Optimal Resetting Strategies for Search Processes in Heterogeneous Environments

New J. Phys. 25 (2023) 113031 https://doi.org/10.1088/1367-2630/ad06da OPEN ACCESS RECEIVED 29 May 2023 REVISED 18 October 2023 ACCEPTED FOR PUBLICATION 25 October 2023 PUBLISHED 20 November 2023 Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. PAPER Optimal resetting strategies for search processes in heterogeneous environments Gregorio García-Valladares1, Carlos A Plata1, Antonio Prados1and Alessandro Manacorda2,∗ 1Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain 2Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg ∗Author to whom any correspondence should be addressed. E-mail: alessandro.manacor[email protected] Keywords: optimal search strategies, stochastic resetting, quenched disorder, first passage time, heterogeneity, non-equilibrium statistical mechanics Abstract In many physical situations, there appears the problem of reaching a single target that is spatially distributed. Here we analyse how stochastic resetting, also spatially distributed, can be used to improve the search process when the target location is quenched, i.e. it does not evolve in time. More specifically, we consider a model with minimal but sufficient ingredients that allows us to derive analytical results for the relevant physical quantities, such as the first passage time distribution. We focus on the minimisation of the mean first passage time (MFPT) and its fluctuations (standard deviation), which proves to be non-trivial. Our analysis shows that the no-disorder case is singular: for small disorder, the resetting rate distribution that minimises the MFPT leads to diverging fluctuations—which impinge on the practicality of this minimisation. Interestingly, this issue is healed by minimising the fluctuations: the associated resetting rate distribution gives first passage times that are very close to the optimal ones. 1. Introduction Stochastic resetting is an intriguing framework in stochastic dynamics because of its game-changing strategy in Brownian search processes, with applications in a wide range of contexts beyond physics, such as animal foraging, site-binding search of biomolecules or chemical reactions [1]. In search processes, a usual goal is the minimisation of the mean first passage time (MFPT), which is the average time—over the different realisations of the dynamics—for finding the target. The simplest situation arises when one considers a particle undergoing Brownian motion in one dimension, with an immobile target at a certain position xT and a Poissonian reset process with rate rto a certain point xr. Resetting makes the MFPT finite and, in addition, there appears an optimal resetting rateer—the MFPT diverges in both the limits of no resetting (r=0) and of infinitely frequent resetting (r→∞). General intermittent strategies, which combine searching reactive phases with faster relocating non-reactive phases, have been shown to be significantly useful in the context of search problems [2–8]. These strategies have been applied in both microscopic and macroscopic contexts, both for single- and multiple-target problems, as discussed in the review paper [9]. In particular, stochastic resetting can be thought of as a particular instance of these intermittent strategies, in which—in its simplest formulation—the non-reactive phase is instantaneous. Typically, the target is considered to be fixed in space, i.e. a single target remains at the same position for all realisations of the search process. Still, the position of the target may vary between realisations, i.e. there may be some degree of randomness in the search process that renders every instance thereof unique. We refer to this situation, in which the location of our goal is uncertain, as a search process with disorder—employing a terminology that is usual in physics. More specifically, we focus on problems for which the location of the © 2023 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 25 (2023) 113031 G García-Valladares et al target is fixed for each realisation of the stochastic process, which thus gives rise to a quenched disorder. This quenched disordered scenario naturally emerges in many contexts, such as environmental phenomena associated with foraging [10–16]. Therein, animals look for nutrient or resources throughout an environment that is heterogeneous; the targets are static (quenched) but spatially distributed. Another classical disordered scenario is that found for the location of minima of a complex energy landscape, which possibly represents the main challenge in disordered systems such as glasses [17–21]; a problem that becomes highly non-trivial especially in high-dimensional contexts. In contrast with quenched disorder, annealed disorder—also called dynamical disorder [22]—can also be considered. In the foraging context, annealed disorder involves a dynamic target, which changes its position over the same time scale of the searcher. Interestingly, stochastic resetting has been very recently introduced for intermittent targets, i.e. targets that fluctuate between a reactive and a non-reactive state [23,24]. This kind of intermittent targets are often found in cell biology, from the accessibility of DNA-binding sites to the opening and closing of ion channels, and they can be regarded as a particular case of annealed disorder. In this work, we mainly address search processes with quenched (or static) disorder, i.e. we consider that the position of the target follows a certain static, time-independent, spatial distribution pT(xT). In the same context of foraging, this can be understood from the dynamics of searcher being much faster than that of the target, which is thus assumed to be immobile after the animal starts the search. The assumption of disorder for the target location raises two questions in the field of stochastic resetting: what is the effect of stochastic resetting on such a disordered search problem? Is it possible to choose a clever, space-dependent reset rate r(x) to optimise the search process, given the distribution of the target position pT(xT)? Note that a homogeneous resetting rate cannot be the optimal choice in general. For a fixed, delta-distributed target, with resetting position xr=0, the optimal resetting rate is heterogeneous: r(x<0)→∞and r(x>0) = 0, since this choice always prevents particles from moving away from the target. The above discussion entails that, physically, it is expected that any quenched heterogeneous distribution of targets pT(xT), stemming from the heterogeneity of the searcher’s environment, has an optimal heterogeneous resetting rate r(x) associated. Despite the intuitive character of this statement, few works have been devoted to the analysis of stochastic resetting in heterogeneous environments. A first attempt involved the optimisation of the distribution of the resetting position in presence of a non-resetting window around the resetting point for exponentially distributed targets [25]. Therein, a transition in the optimal distribution was found out. While a general framework to obtain the stationary distribution in presence of heterogeneous resetting has been developed [26], the MFPT has been considered in works where the resetting takes place only when the particle’s position exceeds some threshold value [27,28], a problem that naturally appears when modelling catastrophic events like power blackouts. In the context of systems with discrete states, a random walker on complex networks with node-dependent resetting rate has been considered as well [29]. Heterogeneity has also been investigated introducing spatially-dependent diffusion coefficients [30–32], in a somewhat alternative direction with respect to the present paper. Rigorous results have been derived for the MFPT of a spatially-dependent resetting problem with a given distribution of the target position, assuming that explicit solutions for the MFPT equation are known [33]. Heterogeneity thus emerges as a promising research ground in stochastic resetting and search processes. The objective of this work is to investigate the effect of quenched disorder on the first passage time problem. Analytical results are typically hindered by the challenging mathematical structure of the optimisation problem. Therefore, we assume here a simple dichotomous time-independent distribution of the target position pT(xT): the distance to the target is fixed and the probability of the target of being to the right (left) of the resetting point is p(1 −p). Consequently, we also assume a constant dichotomous resetting rate, piecewise homogeneous in the two real half-lines. These choices will allow us to answer analytically the following questions: what are the optimal resetting rates? how is the MFPT reduced by this assumption? what are the fluctuations of the first passage time, especially when the disorder is small, i.e. in the p≪1 (or 1−p≪1) limit? what is the asymptotic behaviour of the optimal solution in the same limit and the long time tail of the first passage time distribution? The answer to these questions represents a first step towards the generalisation to more realistic target distributions and to heterogeneous resetting processes. 2. Model We consider a one-dimensional freely diffusing overdamped Brownian particle submitted to stochastic resetting to a given point xr, with heterogeneous resetting rate r(x). Thus, the probability to observe the particle at position xat time tgiven that the initial (t0=0) position was x0, i.e. the propagator P(x,t|x0), follows the (forward) dynamic law 2 New J. Phys. 25 (2023) 113031 G García-Valladares et al ∂tP(x,t,|x0) = D∂2 xP(x,t|x0)−r(x)P(x,t|x0) +δ(x−xr)ˆdyr(y)P(y,t|x0),(1) where Dis the diffusion constant. On the right hand side (rhs) of the evolution equation, each term has a neat physical meaning. Namely, the first term stands for free diffusion, the second one is a loss term due to resetting, and the last one is a gain term due to the particles that have reset from any point to x=xr. To the purpose of the work, a piecewise constant resetting rate is assumed r(x) = r+Θ(x−xr) + r−Θ(xr−x) = r+,if x>xr, r−,if x<xr,(2) where Θstands for the Heaviside function. Our assumption of a fixed resetting location at x=xris especially appealing—in addition to being the simplest—in the context of foraging problems, where the searching animal has a single home to come back. Furthermore, the physically relevant information for our purposes is the distribution of the target location with respect to the resetting location; one can therefore safely fix the resetting location at the origin without any loss of generality. Equation (1) has to be complemented with initial and boundary conditions. The initial condition for the propagator is, of course, P(x,0|x0) = δ(x−x0). Boundary conditions depend on the physical constraints. On the one hand, in the absence of any constraint, the probability current vanishes at x→±∞, i.e. lim x→±∞ ∂xP(x,t|x0) = 0.(3) In the long-time limit, the corresponding solution P(x,t|x0)tends to the non-equilibrium stationary state Ps(x)≡α+α− α++α−he−α+(x−xr)Θ(x−xr) + eα−(x−xr)Θ(xr−x)i,(4) where the competition between diffusion and resetting defines the length scales α−1 ±=pD/r±. Note that (4) can be explicitly obtained due to the simple choice for r(x) in (2). An explicit solution of equation (1) is not available for arbitrary r(x), which would define an associated space-dependent length scale α(x)−1=pD/r(x). On the other hand, if there is just one absorbing boundary at x=xT, the boundary conditions to impose are P(xT,t|x0) = 0,lim x→sgn(xr−xT)∞∂xP(x,t|x0) = 0,(5) standing sgn(x)for the sign function. In our study, it suffices to impose the second condition in equation (5) and not necessarily limx→±∞ ∂xP(x,t|x0) = 0 because we will only consider situations where x0and xrare on the same side with respect to xT. In a more general case, both limit conditions should be enforced. The case with just one absorbing boundary is the one addressed in this article, since we will look into the first passage problem to a target position with no further obstacles. Now we introduce the disorder, which is a fundamental and novel ingredient in our modelling. We assume that xTis static but not fixed; instead, it is distributed according to a certain distribution pT(xT)—i.e. the disorder is quenched. Note that equation (1), complemented with equation (5), is still valid, but governing the evolution of the propagator P(x,t|x0;xT), where it is important to make the dependence on xT explicit. The propagator for the ensemble is achieved by integrating over xT, P(x,t|x0) = ˆdxTpT(xT)P(x,t|x0;xT).(6) As already commented in the introduction, we use the terminology quenched disorder throughout our work, which is quite usual in the context of statistical physics, in opposition to annealed disorder. The disorder is quenched in our model, since xTis ‘frozen’ for each realisation of the dynamics. Instead, annealed disorder would involve a target xTevolving with a certain dynamics, even within a single realisation. In our context, a simple example of annealed disorder would be obtained if the target were relocated to a new random position extracted from pT(xT)after each resetting event. Despite its apparent similarity, this problem with annealed disorder is not equivalent to the quenched disorder situation considered here. For the sake of simplicity, we use the simplest non-trivial quenched disorder pT(xT) = pδ(xT−xr−X) + (1−p)δ(xT−xr+X),(7) 3 New J. Phys. 25 (2023) 113031 G García-Valladares et al with 0 <p<1 and X>0. The distance from the resetting position to the target is fixed and equal to X, whereas there is a certain probability p(1 −p) for the target to be to the right (left) of xr. The heterogeneity of pTmotivates the asymmetric choice with respect to spatial inversion of x−xrfor the resetting rate in (2). It is handy to use dimensionless variables in order to identify the natural units of our problem. Specifically, we take ¯ x= (x−xr)/Xand¯ t=t/(X2/D). Consequently, ¯ P(¯ x,¯ t|¯ x0;±1) = XP(x,t|x0;±X)and ¯α(¯ x) = Xα(x). Then, equation (1) becomes ∂tP(x,t|x0;xT) = ∂2 xP(x,t|x0;xT)−α(x)2P(x,t|x0;xT) +δ(x)ˆdyα(y)2P(y,t|x0;xT),(8) in dimensionless form, where all bars have been removed in order to avoid cluttering our formulas. The boundary conditions in equation (5) become P(xT,t|x0;xT) = 0,lim x→−sgn(xT)∞∂xP(x,t|x0;xT) = 0,(9) where xT=±1 and α(x) = α+Θ(x) + α−Θ(−x).(10) Averaging over the disorder (7), P(x,t|x0) = pP+(x,t|x0) + (1−p)P−(x,t|x0),(11) where we have conveniently introduced the notation P+(x,t|x0)≡P(x,t|x0;xT= +1),P−(x,t|x0)≡P(x,t|x0;xT=−1).(12) As described in the introduction, our focus is not the forward evolution but the first passage problem, which is formulated from the backward evolution in the next section. Nevertheless, the separation introduced in equation (11), where the observable of interest is achieved from the average of the quenched contributions, each of them verifying its own equation, e.g. equation (8) with boundary conditions (9) will be exploited along the rest of this work. 3. First passage problem From now on, we aim at solving the first passage problem to the disordered target. To accomplish this task, we start from the backward evolution equation for P±(x,t|x0). Using an analogous notation to that for P±(x,t|x0)in (12), we define the survival probabilities S±(t|x0) = ˆdxP±(x,t|x0),(13) i.e. S+(t|x0)is the probability for having avoided at time tthe absorption at xT= +1 starting from x0—similarly for S−(t|x0)with xT=−1. The corresponding distributions of first passage times to xTstarting from x0are f±(t|x0) = −∂tS±(t|x0).(14) All these functions, P±(x,t|x0),S±(t|x0), and f±(t|x0), fulfil the same backward equation below. For the sake of concreteness, we write the backward equation just for the first passage time distribution f±(t|x0), which plays a central role in this research, ∂tf±(t|x0) = ∂2 x0f±(t|x0)−α(x0)2f±(t|x0) + α(x0)2f±(t|0).(15) A brief derivation of the backward equation is presented in appendix A. The first passage time distribution has to be complemented with the boundary conditions f±(t|±1) = δ(t),lim x0→∓∞ ∂x0f±(t|x0) = 0,(16) 4 New J. Phys. 25 (2023) 113031 G García-Valladares et al and the initial condition f±(0|x0) = 0, which is a consequence of S±(0|x0) = 1 for any x0. Moreover, since the particle unavoidably reaches the target at long times, we have the normalisation condition ˆ∞ 0 dtf±(t|x0) = S±(0|x0) | {z } 1 −lim t→∞ S±(t|x0) | {z } 0 =1.(17) It is important to keep in mind that the first passage time distribution of the ensemble f(t|x0)is obtained averaging over the disorder, f(t|x0) = pf+(t|x0) + (1−p)f−(t|x0).(18) This is a property that is directly transferred to all moments of the distribution, as well as any linear combination thereof. In what follows, we just focus on f+(t|x0)due to the left-right symmetry of our system: f−(t|−x0)is obtained by exchanging α+←→α−in f+(t|x0). This invariance stems from a mirror symmetry in our system: when simultaneously exchanging xT←→−xTand r+←→r−, the obtained system is just the mirror image of the original one with respect to the resetting point xr=0. For solving the quenched first passage time distribution, we define the moment generating function (Laplace transform) ϕ±(s|x0) = ˆ∞ 0 dte−stf±(t|x0),(19) from which all the moments of the distribution, τ(n), are derived: τ(n) ±(x0) = ˆ∞ 0 dttnf±(t|x0)=(−1)n∂n sϕ±(s|x0)|s=0.(20) Making use of the initial condition f±(0|x0) = 0, equation (15) yields sϕ±(s|x0) = ∂2 x0ϕ±(s|x0)−α(x0)2ϕ±(s|x0) + α(x0)2ϕ±(s|0),(21) which, together with equation (20), gives the recursive equation for the moments −nτ(n−1) ±(x0) = ∂2 x0τ(n) ±(x0)−α(x0)2hτ(n) ±(x0)−τ(n) ±(0)i,(22) with τ(0) ±(x0) = 1. Instead of solving equation (22) for the moments iteratively, we address the first passage time problem in a more compact way by solving equation (21) for ϕ+(s|x0)at both sides of the resetting point. Specifically, the boundary conditions along with continuity for the function and its derivative at x0=0 must be imposed, as well as the consistency condition stemming from the local contribution at the resetting point—see appendix Bfor the detailed solution. Herein, for the sake of concreteness, we focus on the first passage problem starting from the resetting position, i.e. x0=0. The result for the quenched moment generating function is ϕ+(s|0) =    α2 ++scoshqs+α2 + s+α2 + + ssinhqs+α2 + qs+α2 +qs+α2 −    −1 .(23) The behaviour of the tails of the first passage time distribution is derived by carrying out an asymptotic analysis of the moment generating function. Specifically, one gets f+(t|0)∼s+α2 +s+α2 − ∂sζ(s,α+,α−)ests=s∗ + ,t→∞,(24) where ζ(s,α+,α−)≡s+α2 +s+α2 − ϕ+(s|0),(25) 5 New J. Phys. 25 (2023) 113031 G García-Valladares et al and s∗ +is the pole with the highest real part of the moment generating function (23), which is always real and negative. See appendix Cfor the detailed technical derivation. As already stated before, ϕ−(s|0)is obtained from ϕ+(s|0)by exchanging α+↔α−. Therefore, we have that f−(t|0)∼s+α2 −s+α2 + ∂sζ(s,α−,α+)ests=s∗ − ,(26) with s∗ −being the zero of the function ζ(s,α−,α+)with the highest real part. Hence, equations (18), (24) and (26) entail that the first passage time distribution is a weighted sum of two exponential functions for long times. Such a prediction has been validated by simulating the stochastic process for different values of the system parameters, as shown below. Figure 1compares our theoretical predictions for the first passage time distribution with numerical results. In each panel, a different value of the disorder pis displayed; the rest of parameters, i.e. (α+,α−), are those minimising the MFPT for that value of p—derived later in section 4. Simulation data (circles) correspond to 107trajectories for the parameters considered in each panel, integrated with a time step ∆t=10−5up to the first passage time; they show an excellent agreement with our theoretical predictions, both asymptotic (solid green) and numerical Laplace inversion (dashed red). Although the asymptotic prediction is expected to work only in the limit of long times, it fails just at very short times. Therein, numerical inversion of the exact result still perfectly matches the simulations—as particularly shown in the inset of the second panel. An especially interesting feature of figure 1is the apparent crossover between the two exponential decays corresponding to the long time behaviours of f±(t|0)in equations (24) and (26), provided that p={0,1}. For p=0 (first panel), i.e. no disorder, only f−contributes and therefore the long time behaviour is exponential with s∗ −. For 0 <p<0.5 (rest of panels), the largest pole (smallest in absolute value) is s∗ +for our choice of (α+,α−); note that s∗ +vanishes in the limit p→0. This entails the crossover: even for p=0+, the long time behaviour is first dominated by s∗ −(since the contribution from the +mode is weighted with p≪1) but turns later to be dominated by s∗ +. The crossover is not present for p=1/2, because s∗ +=s∗ −for our choice of parameters. Note that each panel follows the same scale, this decision has been made to stress the emergence of a longer time exponential tail for non-zero disorder—on the first panel, it is clearly observed that the first passage distribution decays much faster than on the other panels. The mean and the second moment of the quenched contribution are obtained by using equations (20) and (23)—or, equivalently, solving equation (22) for n=1,2, τ(1) +=α−2 +coshα++α+ α− sinhα+−1≡F1(α+,α−),(27) τ(2) +=1 α3 −α4 +α2 −−α2 +α−+α2 +−α2 −(α−+3)α+sinhα+ −α2 ++2α−−4α+sinhα+α2 −coshα+ +α2 −+α2 +α−cosh(2α+)≡F2(α+,α−).(28) Again, τ(n) −follows from τ(n) +by exchanging α+←→α−. Note that, in order to simplify the notation, we have omitted the argument of τ(n) ±,τ(n); in this work, we only consider the case x0=xr=0. Averaging over the disorder, τ(n)=pτ(n) ++ (1−p)τ(n) −.(29) In the following, we mainly focus on the MFPT and the standard deviation, τ(1)and στ, respectively. The latter cannot be computed as an average between two contributions στ±since it is not a linear combination of moments, we must use στ=qτ(2)−τ(1)2.(30) We have obtained exact results for τ(1)and στas functions of (p,α+,α−), but their explicit formulas have been deliberately obviated since they are not illuminating. For α+=α−=αh(homogeneous resetting case), the expressions for the MFPT and the standard deviation are τ(1) h=eα−1 α2, στ,h=p(2sinhα−α)eα α2,(31) 6 New J. Phys. 25 (2023) 113031 G García-Valladares et al Figure 1. First passage time distribution for the disordered resetting system. Specifically, we plot the results from the numerical simulation (circles), the numerical Laplace inversion of equation (23) (dashed red line), the asymptotic contribution from f+ given by equation (24) (dotted blue line), the asymptotic contribution from f−given by equation (26) (dotdashed orange line), and the sum of the two latter asymptotic contributions (solid green line). On the inset plot of the right top panel, we can check how the numerical Laplace inversion and simulations agree at short times and both lines overlap with the asymptotic estimation from t=2·10−1on. Figure 2. Mean first passage time (left panel) and its standard deviation (right panel) as functions of α+, at fixed α−=2. Simulation data (symbols) is compared with the exact analytical results (solid lines), showing a perfect agreement. Different values of disorder phave been used to produce the different curves, as detailed in the legend. which is a robust result, valid regardless the value of p, due to the left-right symmetry. This result reasonably coincides with that obtained for homogeneous resetting in the absence of disorder [25,34]. In figure 2, we compare our theoretical predictions for τ(1)and στwith numerical simulations, again an excellent agreement is found. For the numerical results, a time step ∆t=10−5and 106trajectories have been used. Specifically, α−has been fixed and the plots show the dependence of the observable of interest as a function of α+, for several values of p. All curves exhibit a minimum value as a function of α+, except those corresponding to p=0. Optimality of the observables is discussed in section 4. Figure 3provides a more complete picture of the behaviour of the MFPT and the standard deviation as functions of (α+,α−), for different values of p. It is worth highlighting some interesting properties. As expected, the panels for p=0.5 show reflection symmetry with respect to the line α+=α−. As a 7 New J. Phys. 25 (2023) 113031 G García-Valladares et al Figure 3. Density plots of the mean first passage time (left panels) and its standard deviation (right panels) on the (α+,α−) plane. They are obtained by the evaluation of our exact results in equations (27)–(30). Different rows stand for different values of the disorder p, namely p={0,0.05,0.5}from top to bottom. In each panel, white is used to indicate values larger than the maximum one in the corresponding colour legend, whereas the filled black star indicates the minimum—for p=0 it is reached at α+→ ∞ and α−=0. consequence of the properties of F1,2(α+,α−), for p<1, both the MFPT and the standard deviation diverge in the limits α+→0 and/or α−→0, except for the case p=0 where they remain finite for α−→0.3Hence, excluding p=0, both objects develop non-trivial dependence on (α+,α−). 3For p=0, the target is certainly to the left of x0=xr, and therefore it suffices to reset the trajectories straying to the right of x0to have a finite MFPT. 8 New J. Phys. 25 (2023) 113031 G García-Valladares et al 4. Optimisation of the physical quantities In this section, the MFPT and its standard deviation are minimised. Moreover, the dependence of both the optimal values and their corresponding optimal resetting rates are looked into as a function of the disorder p. 4.1. Optimal MFPT First, we focus on the MFPT. Mathematically, the problem reduces to finding the minimum of the function τ(1)=pF1(α+,α−) + (1−p)F1(α−,α+),(32) with F1(α+,α−)defined in equation (27). Then, it is needed to solve ∂τ(1) ∂α+α+=eα+ α−=eα− =∂τ(1) ∂α−α+=eα+ α−=eα− =0 (33) for (eα+,eα−). From now on, tilde notation refers to the solution of the minimisation problem of τ(1). In the following, we study both (eα+,eα−)and its associated MFPT eτ(1)as functions of p. It is convenient to start with a physical intuition of the optimal problem. For p=1 (p=0), there is absolute certainty about the target position. Then, we expect the optimal resetting to forbid the exploration of the region on the opposite side of the target with α−→∞(α+→∞), whereas the exploration of the correct direction is facilitated with α+=0(α−=0). This physical intuition is mathematically confirmed by taking into account that (i) F1is a monotonically increasing (decreasing) function with respect to its first (second) argument, and (ii) just one F1survives in equation (32) in the extreme cases for p. This is clearly observed in the panel corresponding to p=0 in figure 3, where the MFPT monotonically decreases (increases) with α+(α−). For any p∈[0,1], there is just one non-negative solution (eα+,eα−)of equation (33). This solution has been numerically found for any p, and also approximately through analytical expansions in the vicinity of p=1/2, p=1, and p=0. For the sake of clearness, the mathematical derivations as well as the specific expressions for the theoretical approximations are given in appendix E. The optimal values of the resetting rates (eα+,eα−)and the resulting minimum MFPT eτ(1), along with their theoretical approximations, are shown in figure 4as functions of p. The accuracy of the analytical approximations are quite good, even for values of pnot so close to the ones used for the expansions. It is just in the limit as p→0 that the approximation for eα−→0 seems to fail, due to the quite rapid variation of eα−in a narrow interval of p—this abrupt variation arises from the emergence of the disorder when changing from p=0 (no disorder) to p→0+(weak disorder). Approximate analytical expressions are obtained for optimal resetting in appendix E. The agreement between the approximation and the numerical results is reconciled when looking at the semi-logarithmic plot shown in the inset. The plots are restricted to the interval p∈[0,1/2] because of left-right symmetry: for p>1/2, the optimal rates are obtained by the exchange α−↔α+in those for 1 −p, and the optimal MFPT eτ(1)is thus symmetric with respect to p=1/2. 4.2. Optimal standard deviation of the first passage time Now we look into the minimisation of the standard deviation, i.e. look for the solution of the system of equations ∂στ ∂α+α+=bα+ α−=bα− =∂στ ∂α−α+=bα+ α−=bα− =0.(34) We use hat notation to refer to the optimisation of the standard deviation, in contrast to the tilde used for the minimisation of the MFPT. Since the analytical expression of στis quite cumbersome, analytical progress does not seem illuminating and we have addressed this minimisation problem numerically. In figure 5, the optimal values of the resetting rates (bα+,bα−)and the resulting minimum standard deviation bστare shown as functions of p. As expected from direct inspection of the case p=0 in figure 3, the asymptotic value for α±obtained in the no-disorder limit p=0 is the same than that found in the minimisation of the MFPT. However, the tendency towards the limit value are much slower here, as discussed in the following section. 4.3. Comparisons between the different optimisations Now we compare the two optimisations we have just analysed, that of the MFPT and that of the standard deviation. In the left panel of figure 6, we present together the optimal resetting rates (eα+,eα−)and (bα+,bα−) 9 New J. Phys. 25 (2023) 113031 G García-Valladares et al target location). The theoretical predictions stemming from this analysis have been plotted in figure 4of the main text. E.1. Maximum uncertainty about the target location When the dichotomous target distribution is almost symmetric, i.e. p=1/2+δpwith δp≪1, it is handy to rewrite the MFPT in equation (32) as τ(1)=Fs(α+,α−) + Fa(α+,α−)δp,(E.1) where the functions Fs(α+,α−) = F1(α+,α−) + F1(α−,α+) 2,(E.2) Fa(α+,α−) = F1(α+,α−)−F1(α−,α+),(E.3) are symmetric and antisymmetric under the exchange α+↔α−. From homogeneous resetting, it is known that Fsis optimal for α+=α−=eαh, where the symmetric optimal rate eαhis given by the equation eαh=21−e−eαh⇒eαh=2+W0−2e−2,(E.4) where W0is the principal branch of the Lambert Wfunction [25,34]. For δp≪1, we write (α+,α−)=(eαh,eαh) + (c+,c−)δpand the MFPT reads τ(1)≃Fs+cT·∇Fsδp+1 2cT·Hs·c(δp)2+Faδp+cT·∇Fa(δp)2(E.5) where cT≡(c+,c−),Hsis the Hessian matrix of Fs. All the coefficients are computed at α+=α−=eαh, thus ∇Fs=0 because of optimality and Fa=0 because of antisymmetry. Taking derivative with respect to c±and equating to zero, we get ec=−H−1 s·∇Fa,(E.6) for the optimal value of c. The minimum MFPT is then eτ(1)≃Fs−1 2(∇Fa)T·H−1 s·∇Fa(δp)2⩽Fs.(E.7) Making use of equation (E.4), we would be able to get the specific values for the coefficients, (∇Fa)T=4−eαh 2eα2 h(2−eαh)(1,−1),(E.8) Hs=1 4eα3 h(2−eαh)eαs(1+eαs)−4+eαs(3−eαs) −4+eαs(3−eαs)eαs(1+eαs),(E.9) ecT=−eαh(4−eαh) 2+eαh(eαh−1)(1,−1),(E.10) 1 2(∇Fa)T·H−1 s·∇Fa=(4−eαh)2 2eαh(2−eαh)(2+eαh(eαh−1)).(E.11) We have cut the expansion for (eα+,eα−)above at linear terms, which provides good estimates in the vicinity of p=1/2. Nonetheless, higher-order corrections can be incorporated to the expansion in a systematic manner, without any technical difficulty. E.2. Minimum uncertainty about the target location In the opposite regime, p=δp≪1 (recall that we have restricted ourselves to 0 ⩽p⩽1/2 because of the symmetry of the problem), it is almost sure that the target lies on the half-line to the left of the resetting point. As physically expected, for p=0, the optimal rates are α−=0 and α+→∞, i.e. a reflecting boundary at the origin and free diffusion for any x<0. Therefore, one expects to have α−≪1 and α+≫1 in the limit as p→0+. Our starting point is equation (32) fot the MFPT, which now reads τ(1)=F1(α−,α+) + δp[F1(α+,α−)−F1(α−,α+)].(E.12) 16 New J. Phys. 25 (2023) 113031 G García-Valladares et al Figure E1. Divergence of the variance at minimum MFPT in the limit as p→0+. The numerical solutions (symbols) are compared with the leading order of the asymptotic prediction p−1(solid line). In the limit α−≪1 and α+≫1, the leading terms in F1are F1(α+,α−) = eα+ 2α+α− +oeα+ α+α−,(E.13) F1(α−,α+) = 1 2+α2 − 24 +1 α+ +oα2 −+oα−1 +.(E.14) To leading order, this entails that ∂τ(1) ∂α+∼− 1 α2 + +α+−1 2α2 +α− eα+δp∼− 1 α2 + +eα+ 2α+α− δp,(E.15) ∂τ(1) ∂α−∼α− 12 −eα+ 2α+α2 − δp.(E.16) For obtaining the optimal values (eα+,eα−), we enforce ∂τ(1)/∂α±to be zero, which gives us the leading terms of the optimal resetting rates for δp≪1, eα+∼2W0 31/4 pδp!,eα−∼2√3 eα+ .(E.17) It is interesting to estimate the value of the pole s∗ +in this limit, for the optimal rates just derived above. It is the vanishing of s∗ +in the limit as p→0+that explains the divergence of the second moment—and thus of the standard deviation—therein. When looking for the poles of the moment generating function (23), a dominant balance argument leads to the asymptotic behaviour s∗ +∼−4√3e−α+; note that |s∗ +|<eα2 −as it should be, in fact |s∗ +|≪ eα2 −∝eα−2 +. 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