∞ a d R R d − (1 − Q ( R, t )) (cid:21) ≡ exp [ − ρI ( t )] . (84)On the other hand, the typical behaviour (the quenched case) P typ s ( t ) =exp [ (cid:104) ln ( P s ( t )) (cid:105) X ] can be expressed as P typ s ( t ) = exp N (cid:88) µ =1 (cid:104) ln (cid:104) Q ( (cid:126)X µ , t ) (cid:105) (cid:105) (cid:126)X µ = exp (cid:110) N (cid:104) ln (cid:104) Q ( (cid:126)X, t ) (cid:105) (cid:105) (cid:126)X (cid:111) (85)where (cid:104) ln (cid:104) Q ( (cid:126)X, t ) (cid:105) (cid:105) (cid:126)X = 1 V (cid:90) R>a d R Γ d R d − ln [ Q ( R, t )] . (86) esetting in arbitrary dimension N, L → ∞ with density of walkers ρ = N/V fixed, we obtain P typ s ( t ) = exp (cid:20) ρ Γ d (cid:90) ∞ a d RR d − ln Q ( R, t ) (cid:21) ≡ exp [ − ρI ( t )] . (87)Thus the determination of the average and typical behaviour reduces to the evaluationof two integrals: I = Γ d (cid:90) ∞ a d R R d − [1 − Q ( R, t )] (88) I = − Γ d (cid:90) ∞ a d R R d − ln [ Q ( R, t )] . (89) The Laplace transform of (88) ˜ I ( s ) = (cid:82) ∞ I ( t ) e − st d t can be determined using (27):˜ I ( s ) = Γ d (cid:90) ∞ a d R R d − (cid:20) s − q ( R, s ) (cid:21) = Γ d ( r + s ) s (cid:90) ∞ a d R R d − (cid:34) r + s (cid:18) aR (cid:19) ν K ν ( αa ) K ν ( αR ) (cid:35) − . (90)We now wish to determine the small s behaviour. First note that for large xK ν ( x ) ∼ (cid:18) π x (cid:19) / e − x . (91)Therefore in the integral (90) the term r in the square bracket will dominate upto somelength scale R ∗ and we can determine the leading behaviour as˜ I ( s ) (cid:39) Γ d ( r + s ) rs (cid:90) R ∗ a d R R d − (92) (cid:39) Γ d ( r + s ) rs ( R ∗ ) d d , (93)where we have assumed R ∗ (cid:29) a .In order to deduce R ∗ we note that it is defined by (cid:18) aR r (cid:19) ν K ν ( αa ) K ν ( αR r ) (cid:39) rs . (94)Using the asymptotic behaviour (91) of K ν ( αR ∗ ) for R ∗ large we havee α R ∗ ( R ∗ ) ν − / (cid:39) (cid:18) π α (cid:19) / a ν K ν ( α a ) rs , (95)therefore as s → R ∗ (cid:39) α ln (cid:18) rs (cid:19) . (96)Hence, ˜ I ( s ) (cid:39) Γ d ds α d (cid:20) ln (cid:18) rs (cid:19)(cid:21) d . (97)Inverting the Laplace transform one then obtains the large time asymptotic behaviourof I ( t ) I ( t ) (cid:39) Γ d dα D [ln t ] d . (98) esetting in arbitrary dimension P av s ( t ) ∼ exp (cid:34) − ρ Γ d dα d (ln t ) d (cid:35) . (99)Expression (99) is the main result of this section. As a check we can take d = 1 torecover the result [1] P av s ( t ) ∼ exp [ − ρα ln t ] = t − ρ ( D/r ) / . (100)Expression (99) shows how the power law decay displayed in d = 1 (99) is generalizedin arbitrary spatial dimension. The scaling form exp (cid:104) − A (ln t ) d (cid:105) is unusual, as far as weare aware. For the typical behaviour (the quenched case) in the long time limit Q ( R, t ) is dominatedby a pole at s in the complex s plane (49), therefore I , given by (89), becomes I (cid:39) Constant − t Γ d (cid:90) ∞ a d R R d − | s ( R ) | . (101)In general dimension d (cid:54) = 1 we were not able to perform the integral in (101) explicitly.However it is easy to show that the integral is convergent. Thus P typ s ( t ) given by (87)asymptotically decays exponentially in time P typ s ( t ) ∼ exp (cid:20) − t Γ d (cid:90) ∞ a d R R d − | s ( R ) | (cid:21) . (102)The correction to the argument of the exponential will come from the branch point at s = − r in (27) and is expected to give a subleading contribution of O ( t / ). The fact thatthe average and typical survival probabilities have distinct asymptotic behaviours—theaverage behaviour (99) decaying far more slowly than the typical behaviour (102)—reflects the strong dependence on the initial conditions, noted above, whose memory isretained through resetting.In dimension d = 1 the integral in (101) can be evaluated in closed form. Thereason is that in this case one can obtain a closed form expression for R as a functionof u from (51) αR = α a − ln(1 − u ) u / . (103)Then one may transform the integration from R to u with range 0 < u < s = r ( u −
1) and letting (cid:15) → (cid:90) ∞ d R | s ( R ) | = rα (cid:90) d u (cid:34) (1 − u ) u / ln(1 − u ) + 1 u / (cid:35) (104)= ( Dr ) / − ln 2) . (105)Thus in one dimension the asymptotic decay of the quenched total survival probabilityis exponential with explicit form [1] P qs ( t ) ∼ exp (cid:104) − tρ ( Dr ) / − ln 2) (cid:105) . (106) esetting in arbitrary dimension In order to understand the asymptotic form of average survival probability (99), weconsider the following simple picture. At long times the absorption probability of thetarget will be dominated by the searcher which started nearest to the target. Denotingthe position of this searcher by y , the average survival probability for the many searcherproblem should then be recovered by averaging the single searcher survival probability(49) over the distribution of the position y ,In order to obtain the distribution of the distance y of the nearest searcher fromthe target (at the origin), we consider first the probability that a single searcher startsat distance R > y from the originProb(
R > y ) = Γ d V (cid:90) Ly d R R d − = Γ d dV (cid:104) L d − y d (cid:105) , (107)where V = Γ d d (cid:104) L d − a d (cid:105) .Then the probability that all N searchers start at distance R > y from the originis given by, in the limit of large
N, L with ρ fixed,Prob( R > y ) N (cid:39) exp (cid:40) N (cid:34) − (cid:18) yL (cid:19) d + (cid:18) aL (cid:19) d (cid:35)(cid:41) → exp (cid:20) ρ Γ d d (cid:16) − y d + a d (cid:17)(cid:21) . (108)Thus the distribution of the distance y of the nearest searcher from the target is P ( y ) = − dd y exp (cid:20) ρ Γ d d (cid:16) − y d + a d (cid:17)(cid:21) (109)= A y d − exp (cid:18) − ρ Γ d d y d (cid:19) , (110)where A is a constant.For a single searcher starting at y the survival probability is given by (49) P s = exp [ − t (1 − u ( y ))] (111)where the function u ( y ) is given by (51).Thus the average survival probability is given approximately by the average of (111)with respect to (110) (cid:104) P s (cid:105) = A (cid:90) ∞ d y y d − exp (cid:20) − ρ Γ d d y d − t (1 − u ( y )) (cid:21) . (112)For large t , we expect the integral to be dominated by the saddle point y ∗ of the integralwith respect to y which yields − ρ Γ d ( y ∗ ) d − + tu (cid:48) ( y ∗ ) = 0 . (113)For large t we expect y ∗ to be large for which the asymptotic behaviour is given by (54) u ( y ) (cid:39) − (cid:15) / − ν e − α y ∗ (1 − (cid:15) ) . The saddle point y ∗ is then given by − ρ Γ d ( y ∗ ) d − + α (1 − (cid:15) ) t e − α y ∗ (1 − (cid:15) ) = 0 which implies that asymptotically y ∗ ∼ ln tα . (114) esetting in arbitrary dimension t the survival probability is dominated by initial arrangements ofsearchers in which the nearest searcher is at distance y ∗ ∼ ln t/α . The integral (112)is then dominated by (cid:104) P s (cid:105) ∼ exp (cid:20) − ρ Γ d d ( y ∗ ) d (cid:21) ∼ exp (cid:34) − ρ Γ d dα d (ln t ) d (cid:35) (115)which recovers the asymptotic result (99) of Subsection 5.2.
6. Conclusion
In this work we have studied diffusion with resetting at rate r in arbitrary spatialdimension d . We have computed the nonequilibrium stationary state of the processgiven by Equation (10), which exhibits non-Gaussian behaviour. Moreover the fulltime-dependence is given by (5). The resetting paradigm thus presents a simple, generalframework in which to study nonequilibrium properties. We have shown in this workthat the spatial dimensionality produces some interesting behaviour.In Section 3 we considered the survival probability of an absorbing target. Ourcentral result is to compute the Laplace transform of the survival probability from whichlong time asymptotic behaviour may be deduced. Then in Section 4 we considered themean time to absorption of the target. The mean time to absorption is finite and has aminimum value at an optimal resetting rate r .Finally in Section 5 we consider the problem of a finite density of diffusive particles,each resetting to its own initial position. While the typical survival probability of thetarget at the origin decays exponentially with time regardless of spatial dimension,the average survival probability decays asymptotically as exp( − A (ln t ) d ) where A is aconstant. We have explain these findings using an interpretation as a renewal processand arguments invoking extreme value statistics.There are many open questions concerning the resetting paradigm. In this work wehave considered a single particle system under diffusion but one could generalise to otherstochastic processes, for example, the Ornstein-Uhlenbeck process describing diffusion ina potential. Moreover one can generalise resetting to extended systems where the config-uration of the system is reset to the initial condition at a constant rate. These dynamicsgenerate a new class of nonequilibrium stationary states. Recent examples that havebeen considered include a class of reaction-diffusion systems in one dimension [24] andalso growth processes described by macroscopic stochastic differential equations suchas the Kardar-Parisi-Zhang and Edwards-Wilkinson equations [25]. Naturally, studyingthe resetting dynamics in these extended systems in higher dimensions would be aninteresting challenge. Acknowledgements
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