Resetting transition is governed by an interplay between thermal and potential energy
RResetting transition is governed by an interplay between thermal andpotential energy
Somrita Ray
1, 2, a) and Shlomi Reuveni b) School of Chemistry, The Center for Physics and Chemistry of Living Systems,The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science,& The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978,Israel Department of Chemistry, Indian Institute of Technology Tirupati, Tirupati 517506,India (Dated: 1 March 2021)
A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated orhindered due to resetting. Tuning system parameters such as temperature, viscosity or concentration, can invertthe effect of resetting on the mean completion time of the process, which leads to a resetting transition. Thoughthe resetting transition was recently studied for diffusion in a handful of model potentials, it is yet unknownwhether the results follow any universality in terms of well-defined physical parameters. To bridge this gap,we propose a general framework which reveals that the resetting transition is governed by an interplay betweenthermal and potential energy. This result is illustrated for different classes of potentials that are used to model awide variety of stochastic processes with numerous applications.PACS numbers: 05.40.-a,05.40.JcResetting refers to a situation where an ongoing dynam-ical process is stopped in its midst and started over .Theoretical study of stochastic dynamics with resetting hasdrawn overwhelming attention in recent years and ap-plications to problems in chemical and biological physics have further amplified activity in this rapidlyemerging field. In addition, experimental realizations ofdiffusion with resetting have been successful very recently,where optical tweezers or laser traps were used to resetthe position of a colloidal particle. These studies openedup further possibilities to explore the dynamics of physicalsystems with stochastic resetting.To illustrate resetting, consider a chemical reaction R → P that owns an energy profile with two minima, where thereactant ( R ) and product ( P ) states are located. This sys-tem can be modeled as diffusion in a double-well potential,where the reaction coordinate is mapped onto the positionof a diffusing particle (Fig. 1). The time to complete thereaction is then given by the random time it takes the par-ticle to get from one minimum of the potential to the other,across a separating energy barrier. This time is commonlyknown as the first-passage time .Chemical reactions, such as the one described in Fig. 1,are naturally subject to stochastic resetting. Resetting hap-pens e.g., when two species that bind to form an acti-vated complex, unbind without actually reacting; but onlyto rebind again at some later time. In a similar way,unbinding acts to reset enzymatic reactions and a one-on-one mapping between first-passage with resetting andthe Michaelis-Menten reaction scheme has recently beenestablished . Chemical reactions can also be reset bymeans of external manipulation, e.g., an electromagneticpulse can bring the reactant to its initial state [( R ) in Fig. 1]. a) Electronic mail: [email protected] b) Electronic mail: [email protected] E n e r gy 𝑥 𝑥 𝑎 𝑥 R P Reaction coordinate
FIG. 1. Schematic illustration of a chemical reaction R → P envi-sioned as diffusion of a Brownian particle in a double well poten-tial. The completion time of the reaction is given by the particle’sfirst-passage time from x (state R ) to x a (state P ). Stochastic re-setting of the reaction brings the reaction coordinate back to x at random time epochs. Depending on parameters, resetting caneither increase or decrease the mean time taken to form a product. Importantly, it has been found that resetting can either hin-der or expedite the completion of a chemical reaction (orany other first-passage process), with the net effect deter-mined by various physical parameters such as temperature,viscosity, and the potential energy landscape underlyingthe reaction at hand. Tuning one (or more) of these gov-erning parameters across some critical value can invert theeffect of resetting on the mean completion time of the re-action, which leads to a “resetting transition” .In recent years, the effect of resetting on diffusion invarious model potentials, e.g., linear, harmonic, power-law a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b and logarithmic was thoroughly explored. In each ofthese studies, however, the resetting transition was charac-terized in terms of a different physical governing param-eter that was specific to the system under consideration.It is thus still unclear whether one can present a unifyingphysical understanding of the resetting transition that ap-plies to diffusion in arbitrary potentials. Motivated by thisquestion, we develop a general approach to show that theresetting transition can be understood in terms of an inter-play between two competing energy scales, viz., the poten-tial energy and the thermal energy. Our results are generaland we illustrate them with three different classes of po-tentials that are used to model a wide variety of stochasticprocesses with numerous applications.We start with a first-passage process, e.g., the chemicalreaction shown in Fig.1, but this time we consider an ar-bitrary energy profile U ( x ) to keep things general. Thisscenario can be modeled with a particle undergoing diffu-sion with a diffusion coefficient D in an arbitrary potential U ( x ) . Letting p ( x , t | x ) denote the conditional probabilitydensity of finding the particle at a position x at time t , pro-vided its initial position was x >
0, we write the backwardFokker Planck equation for the process ∂ p ( x , t | x ) ∂ t = (cid:20) − U (cid:48) ( x ) ζ (cid:21) ∂ p ( x , t | x ) ∂ x + D ∂ p ( x , t | x ) ∂ x , (1)where ζ denotes the friction coefficient and U (cid:48) ( x ) ≡ dU ( x ) / dx . The process ends when the particle hits an ab-sorbing boundary, which is placed on the non-negative axisto the right side of the origin at 0 ≤ x a ≤ x . When thishappens, the particle is removed from the system whichleads to the boundary condition p ( x a , t | x ) =
0. The sur-vival probability, i.e., the probability of finding the particlewithin the interval [ x a , ∞ ) at time t , is given by Q ( t | x ) = (cid:82) ∞ x a p ( x , t | x ) dx . Integrating Eq. (1) with respect to x over [ x a , ∞ ) thus gives ∂ Q ( t | x ) ∂ t = (cid:20) − U (cid:48) ( x ) ζ (cid:21) ∂ Q ( t | x ) ∂ x + D ∂ Q ( t | x ) ∂ x . (2)Here the initial condition is Q ( | x ) = Q ( t | x a ) =
0. Letting T denote the first-passagetime (FPT) to the absorbing boundary placed at x a , we re-call that the probability density function of this randomvariable is given by − ∂ Q ( t | x ) / ∂ t . Thus the n -th mo-ment of the FPT is τ n = (cid:104) T n (cid:105) : = − (cid:82) ∞ dt t n (cid:104) ∂ Q ( t | x ) ∂ t (cid:105) = n (cid:82) ∞ dt t n − Q ( t | x ) , where the second equality comesfrom integrating by parts and further demanding that Q ( t | x ) | t → ∞ =
0. Multiplying Eq. (2) by t n − and inte-grating over [ , ∞ ] with respect to t we get − n τ n − = − (cid:20) U (cid:48) ( x ) ζ (cid:21) d τ n dx + D d τ n dx , (3)where we utilize the previous identities to identify themoments.Eq. (3) is a single-variable recursive differential equa-tion in τ n . Given τ n − , it can be solved with appropriateboundary conditions . This method allows us tocalculate the moments of the FPT directly, bypassing thecalculation of the survival probability Q ( t | x ) . Taking intoaccount the fact that by definition τ =
1, we solve Eq. (3)for n = τ = D − (cid:82) x x a dy e β U ( y ) (cid:82) ∞ y dz e − β U ( z ) [see the SupportingInformation for the derivation], where we utilized the Ein-stein relation, D = ( β ζ ) − . In a similar spirit, setting n = τ into it, weget a differential equation in τ , whose solution reads τ = D − (cid:82) x x a dw e β U ( w ) (cid:82) ∞ w dv e − β U ( v ) (cid:82) vx a dy e β U ( y ) (cid:82) ∞ y dz e − β U ( z ) [see the Supporting Information]. In what follows, wewill discuss how τ and τ together can predict the effectof resetting, and relate the latter to physical governingparameters.The theory of first-passage with resetting allows us toexpress the FPT of a process with resetting in terms ofthe FPT of the same process without resetting . Theeffect of resetting can thus be predicted based on the FPTof the underlying process. In particular, it was shownthat the introduction of stochastic resetting acceleratesfirst-passage whenever the standard deviation of the FPTis greater than its mean, i.e, (cid:113) τ − τ > τ . In starkcontrast, first-passage is delayed due to the introduction ofstochastic resetting whenever (cid:113) τ − τ < τ . In real-lifesystems, the FPT distribution and hence its moments τ and τ will vary when the governing parameters arealtered. This indicates that by tuning physical parameterslike temperature, the effect of resetting on the FPT canbe inverted. The associated transition, i.e., the resettingtransition occurs when (cid:113) τ − τ = τ , which isequivalent to τ = τ . Note that if either the mean or thestandard deviation of the FPT diverges, the introductionof stochastic resetting always accelerates first-passage.Thus, we exclude these cases from the discussion belowand focus only on situations where the mean and secondmoment of the FPT are finite.Utilizing the expressions for τ and τ above, we seethat the resetting transition of diffusion in a potentiallandscape U ( x ) occurs when (cid:90) x x a dw e + β U ( w ) (cid:90) ∞ w dv e − β U ( v ) (cid:90) vx a dy e + β U ( y ) (cid:90) ∞ y dz e − β U ( z ) = (cid:20) (cid:90) x x a dy e + β U ( y ) (cid:90) ∞ y dz e − β U ( z ) (cid:21) . (4)We will now show that Eq. (4) can be used to character- ize the resetting transition in terms of an interplay between � � =- ��� � � =- ��� � � = ��� � � = ��� � � = ��� - � - � - � - � � � � � � - � - ���� � � � ��� | � | ( � ) � � / � � = ��� � � / � � = ��� � � / � � = ��� � � / � � = ��� � � / � � = ��� � � � �������� ν τ � / � τ �� ( � ) � ��� ��� ��� ������ � � / � � ν ( � ) FIG. 2. Panel (a): A logarithmic potential U ( x ) = U log | x | for different values of U . U ( x ) is repulsive for U < U >
0. The potential has a central singularity at x = x a > x a / x . The solutions, denoted ν (cid:63) , are the abscissa corresponding to thecolored circles. Panel (c): Phase diagram showing the effect of resetting on first-passage. The white region marks the part of the phasespace where the introduction of resetting expedites first-passage, while the shaded region marks the opposite. The black line, obtainedby plotting ν (cid:63) vs. x a / x using Eq. (6), separates the two phases and indicates the points of the resetting transition for any given valueof x a / x . Circles of the same colors show same values of x a / x as in panel (b). the thermal energy β − and the potential energy. We dothis by considering diffusion in various canonical poten-tials, starting with the logarithmic potential.Diffusion in a logarithmic potential is a popular choiceto model stochastic phenomena such as the denaturationof double-stranded DNA by bubble formation , interac-tions of colloids and polymers with walls of narrow chan-nels and pores , and spreading of momenta of coldatoms trapped in optical lattices . A simple log poten-tial of the form U ( x ) = U log | x | (where U is the strengthof the potential) owns a singularity at x = U ( x ) = U log | x | is to place the absorbing boundary at aposition x a > , we comprehensively analysed the ef-fect of resetting on diffusion in a log potential (commonlyknown as the Bessel process ) for the special case of x a =
0, i.e, where the absorbing boundary is placed at theorigin. We demonstrated that the system displays differentdynamical behaviors as the dimensionless parameter β U ,the strength of the potential in units of the thermal energy,is tuned. In particular, we proved that resetting expeditesthe first-passage to the origin when β U <
5, but delaysthe same when β U >
5, leading to a resetting transitionat β U =
5. The present framework allows us to gener-alize this result to a regularized log potential, for which0 < x a < x .Setting U ( x ) = U log | x | in Eq. (4), we get e ± β U ( x ) = | x | ± β U , which allows us to analytically evaluate the inte-grals. Doing so, we obtain τ τ = (cid:20) − x a + x ( x a − x )( ν − ) (cid:21) = , (5)where ν : = ( + β U ) / β − and U . In Fig. 2(b), we graphicallysolve Eq. (5) by plotting its left hand side vs. ν . The solu- tions, denoted ν (cid:63) , are functions of the ratio x a / x and aregiven by ν (cid:63) = + (cid:20) + ( x a / x ) − ( x a / x ) (cid:21) . (6)Thus ν (cid:63) is the critical value of ν at which the resettingtransition occurs. Equation (6) clearly indicates that in thelimit x a →
0, the resetting transition occurs at ν (cid:63) = β U = .In Fig. 2(c), we plot ν (cid:63) as a function of x a / x (blackline), which separates the phase space in two regions withrespect to the effect of resetting. Recalling the definitionof ν , we see that ν is large for large values of β U > x thus delays its first-passage by interrupting that driven motion (shaded regionin Fig. 2(c)). In contrast, ν is small for small values of β U >
0, i.e, when the drive towards the origin is weak.In that case, thermal diffusion predominates and introduc-tion of resetting expedites the particle’s first-passage byeffectively reducing the possibilities of it diffusing awayfrom the origin (white region in Fig. 2(c)). The black linegiven by Eq. (6) characterizes the condition for resettingtransition by identifying the precise separatrix between theabove mentioned phases, one where potential energy dom-inates and the other where thermal energy dominates. Inthe present example, we see that the separatrix depends onthe position of the absorbing boundary relative to the par-ticle’s initial position.The logarithmic potential serves as an ideal example toillustrate the present framework, since in this case we canobtain an exact analytical expression of the critical valueof ν , which governs the resetting transition. For most ofthe commonly studied potentials, however, this is not thecase. In what follows, we will discuss a broad class ofpotentials that increase monotonically with x . In general,for such a choice, the integrals given in Eq. (4) can not beevaluated analytically and one needs to utilize numericalsolutions instead. And yet, we show that here too an inter-play between the thermal and the potential energy governs α = ��� α = ��� α = ��� α = ��� α = ��� - � - � � � ���������������������� � � � | � α ( � ) α = α = α = α = α = ���� ��� ��� � � �������������� ϕ � τ � / � τ �� ( � ) ��� ��� � � ���������� α ϕ � ( � ) FIG. 3. Panel (a): A power-law potential U ( x ) = U | x | α for different values of α , and U =
1. Panel (b): Graphical solution ofEq. (8) by plotting the ratio τ / τ (from the left hand side of Eq. (8)) vs. φ for different values of α . The solutions, φ (cid:63) (abscissacorresponding to the colored circles) correspond to the points where the resetting transition occurs. Panel (c): A phase diagram thatshows the effects of resetting on the first-passage to the origin. The white region indicates the parameter space where the introductionof resetting accelerates first-passage. The shaded region displays the parameter space where resetting delays the first-passage. Thepoints of the resetting transition are presented by plotting the numerical solutions of Eq. (8)) as a function of α (black line). Circles ofthe same color display the same values of α as in panel (b) . the resetting transition.Consider a potential U ( x ) = U f ( x ) , where U is thestrength of the potential as before and f ( x ) is a monoton-ically increasing function within the interval x a ≤ x < ∞ . Taking x a =
0, i.e, placing the absorbing boundary at theorigin to simplify the analysis, we perform a general trans-formation of variable as φ ≡ φ ( x ) : = β U f ( x ) , so that φ (cid:48) ≡ d φ dx = β U d f ( x ) dx . Eq. (4) can then be written as (cid:90) φ φ a d φ ( w ) e φ ( w ) φ (cid:48) ( w ) (cid:90) φ ∞ φ ( w ) d φ ( v ) e − φ ( v ) φ (cid:48) ( v ) (cid:90) φ ( v ) φ a d φ ( y ) e φ ( y ) φ (cid:48) ( y ) (cid:90) φ ∞ φ ( y ) d φ ( z ) e − φ ( z ) φ (cid:48) ( z ) = (cid:34) (cid:90) φ φ a d φ ( y ) e φ ( y ) φ (cid:48) ( y ) (cid:90) φ ∞ φ ( y ) d φ ( z ) e − φ ( z ) φ (cid:48) ( z ) (cid:35) , (7)where φ : = φ ( x ) ≡ β U f ( x ) , φ a : = φ ( x a = ) ≡ β U f ( x a = ) and φ ∞ : = φ ( x → ∞ ) ≡ β U f ( x → ∞ ) . Notethat in Eq. (7), φ ( i ) for i = w , v , y , z are integration vari-ables, and the integrals depend only on the dimensionlessparameters φ , φ a and φ ∞ . Next, we utilize this to inves-tigate the important case of power-law potentials, whichfurther showcase the interplay between thermal and poten-tial energy in the resetting transition.Consider a set of power-law potentials, U ( x ) = U f ( x ) with f ( x ) = | x | α and α > φ i ≡ φ ( i ) for i = w , v , y and z wesimplify Eq. (7) to obtain [see Supplemental Material fordetails] τ τ = (cid:82) φ d φ w e + φ w φ − α w (cid:82) ∞ φ w d φ v e − φ v φ − α v (cid:82) φ v d φ y e + φ y φ − α y (cid:82) ∞ φ y d φ z e − φ z φ − α z (cid:34) (cid:82) φ d φ y e + φ y φ − α y (cid:82) ∞ φ y d φ z e − φ z φ − α z (cid:35) = , (8)where φ = β U | x | α is the dimensionless potential energyof the particle at its initial position x .For α =
1, evaluating the integrals in Eq. (8) we obtain φ (cid:16) + φ (cid:17) φ = . (9) The non-trivial solution of Eq. (9) is φ (cid:63) =
2. Recalling thedefinition of the P´eclet number, Pe : = x ( U ζ − ) / D = x β U /
2, we see that the condition φ (cid:63) : = [ β U x ] (cid:63) = Pe =
1, which agrees with previouswork .For α (cid:54) =
1, the integrals in Eq. (8) can not be evaluatedanalytically. This, instead, we numerically evaluate for dif-ferent values of α and plot the left hand side of Eq. (8) inFig. 3(b), to present graphical solutions for this equation.The solutions, denoted φ (cid:63) , correspond to the resetting tran-sition in each case. Fig. 3(c) displays a phase diagram thatpresents the effect of resetting on first-passage. For largervalues of φ (shaded region), the potential strongly drivesthe particle towards the origin. Resetting interrupts suchmotion and thereby delays first-passage to the origin. Incontrast, for smaller values of φ (white region), the poten-tial is weak and the thermal energy dominates. Introduc-tion of resetting thus accelerates first-passage to the origin.These two phases are separated by the points of the reset-ting transition, φ (cid:63) , which varies with α (black line).Note that the points of the resetting transition above, i.e., τ = τ , depend only on φ , which is dimensionless bydefinition. Moreover, since U ( ) : = U f ( ) =
0, we seethat the transition depends solely on the energy releasedby the particle as it moves from it initial position x > β − . Comparing φ with the P´eclet α = α = α = α = α = ( a ) - � - � � � � - ��������� � � � [( � α ) - � | � � α - α - � | � α ] α = α = α = α = α = α = ( b ) ���� ��� � � � ������������������ β � � τ � / � τ �� ( c ) ��� � ��� � � ��������������� α β � � FIG. 4. Panel (a): Double-well potentials U ( x ) = U (cid:2) ( α ) − | x | α − α − | x | α (cid:3) for U = α . U ( x ) owns minimaat x = ± x =
0. Panel (b): Graphical solution of Eq. (10) by plotting τ / τ from its left hand side vs. β U ,for different values of α . The solutions, [ β U ] (cid:63) (abscissa corresponding to the colored circles), denote the strength of the potential atthe point of the resetting transition in units of the thermal energy. Panel (c): A phase diagram that exhibits the effects of resetting onbarrier crossing. The white region indicates the parameter space where the introduction of resetting accelerates barrier crossing. Theshaded region displays the parameter space where resetting delays the same. The points of the resetting transition are presented by thenumerical solutions of Eq. (10) as a function of α (black line). Circles of the same color display the same value of α as in panel (b). number we see that φ can be considered as a general-ized form of Pe for a monotonically increasing (non-linear)potential of the form U ( x ) = U f ( x ) , which vanishes at theorigin. Having thoroughly analysed this case, we now pro-ceed to discuss the application of our framework to a set ofnon-monotonic potentials, viz., double-well potentials, asthe final example of the present study.Consider a class of potentials U ( x ) = U (cid:16) | x | α α − | x | α α (cid:17) ,where U denotes the strength of the potential as beforeand α > U ( x ) owns a couple of minima at x = ± α , which is why one commonly addresses thisset of potentials as double-well potentials. This class ofnon-monotonic potentials are frequently used in the con-text of rate theory and barrier crossing dynamics . Inwhat follows, we will explore the effect of resetting onbarrier crossing by considering the first-passage of the dif-fusing particle from the minimum at x = x a = φ ( x ) : = β U ( x ) ≡ β U (cid:16) | x | α α − | x | α α (cid:17) andutilizing Eq. (4), we see that the condition for resettingtransition translates to τ τ = (cid:82) dw e φ ( w ) (cid:82) ∞ w dv e − φ ( v ) (cid:82) v dy e φ ( y ) (cid:82) ∞ y dz e − φ ( z ) (cid:104) (cid:82) dy e φ ( y ) (cid:82) ∞ y dz e − φ ( z ) (cid:105) = . (10)The integrals in Eq. (10) can not be evaluated analytically,however, it is not difficult to see that the results dependonly on α and β U , the strength of the potential in termsof the thermal energy. We numerically calculate the ra-tio τ / τ from the left hand side of Eq. (10) and plot itin Fig. 4(b) to graphically solve the equation for differentvalues of α . In Fig. 4(c), we construct a phase diagramby plotting the solutions, denoted [ β U ] (cid:63) vs. α (blackline). The area under the curve (white region) shows theparameter space where resetting expedites barrier crossing(smaller values of β U ), whereas the area above the curve(shaded region) marks the parameter space where reset-ting hinders the same (larger values of β U ). Therefore, our analysis of barrier crossing in the double-well potentialproves that even when the underlying system is not exactlysolvable, the general framework outlined in the present pa-per can be utilized to locate the point of the resetting transi-tion based on an interplay of two competing energy scales,the thermal and the potential energy.In conclusion, we presented a general framework to de-termine the resetting transition point of diffusion in a po-tential. The framework applies to diffusion in arbitrary po-tentials, and it was illustrated with three different show-case potentials, viz., a regularized logarithmic potential, aset of power-law potentials of varying exponents and a setof double-well potentials of varying barrier heights, whichare widely used to model a variety of first-passage pro-cesses. In most of the cases involving diffusion in a poten-tial landscape with resetting, solution of the Fokker Planckequation can be quite challenging. There, instead, ourmethodology can be adapted to identify the resetting tran-sition point. More importantly, here we showed that theresetting transition can be understood in terms of an inter-play between the thermal and the potential energies. Theseinterpretations will strengthen the connection between the-ory and experiment in the field of stochastic resetting bypaving new ways of analyzing experimental data and bygenerating new experimentally verifiable predictions. Acknowledgements:
S. Ray acknowledges support fromthe Raymond and Beverly Sackler Center for Computa-tional Molecular and Materials Science, Tel Aviv Univer-sity, Israel and the DST-INSPIRE Faculty Grant, Govt. ofIndia, executed at Indian Institute of Technology Tirupati(project no. CHY/2021/005/DSTX/SOMR). S. Reuveniacknowledges support from the Azrieli Foundation, fromthe Raymond and Beverly Sackler Center for Computa-tional Molecular and Materials Science at Tel Aviv Uni-versity, and from the Israel Science Foundation (grant No.394/19).
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A. Derivation of the exact expressions of the first and second moments ( τ and τ ) of the first-passage time fromEq.(3): Eq.(3) in the main text is a single variable, recursive, second-order differential equation in τ n , the nth moment of FPT.Utilizing the Einstein relation D = ( β ζ ) − in Eq.(3), where β is the thermodynamic beta , we get d τ n dx − β U (cid:48) ( x ) d τ n dx + (cid:104) nD (cid:105) τ n − = , (S.1)Setting n = τ : = − (cid:82) ∞ dt ∂ Q ( t | x ) ∂ t ≡ Q ( t | x ) | t → − Q ( t | x ) | t → ∞ = τ can be reduced to a first order differential equation in τ (cid:48) ≡ d τ / dx that reads d τ (cid:48) dx − β U (cid:48) ( x ) τ (cid:48) + D = . (S.2)With an integrating factor e − (cid:82) dx β U (cid:48) ( x ) ≡ e − β U ( x ) , Eq. (S.2) leads to the general solution τ (cid:48) = D e + β U ( x ) (cid:20) C − (cid:90) x x a dz e − β U ( z ) (cid:21) , (S.3)where C is the arbitrary integration constant. Putting the boundary condition lim x → ∞ e − β U ( x ) τ (cid:48) ( x ) = C = (cid:82) ∞ x a dz e − β U ( z ) , and that in turn leads to the specific solution of Eq. (S.2) τ (cid:48) = D e + β U ( x ) (cid:90) ∞ x dz e − β U ( z ) . (S.4)Recalling that x a < x , we now integrate Eq. (S.4) within the interval [ x a , x ] and utilize the boundary condition τ ( x a ) = τ = D (cid:90) x x a dy e + β U ( y ) (cid:90) ∞ y dz e − β U ( z ) . (S.5)Eq. (S.5) presents an explicit expression for the mean FPT of a particle that diffuses in a potential U ( x ) from its initialposition x to the absorbing boundary at x a .In a similar spirit, we can obtain an exact expression for τ . For this, we set n = τ d τ dx − β U (cid:48) ( x ) d τ dx = − D (cid:20) (cid:90) x x a dy e + β U ( y ) (cid:90) ∞ y dz e − β U ( z ) (cid:21) , (S.6)where we utilized Eq. (S.5) to substitute for τ . Solving for τ in the exact manner as before, we get τ = D (cid:90) x x a dw e + β U ( w ) (cid:90) ∞ w dv e − β U ( v ) (cid:90) wx a dy e + β U ( y ) (cid:90) ∞ y dz e − β U ( z ) . (S.7)Eq. (S.7) presents an explicit expression for the second moment of the FPT. B. Transformation of variable x → φ ( x ) ≡ β U ( x ) for the power-law potential U ( x ) = U | x | α : Derivation of Eq. (8): As discussed in the main text before Eq. (7), we consider a general transformation of variable x → φ ( x ) : = β U ( x ) ,which leads to φ ( x ) = β U | x | α for the power-law potential. Therefore, for 0 ≤ x < ∞ we get x ≡ (cid:20) φ ( x ) β U (cid:21) α . (S.8)Differentiating Eq. (S.8) with respect to φ ( x ) thus gives dx ≡ α ( β U ) α (cid:34) d φ ( x ) φ ( x ) − α (cid:35) . (S.9)Setting x a =
0, we can write Eq. (S.5) in terms of the transformed variable as τ = D α ( β U ) α (cid:90) φ ( x ) d φ ( y ) e + φ ( y ) φ ( y ) − α (cid:90) ∞ φ ( y ) d φ ( z ) e − φ ( z ) φ ( z ) − α . (S.10)Note that φ ( y ) and φ ( z ) are integration variables in Eq. (S.10), and hence the value of the above double integral dependsonly on φ ≡ φ ( x ) = β U | x | α . To simplify notation, we use φ z = φ ( z ) and φ y = φ ( y ) hereafter. In a similar spirit as inthe case of the mean FPT, Eq. (S.7) in terms of the transformed variable reads τ = D α ( β U ) α (cid:90) φ d φ w e + φ w φ − α w (cid:90) ∞ φ w d φ v e − φ v φ − α v (cid:90) φ v d φ y e + φ y φ − α y (cid:90) ∞ φ y d φ z e − φ z φ − α z , (S.11)where φ w = φ ( w ) and φ v = φ ( v ) . From Eqs. (S.10) and (S.11), the condition for resetting transition, τ / τ ==