A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated or hindered due to resetting. Tuning system parameters, such as temperature, viscosity, or concentration, can invert the effect of resetting on the mean completion time of the process, which leads to a resetting transition. Although the resetting transition has been recently studied for diffusion in a handful of model potentials, it is yet unknown whether the results follow any universality in terms of well-defined physical parameters. To bridge this gap, we propose a general framework that reveals that the resetting transition is governed by an interplay between the thermal and potential energy. This result is illustrated for different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications.

Resetting refers to a situation where an ongoing dynamical process is stopped in its midst and started over.^{1–7} The theoretical study of stochastic dynamics with resetting has drawn overwhelming attention in recent years,^{8–37} and applications to problems in chemical^{38–42} and biological^{43–49} physics have further amplified activity in this emerging field. In addition, experimental realizations of diffusion with resetting have been successful very recently, where optical tweezers^{50} or laser traps^{51} were used to reset the position of a colloidal particle. These studies opened up further possibilities to explore the dynamics of physical systems with stochastic resetting.

To illustrate resetting, consider a chemical reaction *R* → *P* that owns an energy profile with two minima, where the reactant (*R*) and product (*P*) states are located. This system can be modeled as diffusion in a double-well potential, where the reaction coordinate is mapped onto the position of a diffusing particle (Fig. 1). The time to complete the reaction is then given by the random time taken by the particle to move from one minimum of the potential to the other across a separating energy barrier. This time is commonly known as the *first-passage time* (*FPT*).^{52–57}

Chemical reactions, such as the one described in Fig. 1, are naturally subject to stochastic resetting. Resetting happens when two species that bind to form an activated complex unbind without actually reacting but only to 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 and the Michaelis–Menten reaction scheme has recently been established.^{38–41} Chemical reactions can also be reset by means of external manipulation, e.g., an electromagnetic pulse can bring the reactant to its initial state [*R* in Fig. 1]. Importantly, it has been found that resetting can either hinder or expedite the completion of a chemical reaction (or any other first-passage process), with the net effect determined by various physical parameters, such as temperature, viscosity, and the potential energy landscape underlying the reaction at hand. Tuning one (or more) of these governing parameters across some critical value can invert the effect of resetting on the mean completion time of the reaction, which leads to a “resetting transition.”^{38,39,58,59}

In recent years, the effect of resetting on diffusion in various model potentials, e.g., linear, harmonic, power-law, and logarithmic potentials,^{59–63} has been thoroughly explored. In each of these studies, however, the resetting transition was characterized in terms of a different physical governing parameter that was specific to the system under consideration. It is thus still unclear whether one can present a unifying physical understanding of the resetting transition that applies to diffusion in arbitrary potentials. Motivated by this question, we develop a general approach to show that the resetting transition can be understood in terms of an interplay between two competing energy scales, viz., the potential energy and the thermal energy. Our results are general, and we illustrate them with three different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications.

We start with a first-passage process, e.g., the chemical reaction shown in Fig. 1, but this time, we consider an arbitrary energy profile *U*(*x*) to keep things general. This scenario can be modeled with a particle undergoing diffusion with a diffusion coefficient *D* in an arbitrary potential *U*(*x*). Letting *p*(*x*, *t*|*x*_{0}) denote the conditional probability density of finding the particle at a position *x* at time *t*, provided that its initial position was *x*_{0} > 0, we write the backward Fokker–Planck equation for the process,^{52–54}

where *ζ* denotes the friction coefficient and *U*′(*x*) ≡ *dU*(*x*)/*dx*. Note that the forward and backward Fokker–Planck descriptions are equivalent here since *D* and *U*(*x*) are taken to be time-independent.

The process ends when the particle hits an absorbing boundary, which is placed on the non-negative axis at 0 ≤ *x*_{a} ≤ *x*_{0}. When this happens, the particle is removed from the system, which leads to the boundary condition *p*(*x*_{a}, *t*|*x*_{0}) = 0. The survival probability, i.e., the probability of finding the particle within the interval [*x*_{a}, ∞) at time *t*, is given by $Q(t|x0)=\u222bxa\u221ep(x,t|x0)dx$. Integrating Eq. (1) with respect to *x* over [*x*_{a}, ∞) thus gives

Here, the initial condition is *Q*(0|*x*_{0}) = 1, and the boundary condition is *Q*(*t*|*x*_{a}) = 0. Letting *T* denote the first-passage time (FPT) to the absorbing boundary placed at *x*_{a}, we recall that the probability density function of this random variable^{52–54} is given by −*∂Q*(*t*|*x*_{0})/*∂t*. Thus, the *n*th moment of the FPT is $\tau n=Tn:=\u2212\u222b0\u221edt\u2009tn\u2202Q(t|x0)\u2202t=n\u222b0\u221edt\u2009tn\u22121Q(t|x0)$, where the second equality comes from integrating by parts and further demanding that $Q(t|x0)t\u2192\u221e=0$. Multiplying Eq. (2) by *t*^{n−1} and integrating over [0, ∞) with respect to *t*, we get^{52,53}

where we utilized the previous identities to identify the moments.

Equation (3) is a single-variable recursive differential equation in *τ*_{n}. Given *τ*_{n−1}, it can be solved with appropriate boundary conditions.^{52,53} This method allows us to calculate the moments of the FPT directly, bypassing the calculation of the survival probability *Q*(*t*|*x*_{0}). Taking into account the fact that by definition *τ*_{0} = 1, we solve Eq. (3) for *n* = 1 to obtain an expression for the mean FPT,^{52} $\tau 1=D\u22121\u222bxax0dy\u2009e\beta U(y)\u222by\u221edz\u2009e\u2212\beta U(z)$ (see Appendix A for the derivation), where we utilized the Einstein relation *D* = (*βζ*)^{−1}. In a similar spirit, setting *n* = 2 in Eq. (3) and plugging in the expression of *τ*_{1} into it, we get a differential equation for *τ*_{2}, whose solution reads $\tau 2=2D\u22122\u222bxax0dw\u2009e\beta U(w)\u222bw\u221edv\u2009e\u2212\beta U(v)\u222bxavdy\u2009e\beta U(y)\u222by\u221edz\u2009e\u2212\beta U(z)$ (see Appendix A). In what follows, we will discuss how *τ*_{1} and *τ*_{2} together can predict the effect of resetting and relate the latter to physical governing parameters.

The theory of first-passage with resetting allows us to express the FPT of a process with resetting in terms of the FPT of the same process without resetting.^{4,5,38} Indeed, a straightforward relation between the two was proved in Ref. 5 for Poissonian resetting with a constant rate *r*, which implies that the time between two consecutive resetting events is taken from an exponential distribution with mean *r*^{−1}. The effect of resetting can thus be predicted based on the FPT of the underlying process. In particular, it was shown that the introduction of a small resetting rate accelerates first-passage whenever the standard deviation of the FPT is greater than its mean, i.e., $\tau 2\u2212\tau 12>\tau 1$. In stark contrast, first-passage is delayed due to the introduction of stochastic resetting whenever $\tau 2\u2212\tau 12<\tau 1$. Note that if the standard deviation of the FPT diverges, the introduction of stochastic resetting always accelerates first-passage. In addition, it can be shown that the same holds true when the mean FPT diverges.^{5} Thus, we exclude these cases from the discussion below and focus only on situations where the mean and second moment of the FPT are finite.

In real-life systems, the FPT distribution and hence its moments *τ*_{1} and *τ*_{2} will vary when the governing parameters are altered. This indicates that by tuning physical parameters such as temperature, the effect of resetting on the FPT can be inverted. The associated transition, i.e., the resetting transition,^{58–61,63} occurs when $\tau 2\u2212\tau 12=\tau 1$, which is equivalent to $\tau 2=2\tau 12$. Utilizing these expressions for *τ*_{1} and *τ*_{2}, we see that the resetting transition of diffusion in a potential landscape *U*(*x*) occurs when

We will now show that Eq. (4) can be used to characterize the resetting transition in terms of an interplay between the thermal energy *β*^{−1} and the potential energy. We do this by considering diffusion in potentials that are used to model a wide variety of stochastic processes, starting with the logarithmic potential.

Diffusion in a logarithmic potential is a popular choice to model stochastic phenomena such as the denaturation of double-stranded DNA by bubble formation,^{64–67} interactions of colloids and polymers with walls of narrow channels and pores,^{68–71} and spreading of momenta of cold atoms trapped in optical lattices.^{72–74} A simple log potential of the form *U*(*x*) = *U*_{0} log |*x*| (where *U*_{0} is the strength of the potential) owns a singularity at *x* = 0 [Fig. 2(a)]. In many physical applications, however, one frequently encounters versions of the logarithmic potential sans the central singularity, commonly known as “regularized” log potentials. One simple trick to regularize the potential *U*(*x*) = *U*_{0} log |*x*| is to place the absorbing boundary at a position *x*_{a} > 0.

In a recent study,^{61} we have comprehensively analyzed the effect of resetting on diffusion in a log potential (commonly known as the Bessel process^{75,76}) for the special case of *x*_{a} = 0, i.e., where the absorbing boundary is placed at the origin. We have demonstrated that the system displays different dynamical behaviors as the dimensionless parameter *βU*_{0}, the strength of the potential in units of the thermal energy, is tuned. In particular, we have proved that resetting expedites the first-passage to the origin when *βU*_{0} < 5 but delays the same when *βU*_{0} > 5, leading to a resetting transition at *βU*_{0} = 5. The present framework allows us to generalize this result to a regularized log potential, for which 0 < *x*_{a} < *x*_{0}.

Setting *U*(*x*) = *U*_{0} log |*x*| in Eq. (4), we get $e\xb1\beta U(x)=|x|\xb1\beta U0$, which allows us to analytically evaluate the integrals. In doing so, we obtain

where *ν* ≔ (1 + *βU*_{0})/2 is the system parameter that governs the resetting transition by capturing the interplay of the energy scales *β*^{−1} and *U*_{0}. In Fig. 2(b), we illustrate the solution of Eq. (5) by plotting its left hand side vs *ν*. The solutions, denoted as *ν*^{⋆}, are functions of the ratio *x*_{a}/*x*_{0} and are given by

Thus, *ν*^{⋆} is the critical value of *ν* at which the resetting transition occurs. Equation (6) clearly indicates that in the limit *x*_{a} → 0, the resetting transition occurs at *ν*^{⋆} = 3 (i.e., *βU*_{0} = 5), which agrees with our earlier result.^{61}

In Fig. 2(c), we plot *ν*^{⋆} as a function of *x*_{a}/*x*_{0} (black line), which separates the phase space into two regions with respect to the effect of resetting. Recalling the definition of *ν*, we see that *ν* is large for large values of *βU*_{0} > 0. The strongly attractive potential then drives the diffusing particle toward the origin, i.e., in the direction of the absorbing boundary. Resetting to *x*_{0} thus delays its first-passage by interrupting that driven motion [shaded region in Fig. 2(c)]. In contrast, *ν* is small for small values of *βU*_{0} > 0, i.e., when the drive toward the origin is weak. In this case, thermal diffusion predominates and introduction of resetting expedites the particle’s first-passage by effectively reducing the possibilities of it diffusing away from the origin [white region in Fig. 2(c)]. The black line given by Eq. (6) characterizes the condition for the resetting transition by identifying the precise separatrix between the above-mentioned phases, one where the potential energy dominates and the other where the thermal energy dominates. In this example, we see that the separatrix depends on the position of the absorbing boundary relative to the particle’s initial position.

The logarithmic potential serves as an ideal example to illustrate the present framework since in this case, we can obtain an exact analytical expression of the critical value of *ν*, which governs the resetting transition. For most of the commonly studied potentials, however, this is not the case. Next, we discuss a broad class of potentials that increase monotonically with *x* (for *x* ≥ *x*_{a}). In general, for such a choice, the integrals given in Eq. (4) cannot be evaluated analytically and one needs to utilize numerical solutions instead. Yet, we show that here too an interplay between the thermal and the potential energy governs the resetting transition.

Consider a potential *U*(*x*) = *U*_{0}*f*(*x*), where *U*_{0} is the strength of the potential as before and *f*(*x*) is a monotonically increasing function in the interval *x*_{a} ≤ *x* < ∞. Taking *x*_{a} = 0, i.e., placing the absorbing boundary at the origin to simplify the analysis, we perform a general transformation of the variable as *ϕ* ≡ *ϕ*(*x*) ≔ *βU*_{0}*f*(*x*) so that $\varphi \u2032\u2261d\varphi dx=\beta U0df(x)dx$. Equation (4) can then be written as

where *ϕ*_{0} ≔ *ϕ*(*x*_{0}) ≡ *βU*_{0}*f*(*x*_{0}), *ϕ*_{a} ≔ *ϕ*(*x*_{a} = 0) ≡ *βU*_{0}*f*(*x*_{a} = 0), and *ϕ*_{∞} ≔ *ϕ*(*x* → ∞) ≡ *βU*_{0}*f*(*x* → ∞). Note that in Eq. (7), *ϕ*(*i*) for *i* = *w*, *v*, *y*, *z* are integration variables. Thus, the values of the integrals depend only on the dimensionless parameters *ϕ*_{0}, *ϕ*_{a}, and *ϕ*_{∞}, which capture the interplay between the thermal and potential energy and determine the resetting transition point. This general observation is now utilized to investigate the important case of power-law potentials.

Consider a set of power-law potentials *U*(*x*) = *U*_{0}*f*(*x*) with *f*(*x*) = |*x*|^{α} and *α* > 0 [Fig. 3(a)]. For such potentials, using the notations *ϕ*_{i} ≡ *ϕ*(*i*) for *i* = *w*, *v*, *y*, and *z*, we simplify Eq. (7) to obtain (see Appendix B for details)

where *ϕ*_{0} = *βU*_{0}|*x*_{0}|^{α} is the dimensionless potential energy of the particle at its initial position *x*_{0}.

For *α* = 1, evaluating the integrals in Eq. (8), we obtain

The non-trivial solution of Eq. (9), marking the resetting transition point, is $\varphi 0\u22c6=2$. Recalling the definition of the Péclet number, *Pe* ≔ *x*_{0}(*U*_{0}*ζ*^{−1})/2*D* = *x*_{0}*βU*_{0}/2, we see that the condition $\varphi 0\u22c6\u2254[\beta U0x0]\u22c6=2$ is essentially the same as *Pe* = 1, which agrees with previous work.^{60}

For *α* ≠ 1, the integrals in Eq. (8) cannot be evaluated analytically. Thus, instead, we numerically evaluate for different values of *α* and plot the left hand side of Eq. (8) vs *ϕ*_{0} to graphically display the solutions for this equation in Fig. 3(b). The solutions, denoted as $\varphi 0\u22c6$, correspond to the resetting transition in each case. Figure 3(c) displays a phase diagram that presents the effect of resetting on first-passage. For larger values of *ϕ*_{0} (shaded region), the potential strongly drives the particle toward the origin. Resetting interrupts such motion and thereby delays first-passage to the origin. In contrast, for smaller values of *ϕ*_{0} (white region), the potential drive is weaker and the thermal energy dominates. Introduction of resettingthus accelerates first-passage to the origin. These two phases are separated by the points of the resetting transition, $\varphi 0\u22c6$, which vary with *α* (black line).

Note that the above-mentioned points of the resetting transition, i.e., $\tau 2=2\tau 12$, depend only on *ϕ*_{0}, which is dimensionless by definition. Moreover, since *U*(0) ≔ *U*_{0}*f*(0) = 0, we see that the transition point depends solely on the potential energy (in the units of the thermal energy *β*^{−1}) lost by the particle as it moves from its initial position *x*_{0} > 0 to the absorbing boundary at the origin. Comparing *ϕ*_{0} with the Péclet number,^{54,60} we see that *ϕ*_{0} can be considered as a generalized form of *Pe* for a monotonically increasing (non-linear) potential of the form *U*(*x*) = *U*_{0}*f*(*x*), which vanishes at the origin. Having thoroughly analyzed this case, we now proceed to discuss the application of our framework to a set of non-monotonic potentials, viz., double-well potentials, as the final example of this study. Notably, Eq. (7) is not valid for this set of potentials and one must hence go back to Eq. (4). Nevertheless, here too we find that the resetting transition is governed by an interplay between the thermal and potential energy.

Consider a class of potentials $U(x)=U0|x|2\alpha 2\alpha \u2212|x|\alpha \alpha $, where *U*_{0} denotes the strength of the potential as before and *α* > 0 [Fig. 4(a)]. For this particular choice, the potential *U*(*x*) owns a couple of minima at *x* = ±1 irrespective of the value of *α*, which is why one commonly addresses this set of potentials as *double-well* potentials. This class of non-monotonic potentials are frequently used in the context of rate theory and barrier crossing dynamics.^{77,78} In what follows, we will explore the effect of resetting on barrier crossing by considering the first-passage of the diffusing particle from the minimum at *x*_{0} = 1 to the top of the barrier at *x*_{a} = 0.

Recalling that $\varphi (x)\u2254\beta U(x)\u2261\beta U0|x|2\alpha 2\alpha \u2212|x|\alpha \alpha $ and utilizing Eq. (4), we see that the condition for the resetting transition translates to

The integrals in Eq. (10) cannot be evaluated analytically; however, it is not difficult to see that the results depend only on *α* and *βU*_{0}, the strength of the potential in terms of the thermal energy. We numerically calculate the ratio $\tau 2/2\tau 12$ from the left hand side of Eq. (10) and plot it in Fig. 4(b) to graphically illustrate the solution of the equation for different values of *α*. In Fig. 4(c), we construct a phase diagram by plotting the solutions, denoted as $[\beta U0]\u22c6$, vs *α* (black line). The area under the curve (white region) shows the parameter space where resetting expedites barrier crossing (smaller values of *βU*_{0}), whereas the area above the curve (shaded region) marks the parameter space where resetting hinders the same (larger values of *βU*_{0}). Therefore, our analysis of barrier crossing in the double-well potential proves that even when the underlying system is not exactly solvable, the general framework outlined in this communication can be utilized to locate the point of the resetting transition based on an interplay of two competing energy scales, the thermal and the potential energy.

In conclusion, we presented a general framework to determine the resetting transition point of diffusion in a potential. The framework applies to diffusion in arbitrary potentials, and it was illustrated with three different showcase potentials, viz., a regularized logarithmic potential, a set of power-law potentials of varying exponents, and a set of double-well potentials of varying barrier heights, which are widely used to model a variety of first-passage processes. In most of the cases involving diffusion in a potential landscape with resetting, the solution of the Fokker–Planck equation can be quite challenging. Instead, our methodology can be adapted to identify the resetting transition point. More importantly, here, we showed that the resetting transition can be understood in terms of an interplay between the thermal and the potential energy. These interpretations will strengthen the connection between theory and experiment in the field of stochastic resetting by paving new ways of analyzing experimental data and by generating new experimentally verifiable predictions.

S. Ray acknowledges support from the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Israel, and the DST-INSPIRE Faculty Grant, Government of India, executed at IIT Tirupati through

Project No. CHY/2021/005/DSTX/SOMR. S. Reuveni acknowledges support from The Azrieli Foundation, the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science at Tel Aviv University, and the Israel Science Foundation (Grant No. 394/19).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its appendixes.

### APPENDIX A: DERIVATION OF THE EXACT EXPRESSIONS OF THE FIRST AND SECOND MOMENTS ($\tau $_{1} AND $\tau $_{2}) OF THE FIRST-PASSAGE TIME FROM EQ. (3)

Equation (3) in the main text is a single-variable, recursive, second-order differential equation in *τ*_{n}, the *n*th moment of the FPT. Utilizing the Einstein relation *D* = (*βζ*)^{−1} in Eq. (3), where *β* is the thermodynamic *beta*, we get

Setting *n* = 1 in Eq. (A1) and taking into account the fact that $\tau 0:=\u2212\u222b0\u221edt\u2202Q(t|x0)\u2202t\u2261Q(t|x0)t\u21920\u2212Q(t|x0)t\u2192\u221e=1$, we see that the resulting second-order differential equation in *τ*_{1} can be reduced to a first-order differential equation in $\tau 1\u2032$ ≡ *dτ*_{1}/*dx*_{0} that reads as

With an integrating factor $e\u2212\u222bdx0\beta U\u2032(x0)\u2261e\u2212\beta U(x0)$, Eq. (A2) leads to the general solution

where *C* is the arbitrary integration constant. Putting the boundary condition $limx0\u2192\u221ee\u2212\beta U(x0)\tau 1\u2032(x0)=0$ in Eq. (A3), we find $C=\u222bxa\u221edz\u2009e\u2212\beta U(z)$, and this, in turn, leads to the specific solution of Eq. (A2),

Recalling that *x*_{a} < *x*_{0}, we now integrate Eq. (A4) within the interval [*x*_{a}, *x*_{0}] and utilize the boundary condition *τ*_{1}(*x*_{a}) = 0 to obtain^{52}

Equation (A5) presents an explicit expression for the mean FPT of a particle that diffuses in a potential *U*(*x*) from its initial position *x*_{0} to the absorbing boundary at *x*_{a}.

In a similar spirit, we can obtain an exact expression for *τ*_{2}. For this, we set *n* = 2 in Eq. (A1) to obtain a second-order differential equation in *τ*_{2},

where we utilized Eq. (A5) to substitute for *τ*_{1}. Solving for *τ*_{2} in the exact manner as before, we get

Equation (A7) presents an explicit expression for the second moment of the FPT.

### APPENDIX B: TRANSFORMATION OF VARIABLE *x* $\u2192$ $\phi $(*x*) $\u2261$ $\beta $*U*(*x*) FOR THE POWER-LAW POTENTIAL *U*(*x*) = *U*_{0}|*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*) ≔ *βU*(*x*), which leads to *ϕ*(*x*) = *βU*_{0}|*x*|^{α} for the power-law potential. Therefore, for 0 ≤ *x* < ∞, we get

Differentiating Eq. (B1) with respect to *ϕ*(*x*) thus gives

Setting *x*_{a} = 0, we can write Eq. (A5) in terms of the transformed variable as

Note that *ϕ*(*y*) and *ϕ*(*z*) are integration variables in Eq. (B3), and hence, the value of the above double integral depends only on *ϕ*_{0} ≡ *ϕ*(*x*_{0}) = *βU*_{0}|*x*_{0}|^{α}. To simplify the notation, we use *ϕ*_{z} = *ϕ*(*z*) and *ϕ*_{y} = *ϕ*(*y*) hereafter. In a similar spirit as in the case of the mean FPT, Eq. (A7) in terms of the transformed variable reads as