Diffusive escape through a narrow opening: new insights into a classic problem
DDiffusive escape through a narrow opening: new insights into a classic problem
Denis S Grebenkov and Gleb Oshanin Laboratoire de Physique de la Mati`ere Condens´ee, CNRS,Ecole Polytechnique, Universit´e Paris Saclay, F-91128 Palaiseau Cedex, France ∗ Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,UPMC, CNRS UMR 7600, Sorbonne Universit´es,4 Place Jussieu, 75252 Paris Cedex 05, France † We study the mean first exit time T ε of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of anangular size ε . Focusing on the effects of an energy/entropy barrier at the EW, and of the long-rangeinteractions (LRI) with the boundary on the diffusive search for the EW, we develop a self-consistentapproximation to derive for T ε a general expression, akin to the celebrated Collins-Kimball relationin chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysisreveals that the barrier-induced contribution to T ε is the dominant one in the limit ε →
0, implyingthat the narrow escape problem is not “diffusion-limited” but rather “barrier-limited”. We presentthe small- ε expansion for T ε , in which the coefficients in front of the leading terms are expressed viasome integrals and derivatives of the LRI potential. On example of a triangular-well potential, weshow that T ε is non-monotonic with respect to the extent of the attractive LRI, being minimal forthe ones having an intermediate extent, neither too concentrated on the boundary nor penetratingtoo deeply into the bulk. Our analytical predictions are in a good agreement with the numericalsimulations. I. INTRODUCTION
The narrow escape problem (NEP) is ubiquitous inmolecular and cellular biology and concerns diverse situ-ations when a particle, diffusing within a bounded micro-domain, has to search for a small specific target on thedomain’s boundary [1–8]. A particle can be an ion, achemically active molecule, a protein, a receptor, a lig-and, etc. A confining domain can be a cell, a microvesicle,a compartment, an endosome, a caviola, etc, while thetarget can be a binding or an active site, a catalytic germ,or a narrow exit to an outer space, from which case thename of the problem originates. The outer space can bean extracellular environment or, as considered in recentanalysis of diffusion and retention of Ca -calmoduline-dependent protein kinase II in dendritic spines [9, 10],be the dendrite itself while the narrow tunnel is a neckseparating the spine and the dendrite. For all these strayexamples, called in a unified way as the NEP, one is gen-erally interested to estimate the time needed for a par-ticle, starting from a prescribed or a random locationwithin the micro-domain, to arrive for the first time tothe location of the target. The history, results, and im-portant advances in understanding of the NEP have beenrecently reviewed [11–15].Early works on NEP were focused on situations whenthe confining boundary is a hard wall, i.e., is perfectlyreflecting everywhere, except for the escape window (orother specific target), which is also perfect in the sensethat there is neither an energy nor even an entropy bar- ∗ Electronic address: [email protected] † Electronic address: [email protected] rier which the particle has to overpass in order to exitthe domain (or to bind to the specific site). In this per-fect, idealised situation, the mean first exit time (MFET)through this window is actually the mean first passagetime (MFPT) to its location. This MFPT was calculatedby solving the diffusion equation with mixed Dirichlet-Neumann boundary conditions [16].Capitalising on the idea that the particle diffusionalong the bounding surface can speed up the searchprocess, set forward for chemoreception by Adam andDelbr¨uck [17] and for protein binding to specific sites ona DNA by Berg et al. [18], as well as on the idea of theso-called intermittent search [19–26], more recent analy-sis of the NEP dealt with intermittent, surface-mediateddiffusive search for an escape window, which also wasconsidered as perfect, i.e. having no barrier at its loca-tion. The MFPTs were determined, using a mean-field[27] and more elaborated approaches [28–37], as a func-tion of the adsorption/desorption rates, and of the sur-face ( D surf ) and the bulk ( D ) diffusion coefficients. It wasfound that, under certain conditions involving the ratiobetween D and D surf , the MFPT can be a non-monotonicfunction of the desorption rate, and can be minimised byan appropriate tuning of this parameter [28–37].In this paper we analyse the problem of a narrow es-cape of a Brownian particle through a small window lo-cated on the boundary of a three-dimensional sphericalor a two-dimensional circular domain (see Fig. 1) takinginto account explicitly a finite energy and/or an entropybarrier at the EW and the presence of long-range (beyondthe usual hard-core repulsion) particle-boundary inter-actions characterised by a radially-symmetric potential W ( r ). We will use the term “escape window” (EW) inwhat follows, but clearly our results will also apply tosituations when this targeted area is an extended bind- a r X i v : . [ c ond - m a t . o t h e r] S e p FIG. 1: Schematic picture of the narrow escape problem : Anescape window (EW) is a cap of polar angle ε and diameter a = 2 R sin ε located at the North pole of a sphere of radius R . A diffusive particle starts from a point ( r, θ ) (small filledcircle) and eventually arrives to the EW. ing region or an active site. Our motivations here are asfollows:i) The adsorption/desorption rates cannot be tuned in-dependently as they are linked on the microscopic levelby the particle-boundary interactions. As a consequence,not all values of the latter parameters are physically pos-sible and it is unclear a priori if the non-monotonic be-haviour of the MFPT observed in earlier works can in-deed take place in physical systems.ii) If diffusion along the surface may indeed speed upthe search for the EW in case of contact[64] attractiveinteractions with the surface, it is natural to expect that,in the presence of long-range attractive interactions withthe surface, these typical search times will be further re-duced since the particle will feel the surface on longerdistances and will experience a drift towards it. More-over, on intuitive grounds, one may expect the existenceof some optimal extent of the interaction potential sincefor potentials with an infinite extent, the problem will re-duce to the one with purely hard-wall interactions. Con-sequently, it may turn out that the mean search time willbe a non-monotonic function of the extent of the inter-action potential.iii) In realistic systems not every passage to the EWresults in the escape from the domain, but this event hap-pens only with some finite probability, due to an energy [64] The term “contact” means that a diffusing particle feels theboundary only appearing in its immediate vicinity. The particlemay then adsorb onto the boundary in a non-localised way, per-form surface diffusion and desorb back to the bulk, re-appear inthe vicinity of the boundary again, and etc. This is precisely thephysical picture underlying the idea of the so-called intermittent,surface-mediated search for the EW. or an entropy barrier, the latter being dominated by thewindow size [2, 42, 43]. Similarly, chemical reactions witha specific site on the boundary of the micro-domain arealso never perfect but happen with a finite probabilitydefining some rate constant κ [44]. This rate constanthas already been incorporated into the theoretical anal-ysis of the NEP for systems with hard-wall or contactinteractions with the surface [34], but to the best of ourknowledge, its effect on the MFET has never been appre-ciated in full detail[65]. We will revisit this analysis forthe situations with (and without) the long-range particle-boundary interaction.These are the focal questions of our analysis. We pro-ceed to show that the MFET naturally decomposes intotwo terms: the mean first passage time to the EW and thetime necessary to overcome the barrier at the entranceto the EW. We realise that the MFPT to the EW is in-deed an optimisable function of the interaction potential,when these interactions are attractive, the effects beingmore pronounced for spherical micro-domains than forcircular micro-domains. More specifically, we find thatwhile the MFPT appears to be a monotonic function ofthe strength of the interaction potential at the boundary,it exhibits a minimum with respect to the spatial extentof the potential, which defines the force acting on the par-ticle in the vicinity of the boundary. We show that theoptimum indeed occurs for interactions of an intermedi-ate extent, such that they are neither localised too closeto the confining boundary, nor extend too deeply intothe bulk of the micro-domain. In a way, the observa-tion that the MFPT exhibits a non-monotonic behaviouras the function of the extent of the potential is consis-tent with the earlier predictions for the optimum of theMFPT with respect to the desorption rate, based on themodels with intermittent motion. As a matter of fact, theextent of the potential defines the barrier against desorp-tion from the confining boundary and hence, controls thedesorption rate. There are, however, some discrepanciesbetween our predictions here and those based on inter-mittent modelling on which we will comment in whatfollows.Further on, focusing on the effect of a barrier at theentrance to the EW, which is always present in realisticsituations, we demonstrate that the contribution to theMFET stemming out from the passage through the bar-rier is more singular in the limit ε → [65] See also [15, 38–40] for the analysis of the NEP with astochastically-gated EW, which situation is equivalent, in thelimit when the stochastic gating process has no memory, to thepresence of a partial reflection [41]. Our analytical approach is based on the backwardFokker-Planck equation with a long-range potential thatgoverns the MFET to an imperfect (partially-reflecting, κ < ∞ ) EW for a particle starting from a given loca-tion within the micro-domain. We obtain an explicit ap-proximate solution of the resulting mixed boundary valueproblem by resorting to an approximation devised orig-inally by Shoup, Lipari and Szabo [45] for the analysisof reaction rates between particles with inhomogeneousreactivity [46]. Within this approximation, the exactboundary conditions are replaced by some effective ones,reducing the problem to finding self-consistent solutions.The original self-consistent approximation was shown tobe in a good agreement with numerical solutions of theoriginal problem [45, 47].We adapt this approximation to the NEP and also in-corporate the long-range potential interactions betweenthe particle and the boundary of the confining domain.Our theoretical predictions, based on this self-consistentapproximation, will be checked against available exactasymptotic results for the case when the boundary is animpenetrable hard wall (long-range interactions are ab-sent) and when there is no barrier at the entrance to theEW. For the general case, we will verify our analyticalpredictions by extensive Monte Carlo simulations and anaccurate numerical solution of the original mixed bound-ary problem by a finite elements method.The paper is organised as follows: In Sec. II we de-scribe our model and introduce the main idea of theself-consistent approximation (SCA). In Sec. III we ob-tain the solution of the NEP with the modified bound-ary conditions, as prescribed by the SCA. In Sec. IVwe first present the general expressions for the MFETfor three-dimensional spherical and two-dimensional cir-cular micro-domains, for arbitrary particle-wall interac-tions, an arbitrary κ and the EW of an arbitrary angularsize. Discussing its physical significance, we highlightthe crucial role of the partial reactivity and its effecton the MFET. Further on, we show that in the narrowescape limit ε → ε expansion for the MFET in which theexpansion coefficients of the leading terms are explicitlydefined via some integrals and derivatives of a rather ar-bitrary interaction potential. Lastly, on example of arepresentative triangular-well potential, we discuss therole of repulsive and attractive particle-boundary inter-actions and also demonstrate that the contribution to theMFET due to the diffusive search for the location of theEW (i.e., the MFPT to the EW), can be optimised byan appropriate tuning of the spatial extent of the inter-action potential. We also analyse here some subtle is-sues related to the applicability of the Adam-Delbr¨uckdimensionality reduction scheme [17]. Section V con-cludes the paper with a brief recapitulation of our mostsignificant results. Mathematical and technical detailsare presented in Supplemental Materials (SM), where wedescribe the numerical approaches used to verify our an-alytical predictions (SM1); derive the asymptotic small- ε expansions and determine the asymptotic behaviourfor short-range potentials (SM2); present solutions in ab-sence of particle-wall interactions (SM3); and discuss theparticular case of a triangular-well interaction potentialfor three-dimensional (SM4) and two-dimensional (SM5)micro-domains. II. MODEL AND BASIC EQUATIONS
Consider a point particle diffusing, with a diffusion co-efficient D , in a three-dimensional sphere (3D case) ofradius R or a two-dimensional disk (2D case) of radius R , containing on the confining boundary a small EWcharacterised by a polar angle ε and having a diameter a = 2 R sin ε , see Fig. 1. We stipulate that, in addition tothe hard-core repulsion at the boundary, the particle in-teracts with the confining wall via a radially-symmetricpotential[66] W ( r ). We hasten to remark that an as-sumption that the interaction potential W ( r ) dependsonly on the radial distance from the origin is only plausi-ble in the narrow escape limit, i.e., when the polar angleis small ( ε (cid:28) π ) or, equivalently, the linear extent a ofthe EW is much smaller than the radius R . Otherwise,the interaction potential can acquire a dependence on theangular coordinates. Therefore, from a physical pointview, this model taking into account the long-range in-teractions with the boundary in a spherically-symmetricform is representative only when ε (cid:28) π . On the otherhand, for situations with W ( r ) ≡ ε .We are mainly interested in the MFET through theEW, T ε , from a uniformly distributed random location,sometimes called the global MFET [48] and defined as T ε = 1 V d (cid:90) Ω dV d t ( r, θ ) , (1)where V d is the volume of the micro-domain Ω ( V =4 πR / V = πR for 2D case) and t ( r, θ ) is the mean time necessary for a particle, startedfrom some fixed location ( r, θ ) inside the domain, to exitthrough the EW. Due to the symmetry of the problem, t ( r, θ ) depends on the radial distance r to the origin(0 ≤ r ≤ R ) and polar angle θ (0 ≤ θ ≤ π ) but isindependent of the azimuthal angle φ for the 3D case.In the 2D case, the reflection symmetry of the circular [66] Note that W ( r ) defines the long-range interaction potential be-yond the hard-wall repulsion. Therefore, the (repulsive) partof realistic interaction potentials, say that of the Lennard-Jones12 − W ( R ) be will be aregular function and all its derivatives at r = R will exist. Werefer to [50] for a more detailed discussion of this issue. micro-domain with respect to the horizontal axis allowsone to restrict the polar angle to (0 , π ) as well. Althoughwe focus in this paper exclusively on the global MFET T ε , the SCA yields the explicit form of t ( r, θ ) too.The function t = t ( r, θ ) satisfies the backward Fokker-Planck equation. For the 3D case, the MFET t obeys t (cid:48)(cid:48) + (cid:18) r − U (cid:48) (cid:19) t (cid:48) + 1 r sin θ ∂∂θ (cid:18) sin θ ∂t∂θ (cid:19) = − D , (2)where the prime here and henceforth denotes the deriva-tive with respect to the variable r , and U ( r ) is a dimen-sionless, reduced potential: U ( r ) = βW ( r ), where β isthe reciprocal temperature measured in the units of theBoltzmann constant. For the 2D case, the MFET t isgoverned by t (cid:48)(cid:48) + (cid:18) r − U (cid:48) ( r ) (cid:19) t (cid:48) + 1 r ∂ t∂θ = − D . (3)In both cases, the backward Fokker-Planck equation isto be solved subject to the reflecting boundary condi-tion which holds everywhere on the wall ( r = R ), ex-cept for the location of the EW, on which a partially-adsorbing boundary condition is imposed. For the back-ward Fokker-Planck equation, these mixed boundary con-ditions have the form[67]: − D t (cid:48) | r = R = (cid:40) κ t ( R, θ ) , ≤ θ ≤ ε, , ε < θ ≤ π, (4)where κ is the above mentioned proportionality factor (inunits length/time), which accounts for an energy or anentropy barrier at the entrance to the EW. If such a bar-rier is absent, κ = ∞ and one has a perfectly adsorbingboundary condition at the location of the EW, in whichcase the first line of (4) becomes t ( R, θ ) = 0 for 0 ≤ θ ≤ ε .Evidently, in this case T ε is entirely determined by thefirst passage to the EW.Within the context of chemical reactions with an activesite located on the inner surface of the micro-domain, thecondition in the first line of (4) can be considered as thepartially-reflecting reactive boundary condition put for-ward in the seminal paper by Collins and Kimball [44],and then extensively studied in the context of chemicalreactions [51–55] and search processes [34, 56, 57]. In thisframework, κ can be written down as κ = K/ (4 πR s g ),where K is the usual elementary reaction act constant(in units of volume times the number of acts per unit of [67] Note that the boundary condition (4) for the forward Fokker-Planck equation will have a different form [58]: − D ( t (cid:48) + t U (cid:48) ) r = R = (cid:40) κ t ( R, θ ) , ≤ θ ≤ ε, , ε < θ ≤ π, i.e., it will include the derivative of the potential. time within this volume) and s g = sin ( ε ) is the geomet-ric steric factor characterising the fraction of the bound-ary area “covered” by the active site, so that 4 πR s g issimply the area of the active site. In turn, K can bewritten[49] as K = f V s (see also the discussion in [50]),where f is the rate describing the number of reaction actsper unit of time within the volume V s of the reaction zonearound an active site. If the reaction takes place withina segment of a spherical shell, defined by R − ρ ≤ r ≤ R and θ ∈ (0 , ε ), where ρ is the capture radius, one has V s = 4 πR ρs g , so that κ = f ρ . Clearly, a similar argu-ment holds for the case of the EW with an energy barrier;in this case f can be interpreted as the rate of successfulpassages through the EW and is dependent on the energybarrier via the standard Arrhenius equation.Last but not least, even when the confining boundaryis a structureless hard wall so that an energy barrier is ab-sent, a particle, penetrating from a bigger volume (micro-domain) to a narrower region of space, will encounter anentropy barrier ∆ S at the entrance to the EW; in thiscase β ∆ S ∼ ln( a/R ) [42, 43], a being the lateral size ofthe escape window. This suggests, in turn, that κ , associ-ated with an entropy barrier, depends linearly on a andhence, linearly on ε . Therefore, for realistic situationsone may expect that κ associated with an entropy bar-rier is rather small, and gets progressively smaller when ε →
0, becoming a rate-controlling factor. However, theabsolute majority of earlier works on the NEP has beendevoted so far to the limit κ = ∞ , which corresponds toan idealised situation when there is neither energy, noreven entropy barrier at the entrance to the EW so thatthe particle escapes from the micro-domain upon the firstarrival to the location of the EW. In this case the MFETis just the MFPT to the EW. As we will show, even if κ is independent of ε , the passage through the barrierprovides the dominant contribution to the MFET in thenarrow escape limit ε → U ( r ) hinges on the SCA developed by Shoup,Lipari and Szabo [45], who studied rates of an associationof particles (e.g., ligands), diffusing in the bath outsideof an impenetrable sphere without LRI potential (i.e., U ( r ) ≡ effective one – a condition of a constant flux through theboundary at the location of the EW. More specifically,the mixed boundary condition in (4) is replaced by aninhomogeneous Neumann boundary condition: Dt (cid:48) | r = R = Q Θ( ε − θ ) , (5)where Θ( θ ) is the Heaviside function and Q is the un-known flux which is to be determined self-consistentlyusing an appropriate closure relation.It is important to emphasise two following points: (i)the solution of the modified problem (2, 5) is defined upto a constant while the solution of the original problem(2, 4) is unique. However, in the narrow escape limit ε →
0, the leading terms of the MFET diverge so thata missing constant would give a marginal contribution.(ii) the replacement of the boundary condition (4) by aneffective one (5) does not guaranty, in principle, that thesolution t will be positive in the vicinity of the EW, sincethe effective boundary condition requires that (4) holdsonly on average (see also SM3). We will show that thisapproximation provides nonetheless accurate results forthe global MFET with zero and non-zero U ( r ). III. SELF-CONSISTENT APPROXIMATION
In this section we adapt the SCA for the NEP and alsoincorporate within this approach the long-range potentialinteractions with the confining boundary [68]. Withinthis extended approach, we derive general expressions for T ε for arbitrary potentials in both 2D and 3D cases, asfunctions of the radius R of the micro-domain, the an-gular size ε of the EW, the constant κ , and the bulkdiffusion coefficient D . A. 3D Case
The general solution of (2) can be written as the fol-lowing expansion t ( r, θ ) = t ( r ) + ∞ (cid:88) n =0 a n g n ( r ) P n (cos( θ )) , (6)where P n (cos( θ )) are the Legendre polynomials, a n arethe expansion coefficients which will be chosen afterwardsto fulfil the boundary conditions, g n ( r ) are the radialfunctions obeying g (cid:48)(cid:48) n ( r ) + (cid:18) r − U (cid:48) ( r ) (cid:19) g (cid:48) n ( r ) − n ( n + 1) r g n ( r ) = 0 , (7)and lastly, t ( r ) is the solution of the inhomogeneousproblem that can be written down explicitly as t ( r ) = 1 D c (cid:90) r dx e U ( x ) x x (cid:90) c dy y e − U ( y ) , (8)where c and c are two adjustable constants. We set c = 0 to ensure the regularity of solution at r = 0. [68] See also recent [50] in which an analogous approach combinedwith a hydrodynamic analysis has been developed to calculatethe self-propulsion velocity of catalytically-decorated colloids. In order to fix the constant c , we impose the Dirichletboundary condition at r = R to get t ( r ) = 1 D R (cid:90) r dx e U ( x ) x x (cid:90) dy y e − U ( y ) . (9)We emphasise that the Dirichlet boundary condition for t ( r ) is chosen here for convenience only. As mentionedearlier, the solution of the modified problem (2, 5) isdefined up to a constant which can be related to theconstant c here. An evident advantage of such a choiceis that t ( r ) in (9) is the exact solution of the original problem in case when ε = π (i.e., the EW is the entireboundary of the sphere).We turn next to the radial functions g n ( r ) defined in(7). For n = 0 the radial function can be defined explic-itly for an arbitrary potential U ( r ) to give g ( r ) = c + c r (cid:90) dx e U ( x ) x . (10)To ensure that this solution is regular at 0, we again set c = 0, so that g ( r ) ≡ c = 1 for convenience).For n >
0, explicit solutions of (7) can be calculated onlywhen one makes a specific choice of the interaction poten-tial. In the next section and in the SM, we will discussthe forms of g n ( r ) for a triangular-well potential U ( r ).In general, we note that g n ( r ) are also defined up to twoconstants. One constant is fixed to ensure the regularityof the solution at r = 0. The second constant can befixed by the choice of their value at r = R . Without anylack of generality, we set g n ( R ) = 1. As a matter of fact,the final results will include only the ratio of g n ( r ) andof its first derivative, and hence, will not depend on theparticular choice of the normalisation.Next, substituting (6) into (5) we get ∞ (cid:88) n =1 a n P n (cos( θ )) g (cid:48) n ( R ) = QD Θ( ε − θ ) − t (cid:48) ( R ) , (11)where we have used g ( r ) ≡
1. Multiplying the latterequation by sin( θ ) and integrating over θ from 0 to π , wefind the following expression for the flux Q : Q = 2 Dt (cid:48) ( R )1 − cos( ε ) , (12)where, in virtue of (9), D t (cid:48) ( R ) = − e U ( R ) R R (cid:90) dr r e − U ( r ) . (13)Note that for U ( r ) ≡
0, the expression in (12) coincideswith the standard compatibility condition for the interiorNeumann problem. Note also that Q depends only on theform of the interaction potential, R and the angular size ε of the EW but it is independent of the kinetic parameters κ and D .Further on, multiplying both sides of (11) by P m (cos( θ )) sin( θ ) and integrating the resulting equationover θ from 0 to π , we get, taking advantage of the or-thogonality of the Legendre polynomials, the followingrepresentation of the expansion coefficients a n : a n = t (cid:48) ( R ) φ n ( ε ) g (cid:48) n ( R ) , n > , (14)where we used (12) and defined φ n ( ε ) = P n − (cos( ε )) − P n +1 (cos( ε ))1 − cos( ε ) , n > . (15)Gathering the above expressions, we rewrite (6) as t ( r, θ ) = t ( r ) + a + t (cid:48) ( R ) ∞ (cid:88) n =1 g n ( r ) g (cid:48) n ( R ) φ n ( ε ) P n (cos( θ )) , (16)in which only a remains undefined. Since the solutionof the modified problem is defined up to a constant, onecould stop here, leaving a as a free constant. To becloser to the original problem, we fix a through the self-consistency condition by plugging (16) into (4, 5), mul-tiplying the result by sin θ and integrating over θ from 0to ε , to get the following closure relation: − κ ε (cid:90) dθ sin( θ ) t ( R, θ ) = D ε (cid:90) dθ sin( θ ) t (cid:48) | r = R = Q ε (cid:90) dθ sin( θ ) . (17)Using the latter relation, noticing that t ( R ) = 0 by con-struction (see (9)) and excluding Q via (12), we arriveat a = − R t (cid:48) ( R ) (cid:20) DRκ (1 − cos( ε )) + R (3) ε (cid:21) , (18)where R (3) ε = ∞ (cid:88) n =1 g n ( R ) Rg (cid:48) n ( R ) φ n ( ε )(2 n + 1) . (19)Equations (9, 15, 18, 19) provide an exact, closed-formsolution (16) of the modified problem in (2, 5) for the3D case. This solution is valid for an arbitrary initiallocation of the particle, an arbitrary reaction rate κ andan arbitrary form of the interaction potential U ( r ). Wealso note that R (3) ε in (19) is a non-trivial function whichencodes all the relevant information about the fine struc-ture of the interaction potential. B. 2D case
We follow essentially the same line of thought like inthe previous subsection. In two dimensions the equation(7) for the radial functions becomes g (cid:48)(cid:48) n + (cid:18) r − U (cid:48) (cid:19) g (cid:48) n − n r g n = 0 . (20)The general solution for t ( r, θ ) then reads t ( r, θ ) = t ( r ) + a + 2 t (cid:48) ( R ) ∞ (cid:88) n =1 g n ( r ) g (cid:48) n ( R ) sin( nε ) nε cos( nθ ) , (21)where cos( nθ ) replace the Legendre polynomials from the3D case, and the solution of the inhomogeneous problemhas the form: t ( r ) = 1 D R (cid:90) r dx e U ( x ) x x (cid:90) dy y e − U ( y ) . (22)The coefficient a in (21) is given explicitly by a = − Rt (cid:48) ( R ) (cid:18) πDRκ ε + R (2) ε (cid:19) , (23)with R (2) ε = 2 ∞ (cid:88) n =1 g n ( R ) Rg (cid:48) n ( R ) (cid:18) sin( nε ) nε (cid:19) . (24)Equations (21) to (24) determine an exact, closed-formsolution t ( r, θ ) in the modified problem for the 2D case.Like in the 3D case, R (2) ε in (24) contains all the relevantinformation about the long-range interaction potential. IV. RESULTS AND DISCUSSION
Capitalising on the results of the previous section,we find the following general expressions for the globalMFET defined in (1): T (3) ε = R D R (3) ε L (3) U ( R ) + R D (cid:90) dx x L (3) U ( xR ) (cid:124) (cid:123)(cid:122) (cid:125) diffusion to the EW + 2 R L (3) U ( R )3 κ (1 − cos( ε )) (cid:124) (cid:123)(cid:122) (cid:125) barrier at the EW , (25)where L (3) U ( r ) is the functional of the potential U ( r ),given explicitly by L (3) U ( r ) = 3 e U ( r ) r (cid:90) r dρ ρ e − U ( ρ ) , (26)and T (2) ε = R D R (2) ε L (2) U ( R ) + R D (cid:90) dx x L (2) U ( xR ) (cid:124) (cid:123)(cid:122) (cid:125) diffusion to the EW + πR L (2) U ( R )2 κ sin( ε ) (cid:124) (cid:123)(cid:122) (cid:125) barrier at the EW , (27)with L (2) U ( r ) = 2 e U ( r ) r (cid:90) r dρ ρ e − U ( ρ ) . (28)For U ( r ) ≡
0, both L (2) U ( r ) and L (3) U ( r ) are simply equalto 1.We note next that R (3) ε and R (2) ε vanish when ε = π ,so that the second terms in the first line in (25) and (27),i.e. T (3) π ( κ = ∞ ) = R D (cid:90) dx x L (3) U ( xR ) (29)and T (2) π ( κ = ∞ ) = R D (cid:90) dx x L (2) U ( xR ) , (30)can be identified as the MFPTs from a random locationto any point on the boundary of the micro-domain, inpresence of the radially-symmetric interaction potential U ( r ). In what follows we will show that these (rathersimple) ε -independent contributions to the overall MFETexhibit quite a non-trivial behaviour as functions of theparameters of the interaction potential U ( r ).Equations (25) and (27) are the main general result ofour analysis. We emphasise that these expressions havethe same physical meaning as the celebrated relation forthe apparent rate constant due to Collins and Kimball[44] and define the global MFET as the sum of two con-tributions: the first one is the time necessary for a diffus-ing particle (starting from a random location within themicro-domain) to find the EW (i.e., the MFPT to theEW), while the second one describes the time necessaryto overcome a finite barrier at the entrance to the EW,once the particle appears in its vicinity. Clearly, the lastcontribution vanishes when κ → ∞ , i.e., in the perfectEW (reaction) case, while the first one vanishes for aninfinitely fast diffusive search, i.e., when D → ∞ . Theadditivity of the two controlling factors will permit usto study separately the effects due to a finite κ , and theeffects associated with the diffusive search for the EW,biased by the long-range potential U ( r ). We proceed toshow that, interestingly enough, the last term in (25)and (27) is always dominant in the limit ε →
0, i.e., therate-controlling factor for the NEP is the barrier at theentrance, not the diffusive search process. To the best ofour knowledge, this important conclusion has not been ever spelled out explicitly, but may definitely have impor-tant conceptual consequences for biological and chemicalapplications.One notices next that the first terms in (25) and (27)contain infinite series R (3) ε and R (2) ε implying that theseterms can be explicitly determined only when one (i)specifies the interaction potential, (ii) manages to solveexactly the differential equations (7) or (20) for the radialfunctions g n ( r ) corresponding to the chosen U ( r ) and,(iii) is able to sum the infinite series. At the first glance,this seems to be a severe limitation of the approach be-cause equations (7) or (20) can be solved exactly onlyfor a few basic potentials. Quite remarkably, however,we managed to bypass all these difficulties and to de-termine the asymptotic behaviour of the MFET in thenarrow escape limit ε → ε behaviour of the infinite series R (3) ε and R (2) ε is dominatedby the terms g n ( R ) /g (cid:48) n ( R ) with n → ∞ , whose asymp-totic behaviour can be directly derived from (7, 20) forpotentials U ( r ) which have a bounded first derivative byconstructing an appropriate perturbation theory expan-sion. For such potentials, we obtain (see SM2 for moredetails): R (3) ε = 323 π ε − + (cid:0) − R U (cid:48) ( R ) (cid:1) ln(1 /ε )+ ln 2 − − (cid:18)
14 + π
12 + ln 2 (cid:19)
R U (cid:48) ( R ) (31)+ ∞ (cid:88) n =1 (2 n + 1) (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n + R U (cid:48) ( R )2 n (cid:19) + O ( ε ) , where U (cid:48) ( R ) denotes the derivative of the interaction po-tential right at the boundary, and the symbol O ( ε ) signi-fies that the omitted terms, in the leading order, vanishlinearly with ε when ε →
0. Note that one needs theprecise knowledge of the radial functions g n ( r ) only forthe calculation of the subdominant, ε -independent termin the third line in (31).Analogous calculations for the 2D case (see the SM2)entail the following small- ε expansion: R (2) ε = 2 ln(1 /ε ) + 3 − ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19) + O ( ε ) . (32)Again, the precise knowledge of the radial functions g n ( r )is only needed for the calculation of the ε -independentterm, which embodies all the dependence on the interac-tion potential; the leading term in this small- ε expansionappears to be completely independent of U ( r ).Combining Eqs. (25) and (31) we find that for an arbi-trary potential U ( r ) possessing a bounded first derivativewithin the domain, the MFET for the 3D case admits thefollowing small- ε asymptotic form T (3) ε = 4 R L (3) U ( R )3 κ ε − + 32 R L (3) U ( R )9 πD ε − + R L (3) U ( R )3 D (cid:0) − RU (cid:48) ( R ) (cid:1) ln(1 /ε ) + ξ (3) U + O ( ε ) , (33)where the ε -independent, sub-leading term ξ U is givenexplicitly by ξ (3) U = T (3) π ( κ = ∞ ) + R L (3) U ( R )9 κ + R L (3) U ( R )3 D (cid:34) ln 2 − − (cid:18) ln 2 + 14 + π (cid:19) RU (cid:48) ( R )+ ∞ (cid:88) n =1 (2 n + 1) (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:18) − RU (cid:48) ( R )2 n (cid:19)(cid:19) (cid:35) , (34)with the MFPT T (3) π ( κ = ∞ ) to any point on the bound-ary being defined in (29).For the 2D case an analogous small- ε expansion reads T (2) ε = πR L (2) U ( R )2 κ ε − + R L (2) U ( R ) D ln(1 /ε ) ++ ξ (2) U + O ( ε ) , (35)where the ε -independent term ξ (2) U is given explicitly by ξ (2) U = T (2) π ( κ = ∞ )+ L (2) U ( R ) R D (cid:32) − ln 2 + ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19)(cid:33) , (36)with T (2) π ( κ = ∞ ) being defined in (30).The expressions in (33) and (35) constitute our secondgeneral result in the narrow escape limit. This result hasseveral interesting features which have to be emphasised.i) For quite a general class of the interaction potentials U ( r ), these expressions make explicit our claim that inthe narrow escape limit ε → κ ∼ ε ) more inverse power of ε , as comparedto the terms defining the MFPT to the EW. Clearly, inthe narrow escape limit ε →
0, a passage through theEW is the dominant rate-controlling factor.ii) Remarkably, it appears that in order to determinethe coefficients in front of the leading terms (diverging inthe limit ε → g n ( r ) but merelyto integrate and to differentiate the interaction potential.The resulting expressions for T ( d ) ε have quite a transpar-ent structure and all the terms entering (33) and (35)have a clear physical meaning. For both 3D and 2D cases, the coefficients in front ofthe leading term associated with the barrier are entirelydefined by L (3) U ( R ) and L (2) U ( R ). For the 3D case, the co-efficient in front of the leading term in the MFPT to theEW, which diverges as 1 /ε , is also entirely defined by theintegrated particle-wall potential U ( r ) via L (3) U ( R ), whilethe sub-leading diverging term ( ∼ ln(1 /ε )) depends alsoon the force, − U (cid:48) ( R ), acting on the particle at the bound-ary of the micro-domain. In the 2D case, the coefficientin front of the leading term in the MFPT to the EW,diverging as ln(1 /ε ), again is defined by the integratedpotential via L (2) U ( R ).Therefore, details of the “fine structure” of the inter-action potential U ( r ) (i.e., possible maxima or minimafor r away from the boundary) which are embodied inthe radial functions, have a minor effect on T (3) ε and T (2) ε appearing only in the subdominant terms, which are in-dependent of ε in the narrow escape limit.iii) The 1 /ε singularity of the MFPT to the EW is aspecific feature of the 3D case and stems from such dif-fusive paths, starting at a random location and endingat the EW, which spend most of the time in the bulk farfrom the boundary. Our analysis shows how the presenceof the long-range interaction potential modifies the am-plitude of the corresponding contribution to the MFET.For the 3D case, the sub-leading, logarithmically diverg-ing term accounts for the contribution of the paths whichare most of the time localised near the confining bound-ary. In what follows, we will discuss the relative weightsof these contributions considering the triangular-well in-teraction potential as a particular example.iv) When the potential U ( r ) is a monotonic functionof r , the coefficients L (2) U ( r ) and L (3) U ( r ) are monotonicfunctions of the amplitude U of the potential U ( r ). Infact, setting U ( r ) = U U ( r ), one gets U ∂ L ( d ) U ( r ) ∂U = dr d e U ( r ) r (cid:90) dρ ρ d − e − U ( ρ ) (cid:2) U ( r ) − U ( ρ ) (cid:3) (37)so that the derivative in the left hand side does notchange the sign. For instance, if U ( r ) is an increasingfunction, then L ( d ) U ( r ) is also increasing for any U (evennegative). As a consequence, the coefficients in front ofeach term in (33) and (35) are monotonic functions of theamplitude U of the potential so that the global MFETis expected to be a monotonic function of U , at leastin the narrow escape limit. In Sec. IV B, on example ofa triangular-well potential, we will quest the possibilityof having a minimum of the MFET with respect to theextent of the interaction potential U ( r ). A. Global MFET for systems without long-rangepotentials
To set up the scene for our further analysis, we con-sider our expressions in (25) and (27) in case when theconfining boundary is a hard wall (i.e. U ( r ) ≡ ε , κ , D and R . This will permit us to check how accurateour approach is, by comparing our predictions againstfew available exact results, and also to highlight in whatfollows the role of the long-range interactions with theconfining boundary.In absence of the interaction potential, equations (25)and (27) considerably simplify: T (3) ε = 2 R κ (1 − cos( ε )) + R D (cid:18) R (3) ε + 15 (cid:19) , (38) T (2) ε = π R κ sin( ε ) + R D (cid:18) R (2) ε + 14 (cid:19) , (39)where R (3) ε and R (2) ε become R (3) ε = ∞ (cid:88) n =1 φ n ( ε ) n (2 n + 1) , (40) R (2) ε = 2 ∞ (cid:88) n =1 n (cid:18) sin nεnε (cid:19) . (41)The asymptotic small- ε behaviour of these series is dis-cussed in detail in SM2.We focus first on the asymptotic behaviour of T ε in thelimit ε →
0. Using the asymptotic small- ε expansion,presented in (S24, S25), we find that in the 3D case T (3) ε = 4 R κ ε − + 32 R πD ε − + R D ln (1 /ε )+ R D (cid:18) ln 2 − (cid:19) + R κ + O ( ε ) . (42)The first term in (42) is the contribution due to a finitebarrier, while the second and the third terms define thecontribution due to the diffusive search for the EW, stem-ming out of the non-trivial term in (40) proportional to R (3) ε .For κ ≡ ∞ (i.e., in an idealised situation when thereis neither an energy nor even an entropy barrier at theentrance to the EW), the first term in (42), proportionalto 1 /κ and exhibiting the strongest singularity in thelimit ε →
0, is forced to vanish, so that the leading ε -dependence of T ε becomes determined by the secondterm, diverging as 1 /ε . The first correction to this dom-inant behavior is then provided by the logarithmicallydiverging term.Such an asymptotic behaviour qualitatively agreeswith the exact asymptotics due to Singer et al. [59]: T (3) ε (cid:39) πR D (cid:18) ε − + ln(1 /ε ) + O (1) (cid:19) , (43) which contains both 1 /ε - and logarithmically divergingterms. We notice however a small discrepancy in thenumerical prefactors: the coefficient 32 / (9 π ) ≈ . π/ ≈ . ε →
0, appears to be π timesless, as compared to the coefficient in the logarithmicallydivergent term in (43). Therefore, for κ ≡ ∞ and ε → T ε on the pertinent parameters but slightlyoverestimates its amplitude, and also underestimates theamplitude of the sub-dominant term, associated with thecontribution of the diffusive paths localised near the con-fining boundary.For the 2D case, the summation in (41) can be per-formed exactly so that (39) can be written explicitly as T (2) ε = πR κ sin( ε ) + R D ξ ( ε ) ε + R D , (44)where ξ ( ε ) = ∞ (cid:88) n =1 sin nεn = 12 ζ (3) − (cid:0) Li (cid:0) e iε (cid:1) + Li (cid:0) e − iε (cid:1)(cid:1) , (45) ζ being the Riemann zeta function, ζ (3) ≈ . ( y ) being the trilogarithm: Li ( y ) = (cid:80) ∞ n =1 y n /n . Inthe limit ε →
0, (44) admits the following asymptoticexpansion T (2) ε = πR κ ε − + R D ln (1 /ε ) + R D (cid:18) − ln 2 (cid:19) + O ( ε ) . (46)Again, we notice that the first term, associated with thebarrier at the entrance to the EW, has a more pronouncedsingularity as ε → /κ and diverges as 1 /ε in the limit ε →
0, is forced to vanishin the idealised case κ ≡ ∞ , and the leading small- ε be-haviour of T (2) ε in (46) becomes determined by the secondterm, which exhibits only a slow, logarithmic divergencewith ε . At this point it is expedient to recall that theoriginal mixed boundary-value problem in (3) and (4) forthe 2D case with κ ≡ ∞ can be solved exactly [60–62]and here T (2) ε reads: T (2) ε = R D (cid:18) − ln(sin( ε/ (cid:19) = R D ln (1 /ε ) + R D (cid:18) ln 2 + 18 (cid:19) + O (cid:0) ε (cid:1) . (47)Comparing the solution of the modified problem in (46)with κ ≡ ∞ , and the exact small- ε expansion in the0 −1 ε D T ε / R (a) SCAasymptFEMMC ε D T ε / R (b) SCAexactFEMMC
FIG. 2: Hard-wall interactions ( U ( r ) ≡
0) and no barrierat the EW ( κ = ∞ ). The dimensionless global MFET, DT ( d ) ε /R , as a function of the target angle ε . (a)
3D Case.Our SCA prediction in (38) and the asymptotic relation in(43) are shown by solid and dashed lines, respectively. Thenumerical solution by the FEM (with the mesh size 0 .
01) isshown by circles, while the Monte Carlo simulations are shownby crosses. (b)
2D Case. Our SCA prediction in (44) (solidline) is compared against the exact result in (47) (dashedline), numerical FEM solution (circles) and MC simulations(crosses). second line in (47), we conclude that the leading termsin both equations coincide. This means that for the 2Dcase the SCA developed in the present work determinesexactly not only the dependence of the leading term onthe pertinent parameters, but also the correct numericalfactor. The subdominant, ε -independent term in (46),appears to be slightly larger, by 3 / − ≈ . ε -dependence of T ε in (38) and (44)for κ ≡ ∞ , as well as the κ -dependence of T ε for fixed ε , and will compare our predictions against the results ofnumerical simulations (see SM1).We first consider the well-studied case when there is κ R/D D T ε / R (a) SCA ( ε = 0.1)FEM ( ε = 0.1)SCA ( ε = 0.2)FEM ( ε = 0.2)SCA ( ε = π /4)FEM ( ε = π /4) κ R/D D T ε / R (b) SCA ( ε = 0.1)FEM ( ε = 0.1)SCA ( ε = 0.2)FEM ( ε = 0.2)SCA ( ε = π /4)FEM ( ε = π /4) FIG. 3: Hard-wall interactions ( U ( r ) ≡
0) with a barrierat the EW. The dimensionless global MFET, DT ( d ) ε /R , asa function of a dimensionless parameter κR/D for ε = 0 . ε = 0 . ε = π/ (a)
3D case and (b)
2D case. The SCA (lines) is compared tonumerical solutions obtained by a FEM (with the mesh size0 . no barrier at the EW ( κ = ∞ ), so that the first term in(38) vanishes. Figure 2a compares the SCA prediction in(38), the asymptotic relation (43) by Singer et al. [59],and the numerical solution of the original problem by a fi-nite elements method (FEM) and by Monte Carlo (MC)simulations, which are described in SM1. We observea fairly good agreement over the whole range of targetangles, for ε varying from 0 to π , which is far beyondthe narrow escape limit, and the dimensionless param-eter DT ε /R varying over more than two decades. Wefind that the SCA provides an accurate solution even forrather large ε , at which the asymptotic relation (43) com-pletely fails. A nearly perfect agreement is also observedfor the 2D case (see Fig. 2b).Next, we study the κ -dependence of the global MFET.As we have already mentioned, the SCA predicts thatthe presence of a finite barrier at the entrance to the EW(and hence, for a finite κ ) is fully captured by the firstterm in (38) and (39). Figure 3 illustrates how accuratelythe SCA accounts for the effect of a partial reactivity κ on the global MFET T ε , even for not too small EW (e.g.,for ε = π/
4) and a broad range of reactivities κ . For1even larger EW, the SCA is still accurate for small κ butsmall deviations are observed at larger κ (not shown). Asexpected, the global MFET diverges as κ → B. Global MFET for a system with atriangular-well potential.
We focus on a particular choice of the interaction po-tential U ( r ) – a triangular-well radial potential – definedas (see also Fig. S2 in SM4) U ( r ) = , ≤ r ≤ r ,U r − r R − r , r < r ≤ R, (48)where 0 ≤ r ≤ R , R − r ≡ l ext is the spatial extentof the potential (a characteristic scale) and U is a di-mensionless strength of the potential at the boundary, U ( R ) ≡ U . Note that − U /l ext can be interpreted asa constant force exerted on the particle when it appearswithin a spherical shell of extent l ext near the boundaryof the micro-domain. The strength of the potential canbe negative (in case of attractive interactions) or positive(in case of repulsive interactions). For this potential, wefind explicit closed-form expressions for the radial func-tions g n ( r ) (see SM4 and SM5) and examine the depen-dence of the global MFET on U , r and other pertinentparameters, such as, ε , D , κ and R .We begin with the analysis of the functionals L (3) U ( R )and L (2) U ( R ), which define the amplitudes of all ε -diverging leading terms in (33) and (35). For thetriangular-well potential in (48), these functionals aregiven explicitly by L (3) U ( R ) = 3 U (cid:26) − − U − U + (6 + 4 U + U ) x − U ) x + 2 x + e U (cid:20) U − x +(6 − U + U ) x − (cid:18) − U + U − U (cid:19) x (cid:21)(cid:27) , (49)where x = r /R , and L (2) U ( R ) = 2 U (cid:26) − − U + ( U + 2) x − x + e U (cid:20) U − x + (cid:18) − U + U (cid:19) x (cid:21)(cid:27) , (50)for the 3D and the 2D cases, respectively. We plot ex-pressions in (49) and (50) in Fig. 4 as functions of U for different values of r . We observe that for any −2 −1 0 1 2012345 U L U ( ) (a) r /R = 0r /R = 0.25r /R = 0.5r /R = 0.75 −2 −1 0 1 2012345 U L U ( ) (b) r /R = 0r /R = 0.25r /R = 0.5r /R = 0.75 FIG. 4: The amplitudes L (3) U ( R ) (a) and L (2) U ( R ) (b) in (49)and (50) vs U for r = 0 (solid line), r /R = 0 .
25 (dashedline), r /R = 0 . r /R = 0 .
75 (dottedline). fixed r , L (3) U ( R ) and L (2) U ( R ) are monotonically increas-ing functions of U , which agrees with the general ar-gument presented earlier in the text. In turn, for anyfixed U > r which means that theyare decreasing functions of the extent of the potential, l ext = R − r . Therefore, all the contributions to theglobal MFET T ε become larger for stronger repulsion (asthey should) and also get increased upon lowering theextent the potential (i.e., upon an increase of the force − U /l ext pointing towards the bulk and hindering thepassage to and through the EW). For sufficiently largepositive U and for any r >
0, the dominant contribu-tion to L ( d ) U ( R ) is L ( d ) U ( R ) ∼ ( r /R ) d exp( U ) , d = 2 , , (51)so that the coefficients in the small- ε expansions of T ε in(33) and (35) become exponentially large with U .In turn, for negative values of U (attractive interac-tions), L (3) U ( R ) and L (2) U ( R ) decrease upon an increase of | U | and also decrease when r approaches R , i.e., whenthe interactions become short ranged.2 C. Adam-Delbr¨uck scenario: Limit ω = R U / ( R − r ) → −∞ Before we proceed to the general case with arbitrary U and r , we discuss first the situations when the di-mensionless parameter ω = R U (cid:48) ( R ) = R U R − r , (52)has large negative values. This can be realised for eitherbig negative U , which case is more of a conceptual inter-est but is apparently not very realistic, or for short-rangepotentials with fixed U < r close to R , the lattercase being physically quite meaningful.For such values of ω the leading behaviour of the am-plitudes L (3) U ( R ) and L (2) U ( R ) is simply described by L ( d ) U ( R ) ∼ d | ω | , d = 2 , , (53)i.e., the amplitudes vanish as ω → −∞ as a first inversepower of ω . This means, in turn, that all the contribu-tions to T ( d ) ε which are multiplied by L ( d ) U ( R ) (i.e., boththe contribution due to a barrier at the EW and theMFPT to the EW) decrease in presence of attractive in-teractions with the boundary of the micro-domain.We analyse next the behaviour of another key ingre-dient of (33, 35) – the infinite series R (3) ε and R (2) ε . Werealise that there is a subtle point in the behaviour of thelatter which stems from the fact that the limit of largenegative ω and the narrow escape limit ε → ε → ω → −∞ , we arrive at the expressionwhich describes correctly the behaviour of R (3) ε and R (2) ε for small ε and moderately large | ω | such that | ω | (cid:28) /ε .If, on contrary, we first take the limit ω → −∞ for a fixed ε , and then take the limit ε →
0, we obtain the correctlarge- | ω | behaviour, which yields physically meaningfulresults for infinitely large negative ω . This point is dis-cussed in detail in SM2.Accordingly, for the 3D case in the narrow escape limit ε → ω , such that 1 (cid:28)| ω | (cid:28) /ε , we have T (3) ε (cid:39) R D (cid:18) ln(1 /ε ) + ln | ω | + ( γ − / (cid:19) + T (3) π ( κ = ∞ )+ 1 | ω | (cid:18) Rκ ε − + 32 R πD ε − + O (1) (cid:19) , (54)where γ ≈ .
577 is the Euler-Mascheroni constant andthe MFPT T (3) π ( κ = ∞ ) from a random location toany point on the boundary is given explicitly for the triangular-well potential by T (3) π ( κ = ∞ ) = r DR − R Dω (cid:18) x + 3+ 8 ω + 12 ω − ω (cid:19) + R e U D ω (cid:18) x + x (3 − x ) ω + 3 x (2 − x ) ω + 6(1 − x ) ω − ω (cid:19) . (55)Note that in the limit ω → −∞ , the MFPT T (3) π ( κ = ∞ )becomes T (3) π ( κ = ∞ ) = r DR + O (cid:18) | ω | (cid:19) , (56)where the leading in this limit term can be interpreted asthe product of the probability ( r /R ) that the particle’sstarting point is within the spherical region of radius r (in which diffusion is not influenced by the potential),and the MFPT, equal to r / (15 D ), from a random loca-tion within this spherical region to its boundary. Clearly,when a particle reaches the extent of the infinitely strongattractive potential (or if it was started in that region),it is drifted immediately to the boundary, and this stepdoes not increase the MFPT.On the other hand, for small ε and negative ω whichcan be arbitrarily (even infinitely) large by an absolutevalue, we find (see SM2) T (3) ε (cid:39) R D (cid:18) /ε ) + ln 2 − / (cid:19) + T (3) π ( κ = ∞ )+ 1 | ω | (cid:18) Rκ ε − + O (1) (cid:19) , (57)which differs from the expression in (54) in two aspects:the amplitude of the dominant term, which diverges log-arithmically as ε →
0, is twice larger, and the term ln | ω | ,which diverges logarithmically as ω → −∞ , is absent.Therefore, we observe that attractive interactions witheither large negative U , or fixed negative U and r ∼ R (both giving a large negative ω according to (52)), effec-tively suppress the contribution due to a finite barrier atthe entrance to the EW, and also the contribution dueto the diffusive paths which find the EW via 3D diffu-sion. On contrary, such interactions do not affect theterm which is logarithmically diverging as ε → r / (15 DR ), see the first term in (55)), andthen diffuses along the boundary, not being able to sur-mount the barrier against desorption and to escape backto the bulk, until it ultimately finds the EW (the term2 R ln(1 /ε ) /D ).3It is worthwhile to mention that our SCA approachreproduces the leading in the limit ε → ε -dependence but also the numerical factor inthe amplitude. In fact, the mean time T s needed for aparticle, diffusing on a surface of a 3D sphere and startingat a random point[69], to arrive for the first time to a discof an angular size ε located on the surface of this spherehas been calculated exactly[51]: T s = R D (cid:18) ln (cid:18) − cos ε (cid:19) − ε (cid:19) (58) (cid:39) R D (cid:18) /ε ) + 2 ln 2 − O ( ε ) (cid:19) . (59)Comparing (57) and (59), we notice that the leadingterms in both expressions are identical, and that bothexpressions differ slightly only by numerical constants inthe ε -independent terms.In the 2D case with large negative ω , the dominantcontribution to the MFET comes not from the logarith-mically diverging term in the limit ε →
0, as one couldexpect, but rather from the infinite sum in the secondline in (36), which is independent of ε . We discuss thisobservation in SM2 and show that this sum diverges inproportion to | ω | when ω → −∞ . Using the asymptoticrelation (S60), which presents this contribution to R (2) ε in an explicit form, we find T (2) ε = T (2) π ( κ = ∞ ) + π R D + O (1)+ 1 | ω | (cid:18) πRκ ε − + 2 R D ln(1 /ε ) (cid:19) , (60)where T (2) π ( κ = ∞ ) is the MFPT from a random locationwithin the disc to any point on its boundary. For thetriangular-well potential the latter is given explicitly by T (2) π ( κ = ∞ ) = r DR − R Dω (cid:18) x + 2 + 3 ω − ω (cid:19) + R e U D ω (cid:18) x + x (2 − x ) ω + 2(1 − x ) ω − ω (cid:19) . (61)For large negative ω , the leading term in (61) is r / (8 DR ). This term can be simply interpreted as the [69] For a starting point fixed by an angular coordinate θ , the MFPTfor surface diffusion is known to be t s ( θ ) = R D ln (cid:18) − cos θ − cos ε (cid:19) for ε ≤ θ ≤ π , and 0 otherwise. Averaging the latter expressionover the uniformly distributed starting point, i.e., performing theintegral T s = 14 π π (cid:90) dϕ π (cid:90) dθ sin θ t s ( θ ) , one obtains the expression in (58). mean time, equal to r / (8 D ), needed to appear for thefirst time within the reach of the triangular-well poten-tial for a particle whose starting point is uniformly dis-tributed within the subregion r < r (in which diffusionis free), multiplied by the probability r /R that thisstarting point is within this subregion.Hence, in the limit ω → −∞ , we obtain for the 2Dcase T (2) ε ≈ (cid:16) r R (cid:17) r D + π R D . (62)Again, we observe that the contribution due to a bar-rier at the EW, and the leading, logarithmically diver-gent contribution to the MFPT to the EW due to a two-dimensional diffusive search within the disc, are both sup-pressed by attractive interactions. Indeed, in this case aplausible Adam-Delbr¨uck-type argument [17] states thata diffusive particle first finds the boundary of the disc (thefirst term in (62)), and then diffuses along the boundaryuntil it ultimately finds the EW. For arbitrary ε the latterMFPT is well-known and is given by T s = R D ( π − ε ) π = π R D + O ( ε ) , (63)where the leading term on the right-hand-side is exactlythe same as the second term in (62). This implies thatour SCA approach predicts correctly the values of bothMFPTs.To summarise the results of this subsection, we notethat the behaviour observed in the limit ω → −∞ fol-lows precisely the Adam-Delbr¨uck dimensionality reduc-tion scenario for both 3D and 2D cases. Our expressions(57) and (60) provide the corresponding MFETs, andalso define the correction terms due to finite values of ω .We emphasise, however, that the limit ω → −∞ is an ex-treme case. For modest negative ω the representative tra-jectories for both three-dimensional and two-dimensionssystems will consist of paths of alternating phases of bulkdiffusion and diffusion along the boundary. We analysethis situation below. D. Beyond the Adam-Delbr¨uck scenario: Arbitrary U and r . Consider the behaviour of the global MFET definedin (33) or (35) for arbitrary U and r . Taking into ac-count the explicit expressions for L (3) U ( R ) in (49) and for L (2) U ( R ) in (50), we notice that the coefficients in front ofeach term in (33) and (35) are monotonic functions of U [70]. This implies that in the narrow escape limit ε → [70] In principle, the coefficient in front of the logarithm, which con-tains the factor (1 − R U / ( R − r )), can become negative in caseof repulsive interactions but it appears that it has a little effecton the whole expression and does not cause a non-monotonicbehaviour. r /R D T ε ( ) / R (a) U = −1U = −2U = −5 r /R D T ε ( ) / R (b) FIG. 5: The dimensionless global MFET, DT ( d ) ε /R , as afunction of x = r /R for a triangular-well potential withno barrier at the EW, κ = ∞ . Our analytical predictionsare shown by curves for three values of U and ε = 0 . . (a)
3D case, T (3) ε is given by (33); (b)
2D case, T (2) ε is given by (35). we do not expect any non-monotonic behaviour of theglobal MFET with respect to U . A thorough analyticaland numerical analysis of T ( d ) ε shows that it is indeed thecase.As we expected on intuitive grounds, for attractiveparticle-boundary interactions the global MFET T ( d ) ε ap-pears to be a non-monotonic, and hence, an optimisablefunction of the extent of the potential. This is a newspectacular feature of the NEP unveiled by our modelinvolving spatially-extended interactions with the con-fining boundary. In Figs. 5a and 5b we plot T (3) ε and T (2) ε , defined by (33) and (35) with κ = ∞ , as a functionof r for several negative values of U ( U = − U = − U = −
5) and ε = 0 .
02. We remark that the firstterms in (33) and (35), associated with a finite barrierand dropped by setting κ = ∞ , are monotonic functionsof r .We observe that in the 3D case, the global MFET T (3) ε ,as a function of r , exhibits a rather pronounced mini-mum for r away from r = 0 and from r = R (con-tact interactions). This means that there exists an op- timal extent of the potential: in order to minimise theglobal MFET, the potential should neither extend toodeep into the bulk nor should be too localised on theboundary. Further on, we realise that the precise lo-cation of the minimum of T (3) ε depends, in general, on ε and U . Analysing our result in (33), we observe thatwhen we gradually decrease ε , the minimum moves closerto the boundary (but never reaches it and T (3) ε still ex-hibits a very abrupt growth when r becomes too closeto R ) and becomes more pronounced. Conversely, if weprogressively increase ε , the minimum moves away fromthe boundary and becomes more shallow. Increasing thestrength of attractive interactions (while keeping ε fixed)also pushes the minimum closer to the boundary.For the 2D case the global MFET T (2) ε shows a qual-itatively similar behaviour but the effects are much lesspronounced. In principle, the optimum also exists in thiscase, but the minimum appears to be much more shallow.We also observe that for strong attraction ( U = − r = R , T (2) ε ap-pears to be almost independent of r .As in the analysis of the NEP without the particle-boundary interactions in Sec. IV A, one can observe somesmall discrepancies between the asymptotic relations (33,35) and numerical solutions of the original problem by aFEM. As discussed earlier, they are related to the SCAapproximation, in which the mixed boundary conditionis replaced by an effective inhomogeneous Neumann con-dition. On the other hand, the proposed approximationprovides explicit asymptotic formulae for arbitrary po-tentials that captures qualitatively well all the studiedfeatures of the NEP.Two remarks are now in order:a) Emergence of an optimum at an intermediate ex-tent of the potential implies that the Adam-Delbr¨uckdimensionality reduction scenario (which corresponds to ω → −∞ and hence, to the part of the curves in Figs. 5aand 5b close to r = R where T (3) ε exhibits quite a steepgrowth attaining large values) is not at all the optimal(i.e., less time consuming) way of finding the EW. In real-ity, the optimum corresponds to situations when the bar-rier against desorption is not very large so that the par-ticle does not remain localised near the boundary uponapproaching it for the first time, but rather has a possi-bility to overpass the barrier against desorption and toperform alternating phases of bulk and surface diffusion.The optimum then corresponds to some fine-tuning of therelative weights of bulk and surface diffusion by changingthe extent of the potential and its value on the boundary.b) These findings are compatible, in principle, with theprediction of the non-monotonic behaviour of T ( d ) ε as afunction of the desorption rate λ made earlier in [29],in which the NEP with κ = ∞ has been analysed foran intermittent diffusion model. In this model a particlediffuses in the bulk with a diffusion coefficient D until ithits the boundary of the micro-domain and switches tosurface diffusion of a random duration (controlled by the5 r /R D T π ( ) / R (a) U =2U =1U =−1U =−2 r /R D T π ( ) / R (b) U =2U =1U =−1U =−2 FIG. 6: The dimensionless MFPT, DT ( d ) π ( κ = ∞ ) /R , for atriangular-well potential in (48) versus r /R for several valuesof U and κ = ∞ . (a)
3D case, (b)
2D case. desorption rate λ ) with a diffusion coefficient D surf . Thismodel tacitly presumes that there are some attractive in-teractions with the surface, in addition to the hard-corerepulsion, which are taken into account in some effectiveway (interactions are replaced by effective contact ones).In our settings, the desorption rate λ in this intermittentmodel should depend on both the strength of the inter-action potential U at the boundary and also on the gra-dient of the potential in the vicinity of the surface, whichdefine the barrier against desorption. There is, however,some quantitative discrepancy between our predictionsand the predictions made in [29]: In [29] (see Fig. 10,right panel), it was argued that for such an intermittentmodel with D = D surf (as in our case) surface diffu-sion is a preferable search mechanism so that the globalMFET is a monotonic function of λ . On contrary, ouranalysis demonstrates that T (3) ε is an optimisable func-tion even in case of equal bulk and surface diffusion co-efficients, which means that neither the bulk diffusionnor 2D surface diffusion alone provide an optimal searchmechanism, but rather their combination. This discrep-ancy is related to subtle differences between two models.For the two-dimensional case, illustrated in Fig. 5b, theanalysis in [29] (see Fig. 10, left panel) suggests that thereis an optimum even for D = D surf . Our analysis agreeswith this conclusion. We close with a brief analysis of the behaviour of T ( d ) π ( κ = ∞ ) – the MFPT for a diffusive particle, start-ing at a random location, to arrive at any point on theboundary of the micro-domain, in presence of long-rangeinteractions with the boundary. This MFPT is includedinto T ( d ) ε but has a little effect on it since it enters onlythe constant, ε -independent terms. At the same time,it is an interesting quantity in its own right. For thetriangular-well potential, T ( d ) π ( κ = ∞ ) is defined explic-itly by (55) and (61) for d = 3 and d = 2, respectively.A first intuitive guess is that increasing either U orthe extent of the potential would make the particle feelthe surface stronger, so that for U >
0, the MFPT T ( d ) π ( κ = ∞ ) would be an increasing function of bothparameters, while for U < T ( d ) π ( κ = ∞ ) would de-crease with an increase of both U and r . As far asthe dependence on U is concerned, this guess appearsto be correct. Indeed, we observe that for a fixed r , theMFPTs for both d = 2 and d = 3 are monotonic increas-ing functions of U . Surprisingly enough, this is not thecase for the dependence of the MFPTs T ( d ) π ( κ = ∞ ) onthe extent l ext of the potential. We find that for a fixed U , T ( d ) π ( κ = ∞ ) exhibits a peculiar non-monotonic be-haviour as a function of r , with a minimum for U < U >
0, as illustrated in Fig. 6. Tothe best of our knowledge, this interesting effect has notbeen reported earlier.
V. CONCLUSION
To recapitulate, we have presented here some new in-sights into the narrow escape problem, which concernsvarious situations when a particle, diffusing within abounded micro-domain, has to escape from it through asmall window (or to bind to some target site) of an angu-lar size ε located on the impenetrable boundary. We havefocused on two aspects of this important problem whichhad not received much attention in the past: the effectsof an energy or an entropy barrier at the escape window,always present in realistic systems, and the effects of long-range potential interactions between a diffusing particleand the boundary. Inspired by the self-consistent ap-proximation developed previously in [45] for calculationof the reaction rates between molecules with inhomoge-neous chemical reactivity, we generalised this approach tothe NEP with long-range potential interactions with theboundary. In this self-consistent approach, the originalmixed boundary condition is replaced by an effective in-homogeneous Neumann condition, in which the unknownflux is determined from an appropriate closure relation.This modified problem was solved exactly for an arbi-trary radial interaction potential.We have concentrated on the functional form of theglobal (or volume-averaged over the starting point) meanfirst exit time, T ε , for which we derived a general expres-sion analogous to the celebrated Collins-Kimball relation6in chemical kinetics, incorporating both the contributiondue to a finite barrier at the escape window (or a bind-ing site) and the contribution due to a diffusive searchfor its location, for an arbitrary radially-symmetric po-tential, any size of the escape window, and a barrier of anarbitrary height. We have realised that these two contri-butions naturally decouple from each other, which per-mitted us to study separately their impact on the MFET.The accuracy of our analytical results based on theself-consistent approximation has been confirmed by twoindependent numerical schemes: a numerical solution ofthe backward Fokker-Planck equation by a finite elementmethod, and Monte Carlo simulations of the diffusivesearch for the escape window in presence of particle-boundary potential interactions. We have shown that theself-consistent approximation is very accurate for smallescape windows (i.e., in the true narrow escape limit)but also captures quite well the behaviour of the globalMFET even for rather large escape windows. In the lat-ter case, small deviations were observed, related to thefact that the solution of the modified problem is definedup to a constant.Turning to the narrow escape limit ε →
0, we haveanalysed the relative weights of each contribution to T ε .We have shown that the contribution due to the pas-sage through the escape window (which had been ig-nored in the majority of earlier works) dominates theglobal MFET in the narrow escape limit, since it ex-hibits a stronger singularity as ε → T ε due to the diffusive search for the locationof the escape window, in which the coefficients in front of the terms diverging in the limit ε → exactly in the narrow escape limit without resort-ing to any approximation.On example of a triangular-well interaction potential,we have discussed the dependence of the contributionto the MFET due to a diffusive search for the escapewindow on the parameters of the potential. We haveshown that this contribution is always a monotonicfunction of the value of the potential at the boundary:as expected, repulsive (resp., attractive) interactionsincrease (resp., decrease) the MFET. Curiously enough,it appeared that for attractive interactions T ε is a non-monotonic function of the extent of the potential: thereexists some optimal extent for which T ε has a minimum.This optimal value corresponds to interactions whichare neither localised near the confining boundary norextend to deeply into the bulk. In case of a very smallextent (i.e., in the limit of short-range interactions),with a fixed value of the interactions potential onthe boundary, the force acting on the particle in theimmediate vicinity of the boundary becomes very largeand the narrow escape process proceeds precisely viathe Adam-Delbr¨uck dimensionality reduction scenario:a particle first reaches the boundary at any point andthan, not being able to surmount the barrier againstdesorption, continues a diffusive search for the escapewindow along the boundary until it finds it. For morerealistic moderate values of the extent, typical pathsconsist of alternating, intermittent bulk diffusion toursfollowed by diffusion along the boundary. Acknowledgments
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Supplemental Materials
SM1. NUMERICAL SIMULATIONS
In this part of the Supplemental Materials we briefly discuss two numerical procedures used to check our theoreticalpredictions.
A. Numerical computation by a finite elements method
To verify the accuracy of the SCA, we solve the Poisson equation (2, 3) by using a finite elements method (FEM)implemented in Matlab PDEtool. This tool solves the following equation: −∇ ( c ∇ u ) + au = f, (S1)where c is a 2x2 matrix, and a and f are given functions.In our case, we need to deal with the Laplace operator in radial or spherical coordinates. In two dimensions, theoriginal equation (3) can be written in radial coordinates as1 r ∂ r ( re − U ( r ) ∂ r ) u + e − U ( r ) r ∂ θ u = − e − U ( r ) D , (S2)from which − (cid:18) ∂ r ∂ θ (cid:19) † c (cid:18) ∂ r ∂ θ (cid:19) u = f, (S3)with a = 0, f = re − U ( r ) /D , and c = (cid:18) re − U ( r ) e − U ( r ) /r (cid:19) . (S4)In three dimensions, the Poisson equation (2) reads in spherical coordinates as1 r ∂ r ( r e − U ( r ) ∂ r ) u + e − U ( r ) r θ ∂ θ (sin θ∂ θ ) u + e − U ( r ) r sin θ ∂ ϕ u = − e − U ( r ) D . (S5)Since our solution does not depend on ϕ , the last term on the left hand side can be omitted so that − (cid:18) ∂ r ∂ θ (cid:19) † c (cid:18) ∂ r ∂ θ (cid:19) u = f, (S6)with a = 0, f = r e − U ( r ) sin θ/D , and c = (cid:18) r e − U ( r ) sin θ e − U ( r ) sin θ (cid:19) . (S7)We set the rectangular domain V = [0 , × [0 , π ] with mixed boundary conditions (4), i.e., a zero flux condition for ∂V \ Γ , except for the segment Γ = { } × [0 , ε ] representing the EW (see Fig. S1): (cid:18) ∂ r u + kD u (cid:19) r =1 = 0 (on Γ ) , ( ∂ r u ) r =1 = 0 (on Γ ) , ( ∂ r u ) r =0 = 0 (on Γ ) , ( ∂ θ u ) θ =0 = 0 (on Γ ) , ( ∂ θ u ) θ = π = 0 (on Γ ) . (S8) ........................................................ ............................ ............................ ........................... .......................... ......................... .......................... ............................ ............................ ............................ ............................ ........................... ........................... ............................ ............................ ........................... ........................... .......................... .......................... ........................... r Γ Γ Γ Γ @@I . . . . . . . . . ε -6 rθπ ε Γ Γ Γ Γ Γ FIG. S1: The original domain (half-disc) and the associated computational domain (rectangle). Similar in 3D case.
In Matlab, the generalised Neumann boundary condition has the form (cid:126)n · ( c ∇ u ) + qu = g, (S9)where the matrix c is the same as in the PDE (S1). We set g = 0 and q = Re − U ( R ) κ/D (on Γ ) ,q = 0 (on Γ ∪ Γ ∪ Γ ∪ Γ ) (S10)in two dimensions, and q = R sin θ e − U ( R ) κ/D (on Γ ) ,q = 0 (on Γ ∪ Γ ∪ Γ ∪ Γ ) (S11)in three dimensions. For fully reactive EW ( κ = ∞ ), the Dirichlet boundary condition is imposed on Γ . B. Monte Carlo simulations
We also compute the distribution of first passage times to the EW by simulating diffusion trajectories which endup at the EW. In practice, we solve a Langevin equation by the following iterative procedure: after generating auniformly distributed starting point r , one re-iterates r n +1 = r n + Dδ f ( r n ) + √ Dδ ξ n , (S12)where δ is a one-step duration, r n is the position after n steps, f = − ∂ r U ( r ) e r is the normalised applied force in theradial direction e r , and ξ is the normalised random thermal force. For instance, we have in two dimensions: x n +1 = x n + Dδ f ( | r n | ) x n / | r n | + √ Dδ ξ x,n ,y n +1 = y n + Dδ f ( | r n | ) y n / | r n | + √ Dδ ξ y,n , (S13)where x n / | r n | and y n / | r n | represent cos( θ ) and sin( θ ) in the projection of the radial force, and ξ x,n , ξ y,n are indepen-dent normal variables with zero mean and unit variance.At each step, one checks whether the new position ( x n +1 , y n +1 ) remains inside the disk: x n +1 + y n +1 < R . Ifthis condition is not satisfied, the particle is considered as being on the boundary. If the particle hits the EW, thetrajectory simulation is stopped and nδ is recorded as the generated exit time. Otherwise, the particle is reflectedback and continues to diffuse. The Monte Carlo simulations in three dimensions are similar. Finally, the partialreactivity of the EW can be introduced by partial reflections [53, 63]. SM2. ASYMPTOTIC BEHAVIOUR OF THE SERIES IN (19) AND (24)
We focus on the asymptotic behaviour of the infinite series R (3) ε and R (2) ε , defined in (19) and (24), for potentials U ( r ) which have a bounded first derivative for any r ∈ (0 , R ). Our aims here are two-fold: first we establishthe exact asymptotic expansions for these infinite series in the narrow escape limit ε →
0, and second, we deriveapproximate explicit expressions for R (3) ε and R (2) ε which permit us to investigate their asymptotic behaviour in thelimit R | U (cid:48) ( R ) | → ∞ .Our analysis is based on two complementary approaches. In the first approach we take advantage of the followingobservation. When ε = 0, there is no EW, and the MFET is infinite, whatever the potential is. Since L ( d ) U doesnot depend on ε , the divergence of the MFET as ε → R ( d ) ε . Suppose thatwe truncate the infinite series in (19) and (24) at some arbitrary n = N ∗ . Then, turning to the limit ε →
0, wefind that both R (3) ε and R (2) ε attain some constant values, which depend on the upper limit N ∗ of summation. Asa consequence, these truncated sums should diverge as N ∗ → ∞ while their small- ε behaviour is dominated by theterms with n → ∞ . One needs therefore to determine the asymptotic behaviour of g n ( R ) /g (cid:48) n ( R ) in this limit and toevaluate the corresponding small- ε asymptotics for R (3) ε and R (2) ε . This will be done in the subsection SM2 A.Next, in subsection SM2 B we will pursue a different approach based on the assumption that, once we are interestedin the behaviour of the ratio g n /g (cid:48) n at r = R only, we may approximate the coefficients in the differential equations(7) and (20), which are functions of r , by taking their values at the confining boundary. This will permit us to derivean explicit expression for g n ( R ) /g (cid:48) n ( R ) valid for arbitrary n , not necessarily large, and arbitrary | U (cid:48) ( R ) | < ∞ . Thisexpression will be checked subsequently against an exact solution obtained for a triangular-well potential (see SM4and SM5). We set out to show that an approximate expression for g n ( R ) /g (cid:48) n ( R ) and an exact result for such a choiceof the potential agree very well already for quite modest values of n and the agreement becomes progressively betterwith an increase of | U (cid:48) ( R ) | . On this basis, we also determine the small- ε , as well large- RU (cid:48) ( R ) asymptotic behaviourof R (3) ε and R (2) ε , which agrees remarkably well with the expressions obtained within the first approach, and the exactsolution derived for the particular case of a triangular-well potential. A. Large- n asymptotics of g n ( R ) /g (cid:48) n ( R ) and the corresponding small- ε behaviour of the infinite series R (3) ε and R (2) ε . We introduce an auxiliary function ψ = ψ n ( r ) = g n ( r ) /g (cid:48) n ( r ), which is the inverse of the logarithmic derivative of g n ( r ) and obeys, in virtue of (7) and (20), the following equations: r (1 − ψ (cid:48) ) + r (2 − rU (cid:48) ( r )) ψ − n ( n + 1) ψ = 0 (S14)for the 3D case, and r (1 − ψ (cid:48) ) + r (1 − rU (cid:48) ( r )) ψ − n ψ = 0 (S15)for the 2D case, respectively. We will seek the solutions of these non-linear Riccati-type differential equations in formof the asymptotic expansion in the inverse powers of n in the limit n → ∞ .Supposing that U (cid:48) ( r ) does not diverge at any point within the domain, we find that the leading term of ψ in thelimit n → ∞ is given by ψ ∼ r/n for both 2D and 3D cases, which is completely independent of the potential U ( r ).Pursuing this approach further, we make no other assumption to get the second term in this large- n expansion, whilefor the evaluation of the third term we stipulate that | U (cid:48)(cid:48) ( r ) | < ∞ . We have then for the 3D case ψ = rn − r U (cid:48) ( r )2 n + r ( U (cid:48) ( r ) (4 + r U (cid:48) ( r )) + 2 r U (cid:48)(cid:48) ( r ))8 n + O (cid:18) n (cid:19) , (S16)and hence, g n ( R ) Rg (cid:48) n ( R ) = 1 n − RU (cid:48) ( R )2 n + RU (cid:48) ( R ) (4 + R U (cid:48) ( R )) + 2 R U (cid:48)(cid:48) ( R )8 n + O (cid:18) n (cid:19) , (S17)where the omitted terms decay in the leading order as 1 /n . Similarly, for the 2D case we find ψ = rn − r U (cid:48) ( r )2 n + r ( U (cid:48) ( r ) (2 + r U (cid:48) ( r )) + 2 r U (cid:48)(cid:48) ( r ))8 n + O (cid:18) n (cid:19) , (S18)and hence, g n ( R ) Rg (cid:48) n ( R ) = 1 n − RU (cid:48) ( R )2 n + RU (cid:48) ( R ) (2 + R U (cid:48) ( R )) + 2 R U (cid:48)(cid:48) ( R )8 n + O (cid:18) n (cid:19) . (S19)We observe that the asymptotic expansions (S17) and (S19) for the 3D and the 2D cases become different from eachother only starting from the third term; first two terms are exactly the same. As it will be made clear below, wedo not have to proceed further with this expansion and, as an actual fact, just two first terms will suffice us todetermine the leading asymptotic behaviour of R (3) ε and just the first one will be enough to determine the analogousbehaviour of R (2) ε . In what follows, in SM4 and SM5 we will also check these expansions against the exact resultsobtained for the triangular-well potential, in which case the radial functions g n and hence, the ratio g n ( R ) /g (cid:48) n ( R )can be calculated exactly. We proceed to show that the asymptotic forms in (S17) and (S19) coincide with the exactasymptotic expansions at least for the first three terms.Focusing first on the 3D case, we formally write g n ( R ) Rg (cid:48) n ( R ) ≡ n − R U (cid:48) ( R )2 n + (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n + R U (cid:48) ( R )2 n (cid:19) , (S20)where the terms in brackets, in virtue of (S16) and (S17), decay as 1 /n when n → ∞ . Inserting (S20) into (19), wehave R (3) ε = Σ − R U (cid:48) ( R ) Σ + ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n + RU (cid:48) ( R )2 n (cid:19) φ n ( ε )(2 n + 1) , (S21)where Σ = ∞ (cid:88) n =1 φ n ( ε ) n (2 n + 1) , (S22)Σ = 12 ∞ (cid:88) n =1 φ n ( ε ) n (2 n + 1) , (S23)and φ n ( ε ) is defined by (15).Next, we find that in the limit ε → = 323 π ε − + ln (1 /ε ) −
74 + ln 2 + O ( ε ) , (S24)Σ = ln (1 /ε ) + 14 + ln 2 + π
12 + O ( ε ) , (S25) ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n + RU (cid:48) ( R )2 n (cid:19) φ n ( ε )2 n + 1 = O (1) . (S26)Combining (S21) and (S24, S25, S26) renders our central result in (31) for the 3D case. The derivation of theasymptotic forms in (S24, S25) is straightforward but rather lengthy and we relegate it to the end of this subsection.Here we only briefly comment on the term in (S26). We havelim ε → φ n ( ε ) = (2 n + 1) , (S27)so that the sum in (S26) converges as ε → ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n + RU (cid:48) ( R )2 n (cid:19) (2 n + 1) . (S28)In view of the discussion above and (S17), the terms in brackets decay as 1 /n , which implies that this series converges.In turn, it means that the expression in (S26) contributes in the limit ε → ε -independent termin the small- ε expansion of R (3) ε .For the 2D case we use only the first term in the expansion in (S16) to formally represent the ratio g n ( R ) /g (cid:48) n ( R ) as g n ( R ) Rg (cid:48) n ( R ) ≡ n + (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19) , (S29)where the terms in the brackets vanish as 1 /n . Then, inserting the latter identity into (24), we have the followingexpression for R (2) ε : R (2) ε = 2 ∞ (cid:88) n =1 n (cid:18) sin( nε ) nε (cid:19) + 2 ∞ (cid:88) n =1 (cid:18) sin( nε ) nε (cid:19) (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19) . (S30)The first sum evidently diverges as ε →
0, sincelim ε → (cid:18) sin( nε ) nε (cid:19) = 1 and ∞ (cid:88) n =1 n = ∞ . (S31)As a matter of fact, this sum can be calculated in an explicit form (see (45) in the main text) for an arbitrary ε . Inthe small- ε limit it is given by 2 ∞ (cid:88) n =1 n (cid:18) sin( nε ) nε (cid:19) = 2 ln(1 /ε ) + 3 − O (cid:0) ε (cid:1) . (S32)On the other hand, for the sum in the last line in (S30) we havelim ε → ∞ (cid:88) n =1 (cid:18) sin( nε ) nε (cid:19) (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19) = ∞ (cid:88) n =1 (cid:18) g n ( R ) Rg (cid:48) n ( R ) − n (cid:19) . (S33)Since the terms in brackets decay as 1 /n according to (S19), we infer that the sum in (S33) converges, so that thesecond term in (S30) contributes only to the constant, ε -independent term in the small- ε expansion of R (2) ε . Collecting(S30) to (S33) we arrive at the asymptotic expansion in (32).Lastly, we outline the derivation of the asymptotic forms in (S24, S25). For this purpose, we represent the differenceof two Legendre polynomials of orders n − n + 1 as P n − ( x ) − P n +1 ( x ) = (2 n + 1) n ( n + 1) (1 − x ) ddx P n ( x ) . (S34)Using next the standard integral representation of the Legendre polynomials, P n ( x ) = 1 π π (cid:90) dz ν n ( z ) , ν ε ( z ) = x + i (cid:112) − x cos( z ) , (S35)we have P n − ( x ) − P n +1 ( x ) = (2 n + 1)( n + 1) √ − x π π (cid:90) dz µ ε ( z ) ν n − ε ( z ) , µ ε ( z ) = (cid:112) − x − ix cos( z ) . (S36)Plugging the latter representation into (S22), performing the summation over n , and setting x = cos ε , we cast Σ into the form of the following double integral:Σ = π (cid:90) dz π (cid:90) dz Φ (1) ε ( z , z ) , (S37)withΦ (1) ε ( z , z ) = 1 π (cid:18) x − x (cid:19) µ ε ( z ) µ ε ( z ) ν ε ( z ) ν ε ( z ) (cid:18)(cid:0) − ν ε ( z ) ν ε ( z ) (cid:1) ln (cid:0) − ν ε ( z ) ν ε ( z ) (cid:1) + Li (cid:0) ν ε ( z ) ν ε ( z ) (cid:1)(cid:19) , (S38)where Li ( y ) is the dilogarithm: Li ( y ) = ∞ (cid:80) n =1 y n /n .We focus next on the small- ε behaviour of Φ (1) ε ( z , z ). After straightforward but lengthy calculations, we find thatin this limit Φ (1) ε ( z , z ) admits the following expansionΦ (1) ε ( z , z ) = B (1)1 ε − + B (1)2 ε − ln( ε ) + B (1)3 ε − + B (1)4 ln( ε ) + B (1)5 + O ( ε ) , (S39)where B (1) j are functions of both z and z : B (1)1 = −
23 cos z cos z , B (1)2 = 8 iπ cos z cos z (cos z + cos z ) ,B (1)3 = 2 i π (cid:32) z cos z (cid:20) π − − iπ + 6 ln (cos z + cos z ) (cid:21) − π (cid:33) (cos z + cos z ) ,B (1)4 = − π (cid:32) (cid:16) − (cos z − cos z ) (cid:17) cos z cos z + 4 (cos z + cos z ) (1 − z cos z ) (cid:33) , and B (1)5 = 19 (cid:32) −
29 cos z cos z + 24 cos z cos z + 6 (3 cos z cos z − (cid:0) cos z + cos z (cid:1) (cid:33) + 1 π (cid:34) z + cos z ) (1 − z cos z ) − cos z cos z (cid:18) z + cos z + 6 cos z cos z (cid:19)(cid:35) + iπ (cid:34) cos z cos z (cid:18) − (cos z − cos z ) (cid:19) + 4 (cos z + cos z ) (1 − z cos z ) (cid:35) − π (cid:40) cos z cos z (cid:18) − (cos z − cos z ) (cid:19) + 4 (cos z + cos z ) (1 − z cos z ) (cid:41) ln (cos z + cos z ) . (S40)Integrating B (1) j over z and z , we get (cid:90) π (cid:90) π dz dz B (1)1 = (cid:90) π (cid:90) π dz dz B (1)2 = 0 , (cid:90) π (cid:90) π dz dz B (1)3 = 323 π , (cid:90) π (cid:90) π dz dz B (1)4 = − , (cid:90) π (cid:90) π dz dz B (1)5 = ln 2 − . (S41)Collecting the expressions in (S41) we get the expansion in (S24). Note that the coefficient 32 / (3 π ) in front of theleading term in (S24) was obtained earlier in [45].Similarly, using (S36), we represent the infinite series Σ in (S23) asΣ = (cid:90) π (cid:90) π dz dz Φ (2) ε ( z , z ) , (S42)where Φ (2) ε ( z , z ) = 12 π (cid:18) x − x (cid:19) µ ( z ) µ ( z ) ν ( z ) ν ( z ) (cid:32) ( ν ( z ) ν ( z ) − (cid:0) ν ( z ) ν ( z ) (cid:1) + ν ( z ) ν ( z ) (cid:33) , (S43)with ν ( z , ) defined in (S35). The small- ε behaviour of Φ (2) ε ( z , z ) followsΦ (2) ε ( z , z ) = B (2)1 ε − + B (2)3 ε − + B (2)4 ln( ε ) + B (2)5 + O ( ε ) , (S44)where B (2)1 , B (2)3 , B (2)4 and B (2)5 are given explicitly by B (2)1 = − π cos z cos z , B (2)3 = − i π (cid:18) π −
6) cos z cos z (cid:19) (cos z + cos z ) ,B (2)4 = − π cos z cos z (cos z + cos z ) , and B (2)5 = 13 (cid:32) (cid:0) cos z + cos z (cid:1) (1 − z cos z ) + 3 cos z cos z (1 − cos z cos z ) (cid:33) + 13 π (cid:34) − (cid:0) cos z + cos z (cid:1) (1 − z cos z ) −
11 cos z cos z + 18 cos z cos z (cid:35) + iπ cos z cos z (cos z + cos z ) − π cos z cos z (cos z + cos z ) ln (cos z + cos z ) . (S45)Integrating B (2) j over z and z , we obtain (cid:90) π (cid:90) π dz dz B (2)1 = (cid:90) π (cid:90) π dz dz B (2)3 = 0 , (cid:90) π (cid:90) π dz dz B (2)4 = − , (cid:90) π (cid:90) π dz dz B (2)5 = 14 + ln 2 + π . (S46)Collecting these results, we obtain eventually the asymptotic expansion in (S25). B. Approximation for g n ( R ) /g (cid:48) n ( R ) and its limiting behavior for sufficiently large | U (cid:48) ( R ) | . We pursue next a different approach for calculation of the logarithmic derivative of the radial functions at theboundary, and of the corresponding expressions for the infinite series R (3) ε and R (2) ε . This approach is based on theassumption that, once we are only interested in the behaviour on the boundary only, we may replace the coefficientsin the differential equations (7) and (20) by their values at the boundary. Such an approximation is legitimate,of course, only for the potentials for which U (cid:48) ( R ) exists. In doing so, we will be able to derive an explicit, albeitan approximate expression for g n ( R ) /g (cid:48) n ( R ) which is valid, in principle, for arbitrary n and arbitrary[71] | U (cid:48) ( R ) | < ∞ . This approximate expression will be subsequently checked against exact results obtained for the triangular-wellpotential (see SM4 and SM5) and the asymptotic forms in (S17) and (S19).We turn to the differential equations (7) and (20) and replace the coefficients in these equations (which are functionsof r ) by their values at the boundary. This gives the following differential equations with constant coefficients: g (cid:48)(cid:48) n + (cid:18) R − U (cid:48) ( R ) (cid:19) g (cid:48) n − n ( n + 1) R g n = 0 (S47)and g (cid:48)(cid:48) n + (cid:18) R − U (cid:48) ( R ) (cid:19) g (cid:48) n − n R g n = 0 (S48)for the 3D and the 2D cases, respectively. Using the notation ω = R U (cid:48) ( R ) , (S49) [71] Note that the condition that U (cid:48) ( R ) is bounded does not preventus to study the behaviour of g n ( R ) /g (cid:48) n ( R ) in the limit | U (cid:48) ( R ) | →∞ . we write down a general solution of (S47): g n = c exp (cid:18) r R (cid:18) ( ω − − (cid:113) (2 n + 1) + ( ω − − (cid:19)(cid:19) + c exp (cid:18) r R (cid:18) ( ω −
2) + (cid:113) (2 n + 1) + ( ω − − (cid:19)(cid:19) , (S50)where c and c are adjustable constants. We note that since (2 n + 1) − > n >
0, the expression underthe square root is always positive. Further, differentiating (S50) and setting r = R , we get the following approximateexpression for the inverse of the logarithmic derivative at the boundary: g n ( R ) Rg (cid:48) n ( R ) ≈ (cid:32) ω − (cid:112) (2 n + 1) + ( ω − − − c (cid:112) (2 n + 1) + ( ω − − c + c exp (cid:0)(cid:112) (2 n + 1) + ( ω − − (cid:1) (cid:33) − . (S51)We notice that the last term in brackets in (S51), which is the ratio of an algebraic and an exponential function,can be safely neglected because the exponential function becomes large when either (or both) n and/or | ω | are large.This yields the following approximation for the inverse logarithmic derivative, which is independent of the constants c and c : g n ( R ) Rg (cid:48) n ( R ) ≈ ω − (cid:113) (2 n + 1) + ( ω − − (cid:113) (2 n + 1) + ( ω − − − ω + 22 n ( n + 1) . (S52)The same arguments yield an analogous approximation for the 2D case: g n ( R ) Rg (cid:48) n ( R ) ≈ (cid:113) n + ( ω − − ω + 12 n . (S53)In Figs. S3 and S5 in the following sections SM4 and SM5 we compare the expressions in (S52) and (S53) with theexact results for the ratio g n ( R ) / ( Rg (cid:48) n ( R )) derived for the special case of a triangular-well potential in (48). Weobserve a fairly good agreement between the approximate forms in (S52) and (S53) and the exact results in (S106)and (S130) even for very modest values of n (say, for n ≥ n there are some apparent deviations whichhowever become smaller the larger | ω | is.We turn to the limit n → ∞ . We find that in this limit the expressions in ( S
52) and (S53) exhibit the followingasymptotic behaviour g n ( R ) Rg (cid:48) n ( R ) ≈ n − ω − n + ω − n + O (cid:18) n (cid:19) (S54)and g n ( R ) Rg (cid:48) n ( R ) ≈ n − ω − n + ω − ω + 18 n + O (cid:18) n (cid:19) (S55)for the 3D and the 2D cases, respectively. Comparing these expansions with the asymptotic forms in (S17) and (S19),we observe that they are identical in the leading terms for large | ω | . This suggests, in turn, that the approximateexpressions for the inverse of the logarithmic derivatives in (S52) and (S53) are reliable (as well as the assumptionsunderlying their derivation) for | ω | large enough.Further, using (S52) and (S53), we evaluate approximate expressions for the infinite series R (3) ε and R (2) ε , and thecorresponding small- ε expansions. To this end, it is expedient to use an auxiliary integral identity (cid:112) A + B = A + B (cid:90) ∞ dξξ e − Aξ J ( Bξ ) , (S56)where J ( · ) is the Bessel function. This identity is valid for A and B such that | Im B | < Re A .
2D case
We start with the 2D case, which is simpler than the 3D one, and set A = 2 n and B = ω −
1. Such a choiceevidently fulfils the condition of the applicability of the identity in (S56). We have then R (2) ε ≈ ∞ (cid:88) n =1 sin ( nε ) n ε − B ∞ (cid:88) n =1 sin ( nε ) n ε + B (cid:90) ∞ dξξ J ( Bξ ) ∞ (cid:88) n =1 sin ( nε ) n ε e − nξ , (S57)where the symbol ≈ signifies that this expression is obtained via an approximate approach. The asymptotic small- ε behaviour of the first sum is given by (S32), while the second and the third terms converge to ε -independent constants: R (2) ε ≈ /ε ) + 3 − O ( ε ) (cid:124) (cid:123)(cid:122) (cid:125) first sum − B π (cid:124) (cid:123)(cid:122) (cid:125) second sum + B (cid:90) ∞ dξξ J ( Bξ )Li ( e − ξ ) (cid:124) (cid:123)(cid:122) (cid:125) third sum . (S58)Since J ( z ) is an odd function, J ( − z ) = − J ( z ), the integral in the last line in (S58) is an even function of B , i.e.,it depends only on | B | . For large | B | (or large | ω | ), the major contribution to this integral comes from small valuesof ξ , ξ (cid:28)
1, so that this integral is given approximately by B (cid:90) ∞ dξξ J ( Bξ )Li ( e − ξ ) ≈ | B | π | B | + O (1) , (S59)where the omitted terms O (1) are B -independent constants. We therefore obtain R (2) ε ≈ /ε ) + π R (cid:0) | U (cid:48) ( R ) | − U (cid:48) ( R ) (cid:1) + 2 ln( R | U (cid:48) ( R ) | ) + O (1) . (S60)We conclude that in the 2D case, the leading in the limit ε → R (2) ε is independent of the interactionpotential and is identical to the result in (32) based on the large- n expansions. Remarkably, the second term in (S60)is non-zero for attractive potentials (negative U (cid:48) ( R )) only, and becomes identically equal to zero in case of repulsivepotentials (positive U (cid:48) ( R )). As a matter of fact, this term provides the major contribution in the limit of infinitelystrong attractive potentials. For instance, in the case of a triangular-well potential, one has L (2) U ∼ / | ω | from (53)for negative U (cid:48) ( R ) of very large amplitude, so that the MFET in the limit ω → −∞ becomes T (2) ε (cid:39) (cid:16) r R (cid:17) r D + π R D . (S61)As discussed in the main text, the first term is the time for a particle started uniformly to reach the boundary (inpresence of an infinitely strong attractive potential in the region r < r < R ), whereas the second term represents theMFPT from a uniform starting point on a circle of radius R to a point-like target ( ε = 0).
3D case
In the 3D case we set A = 2 n + 1, which is real and positive, and B = (cid:112) ( ω − −
1. Note that the maximumimaginary value of B is 1, and it is less than the minimal value of A = 3, attained for n = 1, so that the identity in(S56) is valid for any n and ω . Using this identity, we can cast R (3) ε into the following form R (3) ε ≈ − ω − ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1)(2 n + 1) + 12 ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1) + F ε ( B ) , (S62)where F ε ( B ) = B (cid:90) ∞ dξξ e − ξ J ( Bξ ) ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1)(2 n + 1) e − nξ . (S63)0For the infinite series entering the first term on the right-hand-side of (S62) we have12 ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1)(2 n + 1) = Σ + 12 ∞ (cid:88) k =1 ( − k ∞ (cid:88) n =1 φ n ( ε ) n k (2 n + 1) , (S64)where Σ and its asymptotic behaviour are defined in (S23) and (S25). Noticing that the second term on the right-hand-side of (S64) converges to an ε -independent constant as ε →
0, i.e.,lim ε → ∞ (cid:88) k =1 ( − k ∞ (cid:88) n =1 φ n ( ε ) n k (2 n + 1) = ∞ (cid:88) k =1 ( − k ∞ (cid:88) n =1 (2 n + 1) n k = − − π , (S65)we infer that 12 ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1)(2 n + 1) = ln(1 /ε ) + O (1) . (S66)The sum in the second term on the right-hand-side of (S62) can be formally rewritten as12 ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1) = ∞ (cid:88) n =1 φ n ( ε ) n (2 n + 1) (cid:18) − n + 1) (cid:19) = Σ − ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1)(2 n + 1) , (S67)where Σ and its asymptotic behaviour are defined in (S22) and (S24). Consequently, we have12 ∞ (cid:88) n =1 φ n ( ε ) n ( n + 1) = 323 π ε − + O (1) . (S68)Lastly, we consider the contribution in (S63). For large | B | , the major contribution to the integral comes from ξ close to 0. Since φ n ( ε ) → (2 n + 1) as ε →
0, the sum would logarithmically diverge if both ε and ξ were set to 0.This simple observation suggests that this sum may exhibit a logarithmic dependence either on ε , or on ξ . In order toevaluate the contribution F ε ( B ), we adopt the summation technique used in the previous subsection. Recalling theintegral representations in (S35, S36), we have F ε ( B ) = B (cid:90) ∞ dξξ J ( Bξ ) G ε ( ξ ) , (S69)where, explicitly, G ε ( ξ ) = e − ξ ∞ (cid:88) n =1 φ n ( ε ) e − nξ n ( n + 1)(2 n + 1) = π (cid:90) dz π (cid:90) dz Φ (3) ε ( z , z , ξ ) (S70)and Φ (3) ε ( z , z , ξ ) = e − ξ π (cid:18) ε − cos ε (cid:19) µ ε ( z ) µ ε ( z ) ∞ (cid:88) n =1 n + 1 n ( n + 1) (cid:2) ν ε ( z ) ν ε ( z ) (cid:3) n − e − nξ , (S71)with µ ε ( z ) and ν ε ( z ) defined in (S35, S36). Denoting ζ = ν ε ( z ) ν ε ( z ) e − ξ , we getΦ (3) ε ( z , z , ξ ) = 1 π (cid:18) ε − cos ε (cid:19) µ ε ( z ) µ ε ( z ) e − ξ (Li ( ζ ) − Li ( ζ ) + (1 − ζ ) ln(1 − ζ ) + ζ ) ζ . (S72)Now, we have two options, either to expand this function first in powers of ε and then in powers of ξ , or to expandit first in powers of ξ and then in powers of ε . These two options correspond to two possible orders of limits: ε → | B | → ∞ .(i) Limit ε → for a fixed | B | .For a fixed ξ >
0, we expand Φ (3) ε ( z , z , ξ ) in powers of ε to getΦ (3) ε ( z , z , ξ ) = C − ( z , z , ξ ) ε − + C − ( z , z , ξ ) ε − + C ( z , z , ξ ) + O ( ε ) . (S73)1Note that this expansion does not contain a term, which logarithmically diverges as ε →
0. Next, each coefficient C j ( z , z , ξ ) has to be expanded in powers of ξ . After integration over z and z , the contributions from C − and C − vanish (as expected), and the leading terms are given by F ( B ) = B (cid:90) ∞ dξξ J ( Bξ ) (cid:20) − ξ − − ξ + O ( ξ ) (cid:21) = | B | ln | B | + | B | ( γ − /
2) + O (1)= | ω | ln | ω | + | ω | ( γ − /
2) + O (ln | ω | ) , (S74)where γ ≈ . ε asymptotic behaviour of R (3) ε for sufficiently large | ω | : R (3) ε ≈ π ε − − ( ω −
2) ln(1 /ε ) + | ω | ln | ω | + ( γ − / | ω | + O (ln | ω | ) . (S75)We note that despite the fact that this small- ε asymptotics is formally valid for sufficiently large | ω | , it predicts aspurious logarithmic divergence of the MFET in the limit | ω | → ∞ . This divergence is clearly unphysical (a strongerattractive potential should reduce the MFET, instead of increasing it) and indicates that (S75) holds for large butbounded ω . Upon a more detailed analysis, we infer that (S75) is only applicable for 1 (cid:28) | ω | (cid:28) /ε .(ii) Limit | B | → ∞ for a fixed small ε .Expanding Φ (3) ε ( z , z , ξ ) in (S72) in powers of ξ , we haveΦ (3) ε ( z , z , ξ ) = Φ (3) , ε ( z , z ) + ξ Φ (3) , ε ( z , z ) + O ( ξ ) , (S76)which yields, upon inserting the latter expansion into (S69), F ε ( B ) = 12 π (cid:90) dz π (cid:90) dz (cid:20) | B | Φ (3) , ε ( z , z ) + Φ (3) , ε ( z , z ) (cid:21) . (S77)Concentrating next on the narrow escape limit ε →
0, we expand Φ (3) , ε ( z , z ) and Φ (3) , ε ( z , z ) in powers of ε to getΦ (3) ,jε ( z , z ) = B (3) ,j ( z , z ) ε − + B (3) ,j ( z , z ) ε − ln( ε )+ B (3) ,j ( z , z ) ε − + B (3) ,j ( z , z ) ln( ε ) + B (3) ,j ( z , z ) + O ( ε ) , (S78)where B (3) ,j ( z , z ) with j = 0 , B (3) , = − (cid:0) ζ (3) − π + 6 (cid:1) π cos z cos z , B (3) , ≡ ,B (3) , = 2 i π (cos z + cos z ) (cid:18) (cid:0) ζ (3) − π + 4 (cid:1) cos z cos z − (cid:0) ζ (3) − π + 6 (cid:1)(cid:19) ,B (3) , = − π cos z cos z (cid:0) cos z + cos z (cid:1) ,B (3) , = − π cos z cos z (cid:0) cos z + cos z (cid:1) ln (cid:0) cos z + cos z (cid:1) ++ 19 π (cid:18)
36 + 36 ζ (3) − π + (cid:0) − π + 18 πi + 108 ζ (3) (cid:1) cos z cos z (cid:0) cos z + cos z (cid:1) + (cid:0) π − − ζ (3) (cid:1)(cid:0) cos z + cos z (cid:1) + (cid:0) π − − ζ (3) (cid:1) cos z cos z + (cid:0) − π + 36 πi (cid:1) cos z cos z (cid:19) , (S79)2and B (3) , = − ζ (3) − π + 2) π cos z cos z , B (3) , = − iπ cos z cos z (cid:0) cos z + cos z (cid:1) ,B (3) , = − iπ cos z cos z (cid:0) cos z + cos z (cid:1) ln (cid:0) cos z + cos z (cid:1) + 2 i π (cid:0) cos z + cos z (cid:1) × (cid:18) cos z cos z (cid:0) ζ (3) + 24 − π + 12 iπ (cid:1) − (cid:0) ζ (3) − π + 2 (cid:1)(cid:19) ,B (3) , = 8 π (cid:18)(cid:0) cos z + cos z (cid:1)(cid:0) − z cos z (cid:1) + 2 cos z cos z (cid:0) − z cos z (cid:1)(cid:19) . (S80)To find an explicit expression for F ε ( B ) in (S77), we now have to integrate all the coefficients B (3) ,j over z and z .This can be done rather straightforwardly and we find that the double integrals b jk = 12 π (cid:90) π (cid:90) dz dz B (3) ,jk ( z , z ) , (S81)are given explicitly by b = b = b = 0 , b = − , b = ln 2 − , (S82)for j = 0, and b = b = b = 0 , b = − π , (S83)for j = 1, respectively. Collecting these explicit expressions for the coefficients b jk , we get F ε ( B ) = | B | ln(1 /ε ) + (ln 2 − / | B | − π ε − + O (1) . (S84)Note that the coefficient in front of the term which diverges as 1 /ε is negative and is equal by the absolute value tothe coefficient of the leading diverging term in (S66), so that these two terms cancel each other. Recalling next thedefinition of B for the 3D case, and combining (S77) with (S66, S68), we obtain the asymptotic behaviour of R (3) ε forvery large | ω | and small fixed ε : R (3) ε ≈ ( | ω | − ω ) ln(1 /ε ) + (ln 2 − / | ω | + O (1) . (S85)Comparing the latter expression with (S75), we note that (S85) does not contain the term | ω | ln | ω | (that caused anunphysical divergence of the MFET in the limit ω → −∞ ), and includes an extra term ( | ω | − ω ) ln(1 /ε ) so that thelogarithmically diverging term in (S85) is twice larger than the one in (S75) in case of negative ω . We note that due tothis additional numerical factor, the expression in (S85) reproduces correctly, in the limit ω → −∞ , the exact resultobtained in [51] for the MFPT to the EW solely due to diffusion along the surface of the 3D spherical micro-domain.We also emphasise that the coefficient in front of ln(1 /ε ) is non-zero only for negative ω (attractive interactions), andvanishes for positive ω (repulsive interactions). SM3. SYSTEMS WITHOUT LONG-RANGE INTERACTIONS
We examine next the simplest case without long-range interactions, U ( r ) ≡
0, so that a particle diffuses freely witha bounded micro-domain.
A. 3D case
The general solution of equation (7) for the radial functions g n ( r ) reads g n ( r ) = c r n + c r − n − . (S86)3We set c = 1 for convenience, and choose c = 0 to ensure the regularity at the origin. Then, the particular solution t ( r ) is t ( r ) = R − r D , (S87)so that t (cid:48) ( R ) = − R/ (3 D ). We therefore obtain t ( r, θ ) = R − r D + a − R D ∞ (cid:88) n =1 φ n ( ε ) n (cid:16) rR (cid:17) n P n (cos θ ) , (S88)where the coefficient a is fixed by the self-consistent condition in (18). This gives a = 2 R κ (1 − cos ε ) + R R (3) ε D , (S89)with R (3) ε defined in (40). By integrating (S88) over the volume of the sphere, we find that the global MFET T ε froma random location is given by (38).We note finally that for a perfect EW (no barrier, κ = ∞ ), such that any arrival of the particle to the EW locationwill result in the escape from the sphere, the condition (17) reduces to ε (cid:90) dθ sin θ t ( R, θ ) = 0 . (S90)In other words, the original Dirichlet boundary condition at each point of the EW, t ( R, θ ) = 0 for 0 ≤ θ ≤ ε , isreplaced by a weaker condition requiring that the MFET vanishes on the EW on average . Hence, the condition (S90)implies that some values of t ( R, θ ) can become negative. As a consequence, the approximation is not expected toyield accurate results for the MFET with the starting point ( r, θ ) close to the EW. One can check numerically (notshown) that the approximation is nonetheless very accurate when the starting point is far from the EW. In general,the SCA is expected to be more accurate for small targets, as well as for weak reactivities κ . B. 2D case
In 2D case, the radial functions g n ( r ) are given by g n ( r ) = c r n + c r − n = r n , where we set c = 1 and c = 0. Wealso have t ( r ) = R − r D , (S91)from which t (cid:48) ( R ) = − R/ (2 D ) follows. We therefore obtain t ( r, θ ) = R − r D + a − R D ∞ (cid:88) n =1 sin( nε ) n ε ( r/R ) n cos( nθ ) , (S92)with a = πR κε + R R (2) ε D , (S93)and R (2) ε defined in (41). Integrating (S92) over the area of the circular micro-domain, we arrive at our result in (39).Lastly, we note that the problem of finding the MFET through the fully reactive arc ( − ε, ε ) of a disk, without LRIpotential ( U ( r ) ≡
0) and without a barrier at the EW ( κ = ∞ ) was solved analytically by Singer et al. [60] (see also[61, 62]).4 U(r)U o r o r R FIG. S2: A sketch of the triangular-well interaction potential in (48).
SM4. TRIANGULAR-WELL POTENTIAL IN 3D CASE
We now make a particular choice of the interaction potential between the diffusive particle and the boundary – atriangular-well radial potential defined in (48) (see Fig. S2). An advantage of such a choice is that i) it is simple butphysically meaningful (see the discussion in [50]), ii) it permits to obtain an exact solution of the modified boundary-value problem and hence to check the accuracy of our predictions in (33) and (35), iii) it allows to verify our argumentsbehind the derivation of the asymptotic series in (S17) and (S19), and finally, iv) it helps to highlight some spectaculareffects of the long-range particle-boundary interactions on the MFET.
A. Solution of the inhomogeneous problem (9).
First, we compute t ( r ) by direct integration of the expression in (9) to get t ( r ) = r − r D + H (3) ( ω ) , ≤ r ≤ r ,H (3) (cid:16) ω rR (cid:17) , r < r ≤ R, (S94)where H (3) ( z ) = R Dω (cid:20)(cid:0) ω / ω + 2 ω + 2 (cid:1) e − ω ω (cid:90) z dx e x x − ω (1 − x ) − − x ) ω + 2 ln x (cid:21) , (S95)with the dimensionless parameters x = r R , ω = RU R − r = U − x , ω = r U R − r = x ω . Next, the derivative of t ( r ) reads t (cid:48) ( R ) = RDω (cid:18) ω + 2 ω + 2 − (cid:0) ω / ω + 2 ω + 2 (cid:1) e ω − ω (cid:19) . (S96)Integrating (S94), one finds the result in (55). B. Radial functions g n ( r ) In order to solve (7), one finds solutions on each of the subintervals g n ( r ) = (cid:40) A − r n + B − r − n − , ≤ r ≤ r ,A + u n (cid:16) ω rR (cid:17) + B + v n (cid:16) ω rR (cid:17) , r < r ≤ R, (S97)5where A ± and B ± are unknown coefficients to be determined, and u n ( z ) = z n M ( n, n + 2 , z ) ,v n ( z ) = z − n − U ( − n − , − n, z ) (S98)are two independent solutions in the presence of a triangular-well potential (48), M ( a, b, z ) and U ( a, b, z ) beingKummer’s and Tricomi’s confluent hypergeometric functions, respectively. The regularity of g n ( r ) at r = 0 requires B − = 0. Requiring the continuity of g n ( r ) and of its derivative g (cid:48) n ( r ) at r = r , one relates A + and B + to A − : A − r n = A + u n ( ω ) + B + v n ( ω ) ,n A − R r n − = A + ω u (cid:48) n ( ω ) + B + ω v (cid:48) n ( ω ) . (S99)These relations can be inverted to get A + = A − (cid:0) r n v (cid:48) n ( ω ) − n R r n − v n ( ω ) /ω (cid:1) u n ( ω ) v (cid:48) n ( ω ) − v n ( ω ) u (cid:48) n ( ω ) ,B + = A − (cid:0) − r n u (cid:48) n ( ω ) + n R r n − u n ( ω ) /ω (cid:1) u n ( ω ) v (cid:48) n ( ω ) − v n ( ω ) u (cid:48) n ( ω ) . (S100)The denominator in the latter expressions is the Wronskian of the solution, which can be calculated explicitly: u n ( z ) v (cid:48) n ( z ) − v n ( z ) u (cid:48) n ( z ) = − (2 n + 1)!( n − e z z . (S101)Note that the Wronskian can be “absorbed” into a prefactor, which will then be factored out. We write then g n ( r ) = A ∗ (cid:20) u n (cid:16) ω rR (cid:17) − v n (cid:16) ω rR (cid:17) w n ( ω ) (cid:21) , (S102)where w n ( z ) = zu (cid:48) n ( z ) − nu n ( z ) zv (cid:48) n ( z ) − nv n ( z ) . (S103)We therefore obtain g n ( R ) Rg (cid:48) n ( R ) = u n ( ω ) − v n ( ω ) w n ( ω ) ω u (cid:48) n ( ω ) − ω v (cid:48) n ( ω ) w n ( ω ) . (S104)Next, using the relations zu (cid:48) n ( z ) = nz n M ( n + 1 , n + 2 , z ) ,zv (cid:48) n ( z ) = − n ( n + 1) z n +1 U ( − n, − n, z ) , one can represent zu (cid:48) n ( z ) − nu n ( z ) = nz n (cid:2) M ( n + 1 , n + 2 , z ) − M ( n, n + 2 , z ) (cid:3) = nz n +1 n + 2 M ( n + 1 , n + 3 , z ) ,zv (cid:48) n ( z ) − nv n ( z ) = − nz − n − (cid:2) ( n + 1) U ( − n, − n, z ) + U ( − n − , − n, z ) (cid:3) = − nz n +1 U ( n + 1 , n + 3 , z ) , so that w n ( z ) = − n + 1) M ( n + 1 , n + 3 , z ) U ( n + 1 , n + 3 , z ) . (S105)6Taking together (S104) to (S105), we obtain g n ( R ) Rg (cid:48) n ( R ) = 1 n M ( n, n + 2 , ω ) M ( n + 1 , n + 2 , ω ) (cid:18) − U ( n, n + 2 , ω ) M ( n, n + 2 , ω ) w n ( ω ) (cid:19) × (cid:18) n + 1) U ( n + 1 , n + 2 , ω ) M ( n + 1 , n + 2 , ω ) w n ( ω ) (cid:19) − , (S106)which is the desired exact expression for the ratio g n ( R ) / ( Rg (cid:48) n ( R )) for the triangular-well potential.For numerical computations, another representation in terms of the modified Bessel functions I n +1 / ( z ) and K n +1 / ( z ) can be convenient. Starting from the identities M ( n + 1 , n + 2 , x ) = Γ( n + 3 / (cid:18) x (cid:19) n +1 / e x/ I n +1 / ( x/ ,U ( n + 1 , n + 2 , x ) = e x/ √ πx n +1 / K n +1 / ( x/ , (S107)one can use the recurrence relations between Kummer’s and Tricomi’s functions to represent all the entries in (S106)in terms of I n +1 / ( z ) and K n +1 / ( z ). This gives g n ( R ) Rg (cid:48) n ( R ) = 1 n (cid:18) ω i n ( ω )2( n + 1) (cid:19) (cid:0) j n ( ω, ωr /R ) (cid:1) − × (cid:18) − j n ( ω, ωr /R ) k n ( ω ) − n +1 ω i n ( ω ) + 2 n +1 ω (cid:19) , (S108)with i n ( z ) = I n +3 / ( z/ I n +1 / ( z/ − , k n ( z ) = K n +3 / ( z/ K n +1 / ( z/
2) + 1 ,j n ( z, z ) = K n +1 / ( z/ K n +3 / ( z / I n +3 / ( z / I n +1 / ( z/ . (S109)Before we proceed with the analysis of the asymptotic large- n behaviour of g n ( R ) / ( Rg (cid:48) n ( R )), it might be expedientto note that in the particular case r = 0, the expression in (S106) simplifies to give g n ( R ) Rg (cid:48) n ( R ) = u n ( ω ) ω u (cid:48) n ( ω ) = 1 n M ( n, n + 2 , ω ) M ( n + 1 , n + 2 , ω ) , (S110)which is just the first factor in (S106) since w n (0) appears to be equal identically to zero. In this particular case, wehave H (3) ( z ) = 1 Dω (cid:20) ω (cid:90) z dx e x − x − x − ω (1 − x ) (cid:21) , (S111)so that the MFPT from a random location to any point on the boundary becomes T (3) π ( κ = ∞ ) = R Dω (cid:20) e ω ( ω −
1) + 1) − ω + 8 ω + 1212 ω (cid:21) . (S112)Consequently, the MFPT to the EW from some fixed location has the form: t ( r, θ ) = t ( r ) + a + R t (cid:48) ( R ) ∞ (cid:88) n =1 M (cid:16) n, n + 2 , ω rR (cid:17) ( r/R ) n nM ( n + 1 , n + 2 , ω ) φ n ( ε ) P n (cos( θ )) , (S113)where a is given by (18) and, explicitly, t (cid:48) ( R ) = RDω (cid:0) ω + 2 ω + 2 − e ω (cid:1) . (S114)7Consider next the behaviour of the inverse logarithmic derivative of the radial functions at the confining boundaryin the limit n → ∞ for arbitrary r . First, we find that the first factor in (S106) obeys1 n M ( n, n + 2 , ω ) M ( n + 1 , n + 2 , ω ) = 1 n − ω n + 4 ω + ω n + O (cid:18) n (cid:19) . (S115)Second, we analyse the large- n behaviour of the second factor in (S106). Taking into account the definition of w n ( z ) in(S105), we observe that the correction term to unity has the form of a product of ratios of two Kummer’s and Tricomi’sfunctions with different arguments. The asymptotic large- n behaviour of the ratio of two Kummer’s functions follows M ( n + 1 , n + 3 , ω )( n + 1) M ( n, n + 2 , ω ) = exp (cid:16) − ω − x ) (cid:17) × (cid:34) n + ω − ω − (4 − ω ) n + O (cid:18) n (cid:19) (cid:35) , (S116)i.e., is an expansion in the inverse powers of n . The ratio of two Tricomi’s functions is given by U ( n, n + 2 , ω ) U ( n + 1 , n + 3 , ω ) = n ω x n (cid:80) n +1 s =0 (cid:0) n +1 s (cid:1) Γ( n + s ) /ω s (cid:80) n +1 s =0 (cid:0) n +1 s (cid:1) Γ( n + s + 1) /ω s . (S117)Noticing that in the latter expression the major contribution to the sums in the numerator and the denominator stemsfrom the terms with s = n + 1, we infer that the leading behaviour of the ratio in (S117) in the limit n → ∞ obeys U ( n, n + 2 , ω ) U ( n + 1 , n + 3 , ω ) ∼ ω n (2 n + 1) x n +10 , (S118)which means that the ratio of two Tricomi’s functions vanishes exponentially fast with n as n → ∞ for x < r < R ). This implies, in turn, that the correction term to unity in the second factor in (S106) is exponentially smallas n → ∞ and hence, can be safely neglected.Next, we consider the behaviour of the third factor in (S106) which is also a product of ratios of two Kummer’sand Tricomi’s functions. We have M ( n + 1 , n + 3 , ω ) M ( n + 1 , n + 2 , ω ) = exp (cid:16) − ω − x ) (cid:17) × (cid:34) − ω + ω (1 − x )16 n + O (cid:18) n (cid:19) (cid:35) (S119)and U ( n + 1 , n + 2 , ω ) U ( n + 1 , n + 3 , ω ) = x n +10 (cid:80) ns =0 (cid:0) ns (cid:1) Γ( n + 1 + s ) /ω s (cid:80) n +1 s =0 (cid:0) n +1 s (cid:1) Γ( n + 1 + s ) /ω s . (S120)Noticing that in the n → ∞ limit the major contribution to the sums in the numerator and the denominator in thelatter expression is provided by the terms with s = n , we find eventually that the leading behaviour of the ratio oftwo Tricomi’s functions is defined by U ( n + 1 , n + 2 , ω ) U ( n + 1 , n + 3 , ω ) ∼ ω (2 n + 1) x n +10 . (S121)Therefore, due to the factor x n +10 , which vanishes exponentially fast as n → ∞ , the third factor in (S106) appearsto be exponentially close to 1 and can be safely neglected.As a consequence, the leading asymptotic behaviour of g n ( R ) / ( Rg (cid:48) n ( R )) in (S106) is entirely dominated by the firstfactor and hence, we have g n ( R ) Rg (cid:48) n ( R ) = 1 n − ω n + 4 ω + ω n + O (cid:18) n (cid:19) . (S122)The expansion in (S122) permits us to verify our general argument on the asymptotic behaviour of the ratio g n ( R ) / ( Rg (cid:48) n ( R )) in the limit n → ∞ , presented in the beginning of SM2 (see (S17)). Recalling that ω = RU (cid:48) ( R ) =8 −2 −1 n g n ( R ) / ( R g ’ n ( R )) U = 2U = 1U = −1U = −2 FIG. S3: The ratio g n ( R ) / ( Rg (cid:48) n ( R )) vs the order n of the radial function for several values of U with r = 0 . R = 1.Comparison of the exact result in (S106) (symbols) and the approximate expression in (S52) (lines). Thin solid line is the 1 /n asymptotics (solution of the problem with U ≡ ε ε R ( ) ε U = 1U = −1U = −2 FIG. S4: Infinite series R (3) ε , multiplied by ε , as a function of the angular size ε of the EW for U = 1, U = − U = − r = 0 and R = 1. Symbols represent the exact result obtained by numerical summation of (19) involving the expression in(S106), while curves show the asymptotic relation (31) without the last infinite sum. U / (1 − x ) and that for such a potential U (cid:48)(cid:48) ( R ) = 0, we observe a perfect coincidence of (S17), based on an intuitive(albeit quite plausible) argument, and the large- n expansion of the inverse of the logarithmic derivative, evaluatedfor an exactly solvable case of a triangular-well potential U ( r ). Further, we note that the large- n behaviour of (S122)is dominated by the first factor, which is the solution for a particular case r = 0. This implies, in turn, that in thelarge- n limit the dependence on r is fully embodied in the dimensionless parameter ω .Lastly, we compare the approximate expression (S52) for g n ( R ) / ( Rg (cid:48) n ( R )) and the exact result in (S106) obtainedfor the triangular-well potential, see Fig. S3. We observe a fairly good agreement between the approximate formula(S52) and the exact result already for quite modest values of n , and notice that the agreement becomes even betterfor larger values of R | U (cid:48) ( R ) | . Accordingly, our approximate small- ε expansion in (31) (without the infinite sum in thelast line) and the exact result for R (3) ε agree well with each other, see Fig. S4. The smaller ε , the better agreement is. SM5. TRIANGULAR-WELL POTENTIAL IN 2D CASEA. Solution of the inhomogeneous problem (22)
Integrating Eq. (22), we get t ( r ) = r − r D + H (2) ( ω ) 0 ≤ r ≤ r ,H (2) (cid:16) ω rR (cid:17) r < r ≤ R, (S123)9where H (2) ( z ) = R Dω (cid:20)(cid:0) ω / ω + 1 (cid:1) e − ω ω (cid:90) z dx e x x − ω (1 − x ) + ln x (cid:21) , (S124)Integrating (S123), one arrives at (61). One also gets t (cid:48) ( R ) = RDω (cid:18) ω + 1 − (cid:0) ω / ω + 1 (cid:1) e ω − ω (cid:19) . (S125) B. Radial functions
In two dimensions, the solutions of (7) for a triangular-well potential read u n ( z ) = z n M ( n, n + 1 , z ) ,v n ( z ) = z − n U ( − n, − n + 1 , z ) . (S126)Using the identities zu (cid:48) n ( z ) = nz n M ( n + 1 , n + 1 , z ) ,zv (cid:48) n ( z ) = − n z − n U ( − n + 1 , − n + 1 , z ) , (S127)one gets zu (cid:48) n ( z ) − nu n ( z ) = nz n +1 n + 1 M ( n + 1 , n + 2 , z ) ,zv (cid:48) n ( z ) − nv n ( z ) = − nz − n U ( − n, − n, z ) , (S128)so that w n ( z ) becomes w n ( z ) = zu (cid:48) n ( z ) − nu n ( z ) zv (cid:48) n ( z ) − nv n ( z ) = − M ( n + 1 , n + 2 , z )(2 n + 1) U ( n + 1 , n + 2 , z ) . (S129)Combining these equations we obtain the following explicit expression for the inverse logarithmic derivative of theradial functions in the 2D case with the triangular-well potential: g n ( R ) Rg (cid:48) n ( R ) = 1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) (cid:32) − U ( n, n + 1 , ω ) M ( n, n + 1 , ω ) w n ( ω ) (cid:33) × (cid:32) nU ( n + 1 , n + 1 , ω ) M ( n + 1 , n + 1 , ω ) w n ( ω ) (cid:33) − . (S130)As earlier in the 3D case, another representation can be obtained using (S107) g n ( R ) Rg (cid:48) n ( R ) = 1 n − i n ( ω ) + (1 + k n ( ω )) j n ( ω, ω )1 + i n ( ω ) + (1 − k n ( ω )) j n ( ω, ω ) , (S131)with i n ( z ) = I n +1 / ( z/ I n − / ( z/ , k n ( z ) = K n +1 / ( z/ K n − / ( z/ ,j n ( z, z ) = K n − / ( z/ K n +1 / ( z / I n +1 / ( z / I n − / ( z/ . (S132)As in the 3D case, we consider first the solution in the particular case when r = 0. One may readily observe thathere w n (0) = 0, which implies that (S130) attains a simpler form g n ( R ) Rg (cid:48) n ( R ) = 1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) , (S133)0 −2 −1 n g n ( R ) / ( R g ’ n ( R )) U = 2U = 1U = −1U = −2 FIG. S5: The ratio g n ( R ) / ( Rg (cid:48) n ( R )) vs the order n of the radial function for several values of U , with r = 0 . R = 1.Comparison of the exact result in (S130) (symbols) and the approximate expression in (S53) (lines). Thin solid line is the 1 /n asymptotics (solution for U ≡ which is again just the first factor in (S130).Turning to the limit n → ∞ , we find that the first factor in (S130) obeys1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) = 1 n − ω n + 2 ω + ω n + O (cid:18) n (cid:19) . (S134)Further, considering the second and the third factors on the right-hand-side of (S130) we use a similar analysis asin the 3D case to find that their deviation from unity is exponentially small. This yields the following result for thebehaviour of g n ( R ) / ( Rg (cid:48) n ( R )) in the limit n → ∞ : g n ( R ) Rg (cid:48) n ( R ) = 1 n − ω n + 2 ω + ω n + O (cid:18) n (cid:19) . (S135)Recalling the definition of ω and noting that for the triangular-well potential U (cid:48)(cid:48) ( R ) = 0, we again observe a perfectagreement between our expansion in (S19) and the exact result in (S135). We note as well that similarly to the3D case, it appears that the large- n behaviour is dominated by the solution with r = 0, which implies that thedependence on this parameter of the interaction potential is fully taken into account by the parameter ω .Lastly, we compare the approximate expression (S53) for g n ( R ) / ( Rg (cid:48) n ( R )) and the exact result in (S130) obtainedfor the triangular-well potential. We observe in Fig. S5 a fairly good agreement between the approximate formula(S53) and the exact result already for even smaller than in the 3D case values of n . The agreement becomes evenbetter for larger values of R ||
In two dimensions, the solutions of (7) for a triangular-well potential read u n ( z ) = z n M ( n, n + 1 , z ) ,v n ( z ) = z − n U ( − n, − n + 1 , z ) . (S126)Using the identities zu (cid:48) n ( z ) = nz n M ( n + 1 , n + 1 , z ) ,zv (cid:48) n ( z ) = − n z − n U ( − n + 1 , − n + 1 , z ) , (S127)one gets zu (cid:48) n ( z ) − nu n ( z ) = nz n +1 n + 1 M ( n + 1 , n + 2 , z ) ,zv (cid:48) n ( z ) − nv n ( z ) = − nz − n U ( − n, − n, z ) , (S128)so that w n ( z ) becomes w n ( z ) = zu (cid:48) n ( z ) − nu n ( z ) zv (cid:48) n ( z ) − nv n ( z ) = − M ( n + 1 , n + 2 , z )(2 n + 1) U ( n + 1 , n + 2 , z ) . (S129)Combining these equations we obtain the following explicit expression for the inverse logarithmic derivative of theradial functions in the 2D case with the triangular-well potential: g n ( R ) Rg (cid:48) n ( R ) = 1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) (cid:32) − U ( n, n + 1 , ω ) M ( n, n + 1 , ω ) w n ( ω ) (cid:33) × (cid:32) nU ( n + 1 , n + 1 , ω ) M ( n + 1 , n + 1 , ω ) w n ( ω ) (cid:33) − . (S130)As earlier in the 3D case, another representation can be obtained using (S107) g n ( R ) Rg (cid:48) n ( R ) = 1 n − i n ( ω ) + (1 + k n ( ω )) j n ( ω, ω )1 + i n ( ω ) + (1 − k n ( ω )) j n ( ω, ω ) , (S131)with i n ( z ) = I n +1 / ( z/ I n − / ( z/ , k n ( z ) = K n +1 / ( z/ K n − / ( z/ ,j n ( z, z ) = K n − / ( z/ K n +1 / ( z / I n +1 / ( z / I n − / ( z/ . (S132)As in the 3D case, we consider first the solution in the particular case when r = 0. One may readily observe thathere w n (0) = 0, which implies that (S130) attains a simpler form g n ( R ) Rg (cid:48) n ( R ) = 1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) , (S133)0 −2 −1 n g n ( R ) / ( R g ’ n ( R )) U = 2U = 1U = −1U = −2 FIG. S5: The ratio g n ( R ) / ( Rg (cid:48) n ( R )) vs the order n of the radial function for several values of U , with r = 0 . R = 1.Comparison of the exact result in (S130) (symbols) and the approximate expression in (S53) (lines). Thin solid line is the 1 /n asymptotics (solution for U ≡ which is again just the first factor in (S130).Turning to the limit n → ∞ , we find that the first factor in (S130) obeys1 n M ( n, n + 1 , ω ) M ( n + 1 , n + 1 , ω ) = 1 n − ω n + 2 ω + ω n + O (cid:18) n (cid:19) . (S134)Further, considering the second and the third factors on the right-hand-side of (S130) we use a similar analysis asin the 3D case to find that their deviation from unity is exponentially small. This yields the following result for thebehaviour of g n ( R ) / ( Rg (cid:48) n ( R )) in the limit n → ∞ : g n ( R ) Rg (cid:48) n ( R ) = 1 n − ω n + 2 ω + ω n + O (cid:18) n (cid:19) . (S135)Recalling the definition of ω and noting that for the triangular-well potential U (cid:48)(cid:48) ( R ) = 0, we again observe a perfectagreement between our expansion in (S19) and the exact result in (S135). We note as well that similarly to the3D case, it appears that the large- n behaviour is dominated by the solution with r = 0, which implies that thedependence on this parameter of the interaction potential is fully taken into account by the parameter ω .Lastly, we compare the approximate expression (S53) for g n ( R ) / ( Rg (cid:48) n ( R )) and the exact result in (S130) obtainedfor the triangular-well potential. We observe in Fig. S5 a fairly good agreement between the approximate formula(S53) and the exact result already for even smaller than in the 3D case values of n . The agreement becomes evenbetter for larger values of R || U (cid:48) ( R ) ||