Yating Wang and Zhanchun Tu

Department of Physics,Beijing Normal University,Beijing 100875,China

Abstract We discuss the escape problem with the consideration of both the activity of particles and the roughness of potentials.We derive analytic expressions for the escape rate of an active Brownian particle in two types of rough potentials by employing the effective equilibrium approach and the Zwanzig method.We find that activity enhances the escape rate,but both the oscillating perturbation and the random amplitude hinder escaping.

Keywords: escape rate,active Brownian particle,rough potential,effective potential

1.Introduction

The escape problem has attracted much attention from researchers in various fields [1–10].The Arrhenius formula indicates that the rate of a chemical reaction depends exponentially on inverse temperature[3,4].Kramers presented the transition state method for calculating the rate of chemical reactions by considering a Brownian particle escaping over a potential barrier [5].Subsequent studies on escape rate are summarized in [10].All of the above studies merely involve passive particles.The research theme has been transferred to active particles with self-propulsion in recent years [11–24].Active systems are intrinsically non-equilibrium since the detailed balance is broken.An effective equilibrium method has been developed to investigate active Brownian particles[25–28].By using this method,Sharmaet aldiscussed an escape problem of active particles in a smooth potential[29].They found that introducing activity increases the escape rate.

The escape problem in the research mentioned above is simplified as a Brownian particle climbing over a smooth potential barrier.However,the potential is not always smooth in reality.Interface area scans of proteins imply that the protein surface is not smooth [30,31].The hierarchical arrangement of the conformational substrates in myoglobin indicates that the potential surface might be rough [32].In addition,the inside of the cell is quite crowded.Thus,diffusion of substance in the cell may not be regarded as Brownian motion in smooth potential.From the biochemical point of view,it is valuable to consider the influence of the roughness of potential diffusion behaviors.The study of diffusion in rough potential offers insight into fields from the transport process in disordered media [33,34] to protein folding [35,36] and glassy systems [37,38].Zwanzig dealt with diffusion in a rough potential and found that the roughness slows down the diffusion at low temperatures[39].Roughness-enhanced transport was also observed in ratchet systems[40–42].Huet aldiscussed diffusion crossing over a barrier in a random rough metastable potential[43].By using numerical simulations,they demonstrate a decrease in the steady escape rate with the an increase of rough intensity.The activity of particles was not considered in these works.

There are a large number of active substances,biochemical reactions,and transport of substances in organisms.Therefore,it is of practical significance to discuss the escape problem with the consideration of both the activity of particles and the roughness of potentials.In order to describe the slow dynamics of a tagged particle in a dense active environment,Subhasishet aldiscussed the escape of a passive particle from an activity-induced energy landscape by using the activity-induced rugged energy landscape approach[44].In this work,we calculate the escape rate of an active Brownian particle (ABP) in rough potentials by using the effective equilibrium approach [25–29] and the Zwanzig method[39].The rest of this paper is organized as follows:In section 2,we briefly introduce the effective equilibrium approach.In section 3,we discuss the escape problems of ABPs in rough potentials with oscillating perturbation or random amplitude.We derive the effective rough potentials following the effective equilibrium approach.Then we analytically calculate the escape rates of ABPs in the effective rough potentials.We find that activity enhances the escape rate,but both the oscillating perturbation and the random amplitude hinder escaping.The last section is a brief summary.

2.Effective equilibrium approach

In this section,we briefly revisit the main ideas of the effective equilibrium approach [25–29].

The motion of the ABP can be described by the following overdamped Langevin equations

where γ is the friction coefficient and F(t) is force on the ABP.r represents the position of the particle.The particle is self-propelling with constant speedv0along orientations n.The dot ‘·’ above a character represents the derivative with respect to timet.The stochastic vectors ξ(t)and η(t)are white noise with correlations 〈ξ(t)ξ(t′)〉=2DtIδ(t-t′)and〈η(t)η(t′)〉=2DrIδ(t-t′),whereDtandDrare the translational and rotational diffusion coefficients,respectively.I is the unit tensor.

A stochastic process with color-noise in equation (3) is non-Markovian.It is impossible to derive an exact Fokker–Planck equation for the time evolution of the probability distribution.Nevertheless,using the Fox approximate method[45,46],we may derive an approximate Fokker–Planck equation

where φ(r,t)is the probability distribution.The current J(r,t)is expressed as

where Feff(r) represents the effective force on the particle.β=(kBT)-1,in whichkBis the Boltzmann constant andTis the temperature.The dimensionless effective diffusion coefficientD(r)=1+Da/(1-τ?·βF(r)),where τ=Dt/(2Dr).The activity parameterDa=The effective force is given by

3.Escape rate of ABP in rough potentials

In this section,we will deduce the effective rough potential and escape rate of ABP in rough potentials.For simplicity,we only consider the case that the bare force depends merely on a one-dimension potentialV=V(x).In this case,where i is the unit vector ofx-coordinate.The prime‘′’on the top right of a character represents the derivative with respect to positionx.From equation (6) we can obtain the effective potential

Now,let us consider a rough potential

where κ0and α are positive constants.The first two terms in equation (9) provide a smooth background with a barrier.The last term in equation(9)is the superposed random or oscillating perturbation.The amplitude ε is assumed to be small,which represents a measure of the ‘roughness’ of the potential.

Now,we look for the effective rough potential βVeff(x)corresponding to equation (9) from equation (7).Assuming κ0τ ?1 and keeping the terms up to the linear order of κ0τ and ε,we obtain the effective rough potential

where

The above three equations and the first three terms in equation (10) have been derived in [29].

The bare and effective rough potentials are schematically depicted in figure 1.xaandxbcorrespond to the minimum and maximum of the potential,respectively.xcis a point on the right ofxb.Passingxc,the particle will not return.In the other words,it is an absorbing boundary condition atxc.In a stationary state,the current(5) can be rewritten as

Following Kramers’s approach [5,47],we obtain the inverse of the escape rate of ABP:

Figure 1.Bare potential and analytic effective potential βVeff(x),equation (28),for different values of Da.For the given parameterτκ0=0.02,ε=0.05 and

wherex1≤xa≤x2≤xb.The detailed derivation of this equation is shown in appendix A.

Considering the rough character of the potential,we use the Zwanzig method [39] to simplify equation (15).The rough potential(10) may be decomposed into two parts.One is the smooth skeleton

the other is the rough perturbation

where〈〉denotes the spatial average during a small interval(x-Δ/2,x+Δ/2).Then equation (15) is transformed into

Next we discuss the spacial situation that ψ±(x)happens to be independent ofx.In this case,the above equation is transformed into

By using the saddle-point approximation and considering κ0τ is small,we derive the escape rate

where

and

The detailed derivation of equation (21) is displayed in appendix B.

For a passive Brownian particle moving in a smooth potential,equation (21) is degenerated into

Obviously,the above equation implies the escape rate

for APB in a smooth potential[29]since ψ+=ψ-=1 for the smooth potential.Then,equation (25) can be further expressed as

3.1.Oscillating perturbation of rough potential

In figure 1,we plot the effective potential for different values of activity parameterDa.We find that the effective barrier decreases with the increase of the activity parameter.Thus,the introduction of activity lowers the effective barrier height so that the particle easily escapes the barrier.

From equation (18),we obtain

whereI0is the modified Bessel function [36].Substituting equation (29) into equation (25),we obtain the escape rate

Figure 2.Dependence of on amplitude ε and active parameter Da,where τκ0=0.02.

3.2.Random amplitude of rough potential

Considering the random amplitude of rough potentialV1with a Gaussian distribution

where σ is the standard deviation.The effective rough potential βVeff(x) is

From equation (18),we obtain

Substituting equation (33) into equation (25),we obtain the escape rate

4.Conclusions

In this work,we have discussed the escape rate of ABPs in rough potentials by using the effective equilibrium approach and the Zwanzig method.We find that activity usually enhances the escape rate.Both the oscillating perturbation and the random amplitude of rough potential hinder escaping.We only discussed the influence of two types of rough potential on the escape rate of ABP.According to equation(21),‘rough potential hinders the escape’ may not hold in all cases.However,‘a(chǎn)ctivity enhances the escape rate’ should be generally applicable.In the theoretical derivation,we need the amplitude ε and κ0τ are small.Our theory is not applicable to large ε and κ0τ.We will develop a new theoretical approach to deal with these situations in the future.

Figure 3.Dependence of on amplitude σε and active parameter Da,where=0.1,τκ0=0.02.

Acknowledgments

The authors are grateful for financial support from the National Natural Science Foundation of China (Grant No.11975050 and No.11735005).We are very grateful for the help of Xiu-Hua Zhao.

Appendix A.Detailed derivation of equation(15)

Following the Kramers method in[47],we derive the inverse escape rate of ABP in the effective rough potential.

Assume βEb?1.In this situation,the system stays the quasi-stationary state such that the probability currentis approximately independent ofx.By integrating equation(14)betweenxaandxcand considering an absorbing boundary conditionx=xc,we obtain

Because the barrier is high,φ(x) nearxamay be approximately given by the stationary distribution

The probabilitypto find ABP nearxais

Appendix B.Saddle-point approximation

The integral expression in equation (20)may be obtained via the saddle-point approximation atxaandxb,respectively.

The effective smooth potential of nearlyxbcan be expanded nearbyxbas:

The second integral of smooth potential on the right-hand side of equation (20) is expressed as

According to the spirit of saddle-point approximation,equation (A5) is transformed into

Substitutingxbinto equation(8)and considering κ0τ is small,we obtain

Similarly,the first integral of smooth potential on the right-hand side of equation (20) may also be obtained by a saddle-point approximation atxa.