Diffusion with Optimal Resetting
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Diffusion with Optimal Resetting
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 (3)
Weizmann Institute of ScienceE-mail: [email protected], [email protected]
Abstract.
We consider the mean time to absorption by an absorbing target of adiffusive particle with the addition of a process whereby the particle is reset to its initialposition with rate r . We consider several generalisations of the model of M. R. Evansand S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106,160601: (i) a space dependent resetting rate r ( x ) ii) resetting to a random position z drawn from a resetting distribution P ( z ) iii) a spatial distribution for the absorbingtarget P T ( x ). As an example of (i) we show that the introduction of a non-resettingwindow around the initial position can reduce the mean time to absorption providedthat the intial position is sufficiently far from the target. We address the problem ofoptimal resetting, that is, minimising the mean time to absorption for a given targetdistribution. For an exponentially decaying target distribution centred at the originwe show that a transition in the optimal resetting distribution occurs as the targetdistribution narrows.
1. Introduction
Search problems occur in a variety of contexts: from animal foraging [1] to the targetsearch of proteins on DNA molecules [2–4]; from internet search algorithms to the moremundane matter of locating one’s mislaid possessions. Often search strategies involvea mixture of local steps and long-range moves [5–9]. For human searchers at least, anatural tendency is to return to the starting point of the search after the length of timespent searching becomes excessive.In a recent paper [10] we modelled such a strategy as a diffusion process with anadditional rate of resetting to the starting point x with rate r . Considering the objectof the search to be an absorbing target at the origin, the duration of the search becomesthe time for the diffusing particle to reach the origin. Statistics such as the mean timeto absorption of the process then give a measure of the efficiency of the search strategy,defined by the resetting rate r . Moreover, the model provides a system where thestatistics of absorption times can be computed exactly.A related model, where searchers have some probabilistic lifetime after whichanother searcher will be sent out, has been studied by Gelenbe [11] and mean times iffusion with Optimal Resetting r to the fixed initial position x . Thus, there isan optimal resetting rate r as a function of the distance to the target x .In this work we address the question of resetting strategies which optimise theMFPT in a wider context. To this end, we make several generalisations of single-particle diffusion with resetting studied in [10]. First, we consider a space dependentresetting rate r ( x ). Second, we consider resetting to a random position z (rather thana fixed x ) drawn from a resetting distribution P ( z ). Finally, we consider a probabilitydistribution for the absorbing target P T ( x ). The general question we ask is: what arethe optimal functions r ( x ), P ( x ) that minimise the MFPT for a given P T ( x )? Althoughwe do not propose a general solution, the examples we study turn up some surprisingresults and illustrate that answers to the problem may be non trivial.The paper is organised as follows. In section 2 we review the calculation of themean first passage time for one-dimensional duffusion in the presence of resetting to theintial position with rate r . In section 3 we introduce spatial dependent resetting r ( x )and work out the example of a non resetting window of width a around the intial point.In section 4 we consider the generalisation to a resetting distribution P ( z ) and to adistribution of the target site P T ( x ). In section 5 we formulate the general problem ofoptimising the mean first passage time with respect to the resetting distribution P ( z ).We consider the example of an exponential target distribution and show that there is atransition in the optimal resetting distribution. We conclude in section 6.
2. First passage time for single particle diffusion with resetting
We begin by briefly reviewing the one-dimensional case of diffusion with resetting tothe initial position x (see Fig. 1), introduced in Ref. [10]. The Master equation for p ( x, t | x ), the probability distribution for the particle at time t having started frominitial position x , reads ∂p ( x, t | x ) ∂t = D ∂ p ( x, t | x ) ∂x − rp ( x, t | x ) + rδ ( x − x ) (1)with initial condition p ( x, | x ) = δ ( x − x ). In Eq. (1) D is the diffusion constant ofthe particle and r is the resetting rate to the initial position x . The second term on theright hand side (rhs) of Eq. 1 denotes the loss of probability from the position x due toreset to the initial position x , while the third term denotes the gain of probability at x due to resetting from all other positions. iffusion with Optimal Resetting (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) x O time space 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 . The stationary state of (1) is the solution of
D ∂ p ∗ ( x | x ) ∂x − rp ∗ ( x | x ) = − rδ ( x − x ) (2)which is determined by the elementary Green function technique, which we now recall.The solutions to the homogeneous counterpart of (2) are e ± α x where α = q r/D . (3)The solution to (2) is constructed from linear combinations of these solutions whichsatisfy the following boundary conditions: p ∗ → x → ±∞ , and p ∗ is continuous at x = x . Imposing these conditions yields p ∗ ( x | x ) = A exp( − α | x − x | ) . (4)Note that (4) has a cusp at x = x . The constant A is fixed by the discontinuity ofthe first derivative at x = x which is determined by integrating (2) over a small regionabout x ∂p ∗ ( x | x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x +0 − ∂p ∗ ( x | x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − = − α . (5)Carrying this out yields A = α / p ∗ ( x | x ) = α − α | x − x | ) . (6)Alternatively, the constant A in (4) could be fixed by the normalisation of the probabilitydistribution (4).Note that (6) is a non-equilibrium stationary state by which it is meant that thereis circulation of probability even in the one-dimensional geometry. At all points x thereis always a diffusive flux of probability in the direction away from x given by − D∂p/∂x ,and a nonlocal resetting flux in the opposite direction from all points x = x to x . iffusion with Optimal Resetting We now consider the mean first passage time for the diffusing particle to reach the origin.One can think of an absorbing target at the origin which instantaneously absorbs theparticle (see e.g. [13]).A standard approach to first-passage problems is to use the backward Masterequation where one treats the initial position as a variable (for a review see Ref. [14]). Let Q ( x, t ) denote the survival probability of the particle up to time t (i.e. the probabilitythat the particle has not visited the origin up to time t ) starting from the initial position x . The boundary and initial conditions are Q (0 , t ) = 0, Q ( x,
0) = 1 (see e.g. [15] formore general reaction boundary conditions).The backward Master equation (where the variable x is now the initial position)reads for the survival probability Q ( x, t ) ∂Q ( x, t ) ∂t = D ∂ Q ( x, t ) ∂x − rQ ( x, t ) + rQ ( x , t ) . (7)Note that Q ( x, t ) depends implicitly on the resetting position x due to the third termon the right hand side of (7). The second and third terms on the rhs correspond to theresetting of the initial position from x to x , which implies a loss of probability from Q ( x, t ) and a gain of probability to Q ( x , t ).Equation (7) may be derived as follows. We consider the survival probability Q ( x, t + ∆ t ) up to time t + ∆ t , where ∆ t is a small interval of time. We divide the timeinterval [0 , t + ∆ t ] into two intervals: [0 , ∆ t ] and [ t, t + ∆ t ]. In the first interval [0 , ∆ t ],there are two possibilities: (i) with probability r ∆ t , the particle may be reset to x andthen for the subsequent interval [∆ t, t + ∆ t ] this x will be the new starting positionand (ii) with probability (1 − r ∆ t ), no resetting takes place, but instead the particlediffuses to a new position ( x + ξ ) in time ∆ t where ξ is a random variable distributedaccording to a gaussian distribution P ( ξ ) = (4 πD ∆ t ) − / exp( − ξ / D ∆ t ). This newposition ( x + ξ ) then becomes the starting position for the subsequent second interval[∆ t, t + ∆ t ]. One then sums over all possible values of ξ drawn from P ( ξ ). Note that weare implicitly using the Markov property of the process whereby for the second interval[∆ t, t +∆ t ], only the end position of the first interval [0 , ∆ t ] matters. Taking into acountthese two possibilities, one then gets Q ( x, t + ∆ t ) = r ∆ tQ ( x , t ) + (1 − r ∆ t ) Z d ξP ( ξ ) Q ( x + ξ, t ) (8)which can be rewritten as Q ( x, t + ∆ t ) − Q ( x, t )∆ t = Z d ξ ∆ t P ( ξ )( Q ( x + ξ, t ) − Q ( x, t ))+ rQ ( x , t ) − rQ ( x, t )+ O (∆ t ) . (9)Taking the limit ∆ t → T to the origin beginning from position x is obtainedby noting that − ∂Q ( x,t ) ∂t d t is the probability of absorption by the target in time t → t +d t .Therefore, on integrating by parts, we have T = − Z ∞ t ∂Q ( x, t ) ∂t d t = Z ∞ Q ( x, t )d t (10) iffusion with Optimal Resetting tQ ( x, t ) → t → ∞ ). Integrating (7) with respect to time yields − D ∂ T ( x ) ∂x − rT ( x ) + rT ( x ) (11)with boundary conditions T (0) = 0 and T ( x ) finite as x → ∞ .To solve for the mean first passage time beginning at the resetting position x = x we first consider the initial position to be at x >
0, different from the resetting position x , then solve (11) with arbitrary x and x . Once we have this solution we set x = x to determine T ( x ) self-consistently.The general solution to (11) is T ( x ) = A e α x + B e − α x + 1 + rT ( x ) r (12)where α = q r/D . The boundary condition that T ( x ) is finite as x → ∞ implies A = 0and the boundary condition T (0) = 0 fixes B . Thus T ( x ) = 1 + rT ( x ) r h − e − α x i . (13)Solving for T ( x ) self-consistently yields T ( x ) = 1 r [exp( α x ) −
1] = 1 r (cid:20) exp (cid:18)q r/D x (cid:19) − (cid:21) . (14)Note from (14) that, for fixed x , T is finite for 0 < r < ∞ . As a function of r forfixed x , T diverges when r → T ≃ x ( Dr ) / . (15)This is expected since as r →
0, one should recover the pure diffusive behaviour (noresetting) for which the T is divergent–due to the large excursions that take the diffusingparticle away from the target at the origin. Also T diverges rapidly as r → ∞ , theexplanation being that as the reset rate increases the diffusing particle has less timebetween resets to reach the origin. In other words, the high resetting rate to x cuts offthe trajectories that bring the diffusing particle towards the target.We now consider T as a function of r for a given value of x and define the reducedvariable z = α x = (cid:18) rD (cid:19) / x . (16)Since T diverges as r → r → ∞ it is clear that there must be a minimum of T with respect to r (see Fig. 2). The condition for the minimum, d T d r = 0, reduces to thetranscendental equation z − e − z (17)which has a unique non-zero solution z ∗ = 1 . ... . In terms of the restting rate,this means an optimal resetting rate r ∗ = ( z ∗ ) D/x = (2 . . . . ) D/x , for which themean first passage time T ( x ) is minimum. The dimensionless variable z (16) is a ratioof two lengths: x , the distance from the resetting point to the target, and ( D/r ) / , iffusion with Optimal Resetting r T Figure 2.
The mean first passage time T = r h exp (cid:16)p r/D x (cid:17) − i plotted as afunction of r for fixed x = 1 and D = 1. clearly T diverges as r → r → ∞ with a single minimum at r ∗ = 2 . . . . . which is the typical distance diffused between resetting events. Thus, for fixed D and x the mean first passage time of the particle can be minimised by choosing r so thatthis ratio takes the value z ∗ .
3. Space-dependent resetting rate
In this section we generalise the model of section 2 to the case of space-dependentresetting rate r ( x ).The master equation for the probability distribution p ( x, t | x ) is generalised from(1) to ∂p ( x, t | x ) ∂t = D ∂ p ( x, t | x ) ∂x − r ( x ) p ( x, t | x )+ Z d x ′ r ( x ′ ) p ( x ′ , t | x ) δ ( x − x )(18)The third term on the right hand side now represents the flux of probability injected at x through resetting from all points x = x .The stationary distribution p ∗ ( x | x ) satisfies D ∂ p ∗ ( x | x ) ∂x − r ( x ) p ∗ ( x | x ) = − Z d x ′ r ( x ′ ) p ∗ ( x ′ | x ) δ ( x − x ) . (19)In general the stationary state is difficult to determine unless r ( x ) has some simple iffusion with Optimal Resetting − D ∂ T ( x ) ∂x − r ( x ) T ( x ) + r ( x ) T ( x ) (20)Again, this is difficult to solve generally for arbitrary r ( x ).In the following we consider a solvable example where r ( x ) is zero in a windowaround x and is constant outside this window. We consider the case of a non-resetting window of width a about | x | , within which theresetting process does not occur: r ( x ) = 0 for | x − x | < a (21)= r for | x − x | ≥ a . (22)This choice is a rather natural one in the sense that a typical searcher usually doesn’treset when it is close to its starting point, but rather the resetting event occurs when itdiffuses a certain threshold distance a away from its initial position.The Master equation reads ∂p ( x, t | x ) ∂t = D ∂ p ( x, t | x ) ∂x + rh ( t ) δ ( x − x ) | x − x | < a (23)= D ∂ p ( x, t | x ) ∂x − rp ( x, t | x ) | x − x | ≥ a (24)where h ( t ) = Z d x p ( x, t | x ) θ ( | x − x | − a ) , (25)with initial condition p ( x,
0) = δ ( x − x ). Thus h ( t ) is the probability that the particleis outside the non-resetting window, i.e., in the resetting zone at time t ; the particle isreset to the origin with a total rate h ( t ) r .First, we consider the stationary state. One can solve for the stationary probabilityusing the Green function technique of section 1. For | x − x | > a (outside the window), p ∗ ( x | x ) satisfies D ∂ p ∗ ( x | x ) ∂x = rp ∗ ( x | x ) and should tend to zero as | x | → ∞ . For0 < | x − x | < a (inside the window), p ∗ ( x | x ) satisfies D ∂ p ∗ ( x | x ) ∂x = 0 for all x = x .The solution should be continuous at x = x , but its derivative must undergo a jumpat x = x and the jump discontinuity can be computed by integrating Eq. 23 across x = x .Thus, noting that the solution should be symmetric about x = x , one has p ∗ ( x | x ) = A exp − α ( | x − x | − a ) for | x − x | > a (26)= A − B ( | x − x | − a )) for | x − x | < a (27)where α = q r/D and the constants A and B are determined by the discontinuity inthe derivative of p ∗ ( x | x ) at x = x and the continuity of the derivative at | x − x | = a . iffusion with Optimal Resetting −10 −5 0 5 10 x p * ( x | ) Figure 3.
The stationary solution p ∗ ( x | x ) in Eq. 26-27 plotted as a function of x , forthe choice x = 1, a = 1, r = 1 and D = 1. The nonresetting window is over x ∈ [0 , x = 1. The solution is symmetric around x = 1 with acusp at x = x = 1. The result is A = α aα + a α B = α A (28)The solution has a cusp at x = x and a discontinuity in the second derivative at | x − x | = a (see Fig. 3).We now consider an absorbing trap at the origin. The backward equation for T ( x ),the mean time to absorption beginning from x , reads − D ∂ T ( x ) ∂x − rT ( x ) + rT ( x ) for | x − x | > a (29) − D ∂ T ( x ) ∂x for | x − x | < a . (30)The general solution to (29,30) is T ( x ) = Ax + B − x D (31)and the solution that does not diverge as x → ∞ is T ( x ) = 1 + rT ( x ) r + C e − α ( x − x − a ) for x > x + a (32) T ( x ) = 1 + rT ( x ) r + E e − α x + F e α x for x < x − a . (33) iffusion with Optimal Resetting A, B, C, E, F are determined by the continuity of T ( x ) and T ′ ( x ) at | x − x | = a and the boundary condition T (0) = 0. The result for T ( x ) is T ( x ) = 1 r (1 + aα ) " cosh α ( x − a ) aα + 3 a α a α ! + sinh α ( x − a ) aα + 3 a α ! − r . (34)We now consider the reduced parameters z = α x and y = α a , and T as a function y for z fixed. The allowed values of y are 0 ≤ y ≤ z . At y = z , one can show that d T d y (cid:12)(cid:12)(cid:12) y = z >
0. Therefore the minimum of T with respect to y is either at y = 0 or at anon trivial minimum 0 ≤ y ≤ z .The condition for a minimum d T d y = 0 reduces to2 + y y + 2 y = tanh( z − y ) (35)Therefore the condition for there to be a nontrivial minimum for y > z > / z > (log 3) / . . . . .In summary, the analysis of the condition for T ( y ) to be a minimum reveals that:if z < (log 3) / y = 0 is the minimum of T ( y ); if z > (log 3) / T ( y ) has anontrivial minimum at 0 < y < z . Therefore, when z < (log 3) / Having seen in the previous example that non-trivial behaviour emerges for a simplespatial-dependent resetting rate r ( x ), one can ask for the optimal function r ( x ). Theoptimisation problem would be to minimise T under certain constraints pertaining tothe information available to the searcher. Clearly if there are no constraints, that isone can use full information about the target position, the optimal strategy is to resetimmediately whenever x > x and not reset when x < x . This corresponds to thechoice r ( x ) = 0 for x < x r ( x ) = ∞ for x > x . In this case problem (20) reduces to the mean first passage time of a diffusive particlewith reflecting barrier at x the solution of which is T ∗ ( x ) = x D . (36)Thus, (36) gives the lowest possible mean first passage time for a diffusive process. Onecan then ask about how close simple strategies, such as a spatially constant resettingrate r or non-resetting window, come to approaching this bound. iffusion with Optimal Resetting r considered in section 2yields a minimum MFPT using (17) T = x D (e z − z = x D e z ∗ z ∗ = 3 . ... T ∗ ( x ) (37)As noted in section 3.1 the value 3.0883 may be improved upon by considering a non-resetting window around x .However, (36) uses the crucial information of whether the target (at x = 0) is tothe right or left of the resetting site x . More realistically, the searcher would not havethis information. The relevant optimisation problem is to find the optimal resetting rate r ( | x − x | ) (constrained to be a function of the distance | x − x | from the resetting site)that minimises T ( x ). This remains an open problem.
4. Resetting distribution and target distribution
In this section we consider the generalisation to a system with resetting to pointsdistributed according to P ( z ). We shall also consider a distribution of the target site P T ( x ). We begin by considering again the one-dimensional case of diffusion but this time withresetting to a random position: at rate r the particle is reset to a random position z → z + d z drawn with probability P ( z )d z . We refer to P ( z ) as the reset distribution.For simplicity we take the initial position x to be distributed according to the samedistribution as the reset position p ( x ,
0) = P ( x ).The Master equation for the probability density p ( x, t ) now reads ∂p ( x, t ) ∂t = D ∂ p ( x, t ) ∂x − rp ( x, t ) + r P ( x ) . (38)The stationary solution to (38) is simply found using (6) as the Green function: p ∗ ( x ) = Z d z P ( z ) p ∗ ( x | z ) (39)which, using p ∗ ( x | x ) given by (6), yields p ∗ ( x ) = α Z d z P ( z ) exp( − α | x − z | )) . (40) The mean first passage time, T ( x , x T ), to a target point x T , starting from x withresetting distribution P ( z ), satisfies − D ∂ T ( x , x T ) ∂x − rT ( x , x T ) + r Z d z P ( z ) T ( z, x T ) (41)with boundary condition T ( x T , x T ) = 0. To solve this equation we let F ( x T ) = Z d z P ( z ) T ( z, x T ) (42) iffusion with Optimal Resetting F ( x T ) self-consistently.The general solution of (41) which is finite as x → ∞ is T ( x , x T ) = A e − α | x − x T | + 1 r + F ( x T ) (43)The boundary condition T ( x T , x T ) = 0 implies A = − (cid:16) r + F (cid:17) . Then substituting thisexpression for A in (43) and integrating we find F ( x T ) = (cid:18) r + F ( x T ) (cid:19) (cid:18) − Z d z P ( z ) e − α | z − x T | (cid:19) (44)which yields F ( x T ) = 1 r α p ∗ ( x T ) − ! . (45)Inserting this into (43) we obtain T ( x , x T ) = α rp ∗ ( x T ) [1 − exp( − α | x − x T | )] . (46)As noted above it is convenient to choose the same distribution for x as the resettingdistribution. Averaging over x then gives using (39) T ( x T ) = 1 r " α p ∗ ( x T ) − . (47)Equation (47) gives the expression for the mean first passage time to a target positionedat x T . Let us check the case of a single position x to which the particle is reset P ( z ) = δ ( z − x ). In this case (47) becomes T ( x T ) = 1 r " α p ∗ ( x T ) − (48)which recovers (14) when x T is set to 0.Finally, we average over possible target positions drawn from a target distribution: P T ( x T ) T = 1 r " α Z d x T P T ( x T ) p ∗ ( x T ) − . (49)Equation (49) gives the main result of this section— the MFPT for a resettingdistribution P ( x ) and averaged over target distribution P T ( x T ).
5. Extremisation of mean first passage time
Let us now consider the problem of extremising T given by (49), for a given targetdistribution P T ( x ), with respect to the resetting distribution P ( z ). Throughoutthis section we will assume a symmetric target distribution: P T ( x ) = P T ( − x ) and P ′ T ( x ) = − P ′ T ( − x ).The problem is to minimize the functional appearing in (49): R d x P T ( x ) p ∗ ( x ) where p ∗ ( x ) = α Z d z P ( z )e − α | z − x | , (50) iffusion with Optimal Resetting R d z P ( z ) = 1. The functional derivative to be satisfied is δδ P ( y ) "Z d x P T ( x ) p ∗ ( x ) + λ Z d x P ( x ) = 0 (51)where λ is a Lagrange multiplier. Condition (51) yields Z d x P T ( x )[ p ∗ ( x )] e − α | y − x | = 2 λα . (52)For (52) to hold for all y requires that P T ( x )[ p ∗ ( x )] = λ , (53)or fixing λ through the normalisation of p ∗ ( x ) p ∗ ( x ) = P / T ( x ) R d zP / T ( z ) . (54)Equation (54) implies that to minimise T the stationary probability distributionshould be given by the square root of the target distribution. This result has been derivedin [16] for the case of searching for the target by sampling a probability distribution P ( x ).This corresponds to the limit r → ∞ of our model. For r < ∞ we have the additionalconstraint that the optimal p ∗ should be realisable from a resetting distribution P ( z )through formula (50).Equation (50) may be solved for P ( z ) for a desired p ∗ ( x ) by taking the Fouriertransform and using the convolution theorem to give e P ( k ) = k α ! e p ∗ ( k ) (55)where e P ( k ) is the Fourier transform of P ( x ) and e p ∗ ( k ) is the Fourier transform of p ∗ ( x ).We may invert the Fourier transformation to find P ( x ) = p ∗ ( x ) − α d p ∗ ( x )d x . (56)However this solution may become negative in which case the solution to theoptimisation problem is unphysical. As a simple example, we consider an exponentially decaying target distribution peakedat x = 0: P T ( x ) = β − β | x | . (57)We first note that for a delta function resetting distribution P ( z ) = δ ( z − x ) the meanfirst passage time (49) diverges when α > β . Therefore, for small β (a broad targetdistribution) one expects an optimal resetting distribution (for fixed α ) that differsfrom a delta function. iffusion with Optimal Resetting β < α , the optimal stationary distribution is from (54) p ∗ ( x ) = β − β | x | / (58)This expression yields from (56) a resetting distribution that is always positive, thusthe optimal resetting distribution P ( z ) = β − β | z | / " − β α + β α δ ( z ) (59)For β > α , (59) always gives negative probabilities due to the first term. Thereforewe anticipate that P ( x ) = δ ( x ) is at least a locally optimal solution. In fact one can provethis is the case by showing that any distribution of the form P ( x ) = (1 − ǫ ) δ ( x ) + ǫf ( x ),where f ( x ) ≥ R d xf ( x ) = 1 leads to an increase in (49) at first order in ǫ when β > α . (As the proof is straightforward but somewhat tedious we did not include ithere.) Thus a transition in the form of the optimal resetting distribution, from a singledelta function to (59), occurs at β = 2 α . p ∗ ( x )As noted above, the constraint P ( x ) ≥ p ∗ ( x ) given by (54)may not be realisable from a physical resetting distribution P ( z ). We are therefore ledto the general question of when a desired stationary distribution (e.g. (54)) which wedenote g ( x ) may be generated from (50) i.e. when can we invert g ( x ) = α Z d z P ( z )e − α | z − x | (60)to obtain a physical P ( z )?Let us first discuss a sufficient condition for the resetting distribution implied by(60) to be physical.Equation (55) relates the characteristic functions of the two distributions P ( x ) and g ( x ) (given there by p ∗ ( x )). In terms of the characteristic function, Polya’s theorem [17]states that if a function φ ( k ) satisfies: φ (0) = 1; φ ( k ) is even; φ ( k ) is convex for k > φ ( ∞ ) = 0; then φ ( k ) is the characteristic function of an absolutely continuoussymmetric distribution. Polya’s theorem therefore gives a sufficient condition for P ( x )implied by p ∗ ( x ) to be physical.The condition for convexity becomes in one dimensiond d k " k α ! g ( k ) ≥ k ≥ . (61)If the function e P ( k ) does not satisfy the conditions of Polya’s theorem, the solution of(60) is invalid as a probability distribution i.e. the desired g ( x ) cannot be realised fromany resetting probability distribution P ( z ).In the case where (60) may not be inverted to give a physical P ( x ), it may bepossible to generate the desired form for g ( x ) on a finite region by choosing a compact iffusion with Optimal Resetting P ( z ). Let us assume g ( x ) to be a symmetric function of x . Then if wechoose P ( z ) = λ " g ( z ) − α d g ( z )d z for | z | ≤ y (62)= 0 for | z | > y , (63)where λ is a normalising constant, we find p ∗ ( x ) = λg ( x ) for | x | ≤ y (64) p ∗ ( x ) = λg ( y )e α ( y −| x | ) for | x | ≥ y (65)provided that y is chosen so that g ( y ) + 1 α g ′ ( y ) = 0 (66) g ( − y ) − α g ′ ( − y ) = 0 . (67)(see Appendix A). The second condition follows from the first by the assumed symmetryof g ( x ). As an example, we consider the gaussian distribution g ( x ) = βπ ! / e − βx (68)The inversion of (60) using (56) yields P ( x ) = g ( x ) − α d g ( x )d x = βπ ! / e − βx " βα − β x α (69)which becomes negative for | x | > α β βα ! / (70)However, choosing a compact support for P ( z ) according to (67), yields y = α β (71)and we find that the resulting distribution (70) is positive for all x .
6. Conclusion
In this paper we have considered some generalisations of diffusion with stochasticresetting to the case of spatial-dependent resetting rate and a resetting distribution.We have considered the mean first passage time to a target which may be situated at afixed point (the origin) or distributed according to a distribution and derived the result(49). The minimisation of this quantity may then be formulated as an optimisationproblem of which we have studied some examples.In particular we have seen some perhaps unexpected results. First, the introductionof a non-resetting window around a fixed resetting position reduces the MFPT when thetarget is sufficiently far away. This suggests that the optimal resetting distribution, in iffusion with Optimal Resetting r ( | x − x | ) may be non-trivial. We have also seen that in the case of an exponentiallydistributed target (57) the optimal resetting distribution undergoes a transition from(59) to a pure delta function at the origin.Generally, the computation of an optimal resetting distribution is an open problemsince the resetting distribution that minimises T may be become negative over somedomain and therefore nonphysical. In the case where (56) becomes unphysical, althoughwe do not have a solution to the extremisation problem of minimising T subject to theadditional constraint P ( x ) ≥ P ( x ) ≥
0, one mightexpect that the optimal solution lies on the boundary where P ( x ) = 0 for some regionsof x . However, we have no proof that this is the case.Further considerations for optimising mean first passage times in a more realisticsearch process would be to add a cost to resetting since in the present model thediffusive particle instantaneously resets to its selected resetting position. This couldbe implemented by attributing some time penalty to each resetting event, as is the casein the framework intermittent searching. Appendix A. Proof that (63) yields (65)
We wish to show that expression (63) for P ( x ) P ( z ) = λ " g ( z ) − α d g ( z )d z for | z | ≤ y (A.1)= 0 otherwise , (A.2)yields (64)-(65) for the stationary distribution given by (40), provided that (66)-(67)holds.We begin by inserting (A.1)-(A.2) into (40) in the case | x | < y : p ∗ ( x ) = α λ (Z x − y " g ( z ) − α d g ( z )d z e − α ( x − z ) + Z y x " g ( z ) − α d g ( z )d z e − α ( z − x ) ) (A.3)We use the following integration by parts, valid for all α = 0 Z ba " g ( z ) − α d g ( z )d z e α z d z = " g ( b ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = b e α b α − " g ( a ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = a e α a α (A.4)Inserting this into (A.3) and cancelling terms yields p ∗ ( x ) = α λ g ( x ) α + e − α ( x + y ) α g ( − y ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = − y + e α ( x − y ) α − g ( y ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = y (A.5) iffusion with Optimal Resetting p ∗ ( x ) = λg ( x ) for | x | < y .In the case x > y we find p ∗ ( x ) = α λ Z y − y " g ( z ) − α d g ( z )d z e − α ( x − z ) d z = α λ − α x e α y α g ( y ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = y − e − α y α g ( − y ) − α d g ( z )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = − y = λ e − α ( x − y ) g ( y ) (A.6)where conditions (66)-(67) have been used.Similarly in the case x < − y we obtain p ∗ ( x ) = λg ( − y )e α ( y + x ) . Acknowledgements
MRE and SNM thank the Weizmann Institute for Weston Visiting Professorships
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