Resonant activation in a simple kinetic model

PHYSICAL REVIEW EVOLUME 50, NUMBER 1JULY 1994 Resonant activation in asimple kinetic model J.J.Brey and J.Casado-Pascual Fisica Teorica, Universidad de Sevilla, Apartado Correos 1066, 41080 Sevilla, Spain (Received 1February 1994) We present avery simple Markovian kinetic model displaying astochastic resonant behavior which is similar to the one found in the escape of aparticle over afluctuating potential barrier. The basic mechanism that is responsible for the existence of resonance is identi6ed. This allows the generalization of the model in different ways, leading to avariety of models where asimilar phenomenon is to be expected. It is also shown that the initial conditions play an important role in determining whether the resonant activation actually shows up. PACS number(s): 05.40.+j, 82.20.Mj I. INTRODUCTION In the past years, much attention has been devoted to the study of resonance efFects in stochastic systems. The most extensively considered phenomenon is the noiseinduced enhancement of asmall systematic periodic signal in nonlinear systems and it is usually referred to as stochastic resonance [1,2]. Recently, Doering and Gadona [3] reported the existence of another resonance efFect in the escape rate over alinear barrier whose slope fluctuates between two values. The mean first passage time (MFPT) as afunction of the Sipping rate of the barrier presents aminimum, which was characterized as aresonant activation over the barrier. In order to understand the origin of the resonant activation, it seems important to determine whether asimilar phenomenon can be observed in simple kinetic models. Bier and Astumian [4] have analyzed aMarkovian kinetic model in which the reactant switches at agiven rate between two internal states. These two states have difFerent rates of decay towards the final absorbing product state. For small flipping rates, the model is closely related to the escape over alinear fluctuating barrier in the limit of large barrier and small fluctuations. Nevertheless, the behavior of both systems for large flipping rates is quite difFerent and the model does not exhibit resonant activation. On the other hand, Van den Broeck [5] had previously shown that the phenomenon can be displayed for non-Markovian variants of the same model. More concretely, he has considered the case of nonexponential waiting time distributions for the internal states of the reactant. The purpose of this paper is to present asimple kinetic model showing, under certain circumstances, aresonant behavior that is similar to that reported in [3]. There are several reasons that render the relevance of the study of this model. First, it is Markovian, unlike those which have been considered previously [5]. Besides, its technical simplicity allows us to determine in aprecise way the conditions under which the resonance efFect arises and, in particular, the important role played by the initial conditions. Finally, the xnodel clearly identifies the mechanism which is responsible for the resonance. It is due to the fluctuations of an intermediate state connecting the initial and final states. These fluctuations must have opposite efFects on the transitions from the intermediate state to the final and to the initial states. From this point of view, our model is aminixnal model in the sense that it can be generalized in many difFerent ways preserving the resonant behavior. In particular, it allows us to explain the behavior of the linear fluctuating barrier analyzed in W. The rest of this paper is organized as follows. In the next section, we describe our model and find exact results for the MFPT from the initial state (reactant) to the final state (product). The conditions under which resonant, activation appears are investigated. In Sec. III we discuss some modified versions of the model that do not lead to resonant behavior. Comparison of the several models clearly shows the origin of stochastic resonance in this kind of models. We conclude with abrief summary of our main results. II. MODEL The model is described by the kinetic scheme depicted in Fig. 1. The transformation of the reactant Ainto the product |takes place through an intermediate substance B. This one, due, for instance, to the influence of some external conditions, randomly switches at arate pbetween two states, denoted by B+ and B,respectively. The substance Bremains in one state for an exponentially distributed random tixne before switching to another and pis the inverse of the average time that B stays in one state before switching. When Bis in state B+,only transitions from Bto Aat rate kq are possible, while for Bbeing in state Bthe possible transitions 8 I Il lf B FIG. 1. Sketch of the kinetic model. The state of the intermediate substance BBuctuates between B+ and Bat rate '~ 1063-651X/94/50(1)/116(5)/$06. 00 50 1994 The American Physical Society 50 RESONANT ACTIVATION IN ASIMPLE KINETIC MODEL 117 are &om Ato B, at rate k2, and &om Bto C, at rate k3. Thus the state of Bacts as acontrol variable for the transformation of Ainto C. Our aim is to find the noise-averaged evolution of the concentrations of Aand B. This is atypical problem of the so-called dynamical disorder and ageneral scheme for treating such problems has been reviewed by Zwanzig [6]. We introduce partiaHy averaged concentrations of A and Bfor agiven state of B. They will be denoted by Pg(A, t) and Py(B, t), where the signs +and —refer to states B+ and B,respectively. Then, for instance, P~(A, t) is the concentration of Aat time t, with B+ the state of B. The actual concentration of Aat time t, P(A, t), is given by P(A, t) =P+(A, t) +P(A, t), and similarly for the concentration of B, P(B,t). The partially averaged concentrations satisfy the equations P+(A,— t) =qP+(A— ,t)+k,P+(B,t)+qP (A, t), Bt (2a) 8P(A, t— )=(k +p— )P (A, t) +pPg(A, t), (2b) I — P+(B,t) =— (kq +p)P+(B,t) +pP (B,t), (2c) 8 — P(B,t) =— (k, +p)P (B,t) 8 Ot +k,P(A, t)+qP+(B,t). (2d) The initial condition is that only reactant Ais present, but nothing wi11 be assumed about the initial distribution of states B+ and B.Therefore, we take P+(A, 0) =a, P(A, 0) =1— a, P+(B,O) =P(B,O) =0, with 0&a&1. The MFPT for the transformation of A into Cis given by [7] (~) =dt [P(A, t) +P(B,t)] 0 and asimple calculation using, for instance, the Laplace transformed of Eq. (2) yields 2p (kg +k2 +ks) +2pkg (k2 +ks) +apk2ks +akim k2ks 7'7 k2k3 +Pklk2k3 (6) The behavior of (r) in the two limiting values of pis easily understood. When pis very large, the above result becomes ( ) ( ) 2(kg +k2 +ks) 23 which does not depend on the value of the initial condition a. In fact, this is the expression one gets by computing the MFPT for the process and substituting the transition rates by their equilibrium values, which are kq/2 for the transition from Bto A, k2/2 for the transition from Ato B, and ks/2 for the transition &om Bto C. On the other hand, in the static disorder limit p~0, (r) goes to infinity (except in the particular case a=0) because the transition from Ato Cis not possible when Bis in the state B+. The question now is whether the MFPT presents aminimum between the above two limiting values. From Eq. (6) it follows that such aminimum exists if k, +k, +k, +(k,k,)& Therefore, aresonance effect occurs when the initial probability distribution of the Buctuating states of B, characterized by the parameter a, and the rates kq, k2, and ks verify the relation given in Eq. (9). The relevant role played by the initial condition to determine whether there is resonance must be noticed. In particular, for given values of the transition rates, there will always be resonance for small enough values of a. As an example, we have plotted in Fig. 2the MFPT as afunction of pfor kq — — k2 — — ks — —1. The initial condition is a=1/2, which 6.8 6.6 6.4- (~l 622k& &ak2k3. (9) Moreover, the minim»m is attained for aswitching rate 5.4010 20 and it is given by kg (ak2ks) ~~2 2~~ ky — (ak2ks) ~(10) FIG. 2. Mean Srst passage time as afunction ofthe switching rate for the model sketched in Fig. 1. The values of the parameters are kq — —kq — —k3 — —1and the initial condition is a=1 2. 118 J.J.BRRY AND J.CASADO-PASCUAL corresponds to the steady distribution of the switching mechanism if it were isolated. As predicted by Eq. (10), the resonance takes place for po — —l. The origin of the resonance shows up clearly if we rewrite Eq. (6) as A 8+ jI ~l fl il 8 a2k' pt'1 1) (T) =—++2/ — +—/. k2k, q+k, ~k, k, y(12) The flipping rate appears in two summands on the right hand side of this expression. The 6rst one is adecreasing function of pand reduces to zero for p~oo or a= 0, while the second one is an increasing function of p and tends to the asymptotic value 2k'/k2ks. Besides, the latter does not depend on the initial condition a. Therefore, the Bipping plays atwofold role. On the one hand, it facilitates the decay of the initial probability of 6nding the system in the conlguration in which the transition to the product Cis impossible. On the other hand, it makes it that not all the transitions from Ato Bproceed up to C. Some of the substance reaching Bis returned to A. The competition ofthese two roles leads in some cases to resonant activation. The condition is that the second tendency dominates for large enough Bipping rates. If we introduce ameasure Bof the resonance effect as B jl II F 8 (b) FIG. 3. Two variants of the kinetic model discussed in Sec. variant of the model that is sketched in Fig. 3(a). The interpretation of the scheme is similar to the one in Fig. 1and the only difference with the previous model is that now the decay to the product Coccurs when the intermediate substance Bis in state B+. Using the same procedure as above and the initial conditions given by Eqs. (3) and (4), the MFPT for this model is obtained as 2p(kg +k2 +ks) +2kgk2 +(1+a)k2ks ( pk2ks it is seen from Eqs. (7) and (11)that, for agiven value of a, the resonance is stronger the larger the value of kq and the smaller the values of k2 and k3. In the limit kq moo, (14) In the limit p~oo, this expression reduces to Eq. (7). Thus, both models have the same behavior in the high frequency limit. This is the expected result since the discussion given below Eq. (7) also applies in this case. Nevertheless, from Eq. (15) one gets and Bdiverges as aconsequence of the divergence of (r) . B(7.)2k' +(u+ 1)ks Op pk3 (16) III. VARIANTS AND GENERALIZATIONS OF THE MODEL In order to identify the basic mechanism that is responsible for the resonance behavior, we now turn to the I which indicates that the MFPT is amonotonously decreasing function of pfor all values of k~, k~, k3, and a and therefore this model does not present the resonance phenomenon. Let us stil1 consider another variant, namely, that described by Fig. 3(b). The expression of the MFPT for it reads (~) =2p (kq +k2 +ks) +p(2k' k2 +(2 — a)kqks +ak2ks) +akqk2ks pks(kg +k2) +pkgk2ks Taking into account that 0&a&1, it is easily seen that we again have amonotonous decay. Let us point out that the high frequency limit for this model is 2(k, +k, +k&) ks(kg +k2) which is diferent from the result obtained in the same limit for the two previous models, Eq. (?). What is the conc1usion emerging from the comparil son of the three models we have analyzed'? Consider the general model with aBuctuating intermediate state represented in Fig. 4. For kq, k2, and k3 diferent &om zero, the MFPT is acontinuous function of all the transition rates. Therefore, our results suggest that the possibility of getting aresonant behavior requires that k~ &k2, k2 )k», and k3 &k4. The Buctuations ofBmust be such that when the transition rate from Bto Cincreases, the transition rate from Ato Balso increases, while that associated with the transformation froxn Bto Abecomes 50 RESONANT ACTIVATION IN ASIMPLE KINETIC MODEL 119 B gl II II B FIG. 4. Sketch of the general kinetic model with an intermediate 6uctuating state. 0 — P+(1,t) =— (1 — A+ +p)Pi(1,t) +(1+A+)P+(2, t) +pP (l, t), (19a) — P(l,t) =— (1 — A+q)P (l,t) +(1+A)P (2, t) +pP+(l, t), (19b) 8 — P+(»t) =— (2+7)P+(»t) +(1— A+)P+(l, t) +pP (2,t), (19c) 19 — P(2,t) =— (2+q)P (2, t) +(1— A)P (l,t) +pP+(2, t), (19d) where we have introduced areBecting boundary at n=0 and an absorbing one at n=2. Since the model verifies the conditions formulated above, it is expected to present resonant activation in some region of its space of parameters. As for the other models, it is straightforward to obtain the MFPT using the initial conditions P+(1,0) = a, P(1,0) =1— a, and P+(2,0) =P(2, 0) =0. Nevertheless, the expression is very large and will be omitted here. In the limit pmoo, it reduces to smaller. As an example, let us assume that the process of going &om Ato Ccorresponds to the diffusion of aparticle on aone dimensional lattice &om the site n=0to the site n=2, in the presence of an external potential field. The probability of jumping per unit of time is u(1 —A) for jumps to the right and (d(1 +A) for jumps to the left, with 0(A&1. The parameter Ais proportional to the slope of the external potential [8] and (a is anatural &equency ofthe lattice, which will be used to fix the time scale and therefore it will be taken equal to unity in the following. Now, suppose that ABuctuates, with aBipping rate p, between two values A+ and Aas aconsequence of the Buctuation of the slope of the external potential. The partially averaged probabilities of finding the particle at site n, for agiven value of A, obey the master equation The existence of resonance can be analyzed by studying the first order correction in pito (r) .When this correction is negative, (r) must present aminimum for some finite value of p. For a=1j2, the system always exhibits resonant activation, independent of the values of A+ and A.Nevertheless, this is not true for other values of a. For instance, for a=0.8, A+ — — 0.2, and A=0.1, (r) decays monotonously to (r) If the number of intermediate states Bis increased, the transition probabilities are scaled with the density of sites, and the continuous limit is taken, the system reduces [8] to the Brownian motion in aHuctuating linear barrier studied by Doering and Gadona [3]. Let us remark that their calculations correspond to the initial condition a=1/2 and therefore it is not surprising that they found resonance for all pairs of values of the slope of the barrier. In view of the analysis presented here, we speculate that the result could be different for other initial conditions. IV. SUMMARY AND CONCLUSIONS In this paper we have shown that the resonance phenomenon referred to as resonant activation can be understood in terms of asimple kinetic model. The main feature of the model is the presence of an intermediate Buctuating state. The technical simplicity of the master equation governing the time evolution of the system allows us to obtain an explicit exact expression for the mean first-passage time in terms of the parameters defining the model and the initial conditions. In this way, a detailed analysis of the conditions required for the display of resonant behavior has been possible. Anecessary condition seems to be that the Buctuations of the intermediate state have different qualitative effects on the transition rates corresponding to processes pointing in the direction of the final state and on those associated with processes in the opposite direction. An interesting feature shown here is the effect of the initial conditions on the resonance. They are not only inBuencing the amplitude of the phenomenon, but actually determining whether it will take place or not. This is something to be taken into account upon designing experiments looking for resonant activation. In this context, although we have not tried to relate our model to any real system, it is clear from its structure that it can be useful for the description of some chemical transitions. Besides, generalizations of the model, like the one described in Sec. III, can be applied to avariety of physicaI problems. ACKNOWLEDGMENTS 2(6 — A+ — A) "--(A. +A 2)' (20) Partial support from the Direccion General de Investigacion Cientifica yTecnica (Spain) through Grant No. PB92-0683 is gratefully acknowledged. 120 J.J.SREY AND J.CASADO-PASCUAL SG [1] B.McNamara and W. Wiesenfeld, Phys. Rev. A39, 4854 (1989). [2] A. Bulsara and F.Moss, 3.Stat. Phys. Special Issues (1,2) 70 (1993). [3] C. R. Doering and 3. C. Gadona, Phys. Rev. Lett. 69, 2318 (1992);U. Ziircher and C. R. Doering, Phys. Rev. E 47, 3862 (1993). [4] M. Bier and R. D. Astumian, Phys. Rev. Lett. 71, 1649 (1993). [5] C. Van den Broeck, Phys. Rev. E47, 4579 (1993). [6] R. Zwanzig, Acc. Chem. Res. 23, 148 (1990). Processes of the same kind are treated in Ref. [7], Sec. VII.7, where they are called "composite Markov processes. " [7] N. G. van Kampen, Stochastic Processes in Physics and Chemist y(North-Holland, Amsterdam, 1992). [8] See, for instance, Stochastic Processes in Physics and Chemistry (Ref. [7]), Sec. XI.2.