Resonant activation in a simple kinetic model

PHYSICAL REVIEW EVOLUME 50, NUMBER 1JULY 1994 Resonan ac i a ion in asimple kine ic model J.J.B ey and J.Casado-Pascual Fisica Teo ica, Uni e sidad de Se illa, Apa ado Co eos 1066, 41080 Se illa, Spain (Recei ed 1Feb ua y 1994) We p esen a e y simple Ma ko ian kine ic model displaying as ochas ic esonan beha io which is simila o he one ound in he escape o apa icle o e a luc ua ing po en ial ba ie . The basic mechanism ha is esponsible o he exis ence o esonance is iden i6ed. This allows he gene aliza ion o he model in di e en ways, leading o a a ie y o models whe e asimila phenomenon is o be expec ed. I is also shown ha he ini ial condi ions play an impo an ole in de e mining whe he he esonan ac i a ion ac ually shows up. PACS numbe (s): 05.40.+j, 82.20.Mj I. INTRODUCTION In he pas yea s, much a en ion has been de o ed o he s udy o esonance e Fec s in s ochas ic sys ems. The mos ex ensi ely conside ed phenomenon is he noise- induced enhancemen o asmall sys ema ic pe iodic sig- nal in nonlinea sys ems and i is usually e e ed o as s ochas ic esonance [1,2]. Recen ly, Doe ing and Gadona [3] epo ed he exis ence o ano he esonance e Fec in he escape a e o e alinea ba ie whose slope luc ua es be ween wo alues. The mean i s passage ime (MFPT) as a unc ion o he Sipping a e o he ba ie p esen s aminimum, which was cha ac e ized as a esonan ac i a ion o e he ba ie . In o de o unde s and he o igin o he esonan ac i- a ion, i seems impo an o de e mine whe he asimi- la phenomenon can be obse ed in simple kine ic mod- els. Bie and As umian [4] ha e analyzed aMa ko ian kine ic model in which he eac an swi ches a agi en a e be ween wo in e nal s a es. These wo s a es ha e di Fe en a es o decay owa ds he inal abso bing p od- uc s a e. Fo small lipping a es, he model is closely ela ed o he escape o e alinea luc ua ing ba ie in he limi o la ge ba ie and small luc ua ions. Ne e - heless, he beha io o bo h sys ems o la ge lipping a es is qui e di Fe en and he model does no exhibi esonan ac i a ion. On he o he hand, Van den B oeck [5] had p e iously shown ha he phenomenon can be displayed o non-Ma ko ian a ian s o he same model. Mo e conc e ely, he has conside ed he case o nonexpo- nen ial wai ing ime dis ibu ions o he in e nal s a es o he eac an . The pu pose o his pape is o p esen asimple kine ic model showing, unde ce ain ci cums ances, a esonan beha io ha is simila o ha epo ed in [3]. The e a e se e al easons ha ende he ele ance o he s udy o his model. Fi s , i is Ma ko ian, unlike hose which ha e been conside ed p e iously [5]. Besides, i s echni- cal simplici y allows us o de e mine in ap ecise way he condi ions unde which he esonance e Fec a ises and, in pa icula , he impo an ole played by he ini ial condi- ions. Finally, he xnodel clea ly iden i ies he mechanism which is esponsible o he esonance. I is due o he luc ua ions o an in e media e s a e connec ing he ini- ial and inal s a es. These luc ua ions mus ha e oppo- si e e Fec s on he ansi ions om he in e media e s a e o he inal and o he ini ial s a es. F om his poin o iew, ou model is aminixnal model in he sense ha i can be gene alized in many di Fe en ways p ese ing he esonan beha io . In pa icula , i allows us o explain he beha io o he linea luc ua ing ba ie analyzed in W. The es o his pape is o ganized as ollows. In he nex sec ion, we desc ibe ou model and ind exac esul s o he MFPT om he ini ial s a e ( eac an ) o he inal s a e (p oduc ). The condi ions unde which esonan , ac i a ion appea s a e in es iga ed. In Sec. III we discuss some modi ied e sions o he model ha do no lead o esonan beha io . Compa ison o he se e al models clea ly shows he o igin o s ochas ic esonance in his kind o models. We conclude wi h ab ie summa y o ou main esul s. II. MODEL The model is desc ibed by he kine ic scheme depic ed in Fig. 1. The ans o ma ion o he eac an Ain o he p oduc | akes place h ough an in e media e sub- s ance B. This one, due, o ins ance, o he in luence o some ex e nal condi ions, andomly swi ches a a a e pbe ween wo s a es, deno ed by B+ and B, espec- i ely. The subs ance B emains in one s a e o an ex- ponen ially dis ibu ed andom ixne be o e swi ching o ano he and pis he in e se o he a e age ime ha B s ays in one s a e be o e swi ching. When Bis in s a e B+,only ansi ions om B o Aa a e kq a e possible, while o Bbeing in s a e B he possible ansi ions 8 I Il l B FIG. 1. Ske ch o he kine ic model. The s a e o he in e - media e subs ance BBuc ua es be ween B+ and Ba a e '~ 1063-651X/94/50(1)/116(5)/$06. 00 50 1994 The Ame ican Physical Socie y 50 RESONANT ACTIVATION IN ASIMPLE KINETIC MODEL 117 a e &om A o B, a a e k2, and &om B o C, a a e k3. Thus he s a e o Bac s as acon ol a iable o he ans o ma ion o Ain o C. Ou aim is o ind he noise-a e aged e olu ion o he concen a ions o Aand B. This is a ypical p oblem o he so-called dynamical diso de and agene al scheme o ea ing such p oblems has been e iewed by Zwanzig [6]. We in oduce pa iaHy a e aged concen a ions o A and B o agi en s a e o B. They will be deno ed by Pg(A, ) and Py(B, ), whe e he signs +and — e e o s a es B+ and B, espec i ely. Then, o ins ance, P~(A, ) is he concen a ion o Aa ime , wi h B+ he s a e o B. The ac ual concen a ion o Aa ime , P(A, ), is gi en by P(A, ) =P+(A, ) +P(A, ), and simila ly o he concen a ion o B, P(B, ). The pa ially a e aged concen a ions sa is y he equa ions P+(A,— ) =qP+(A— , )+k,P+(B, )+qP (A, ), B (2a) 8P(A, — )=(k +p— )P (A, ) +pPg(A, ), (2b) I — P+(B, ) =— (kq +p)P+(B, ) +pP (B, ), (2c) 8 — P(B, ) =— (k, +p)P (B, ) 8 O +k,P(A, )+qP+(B, ). (2d) The ini ial condi ion is ha only eac an Ais p esen , bu no hing wi11 be assumed abou he ini ial dis ibu ion o s a es B+ and B.The e o e, we ake P+(A, 0) =a, P(A, 0) =1— a, P+(B,O) =P(B,O) =0, wi h 0&a&1. The MFPT o he ans o ma ion o A in o Cis gi en by [7] (~) =d [P(A, ) +P(B, )] 0 and asimple calcula ion using, o ins ance, he Laplace ans o med o Eq. (2) yields 2p (kg +k2 +ks) +2pkg (k2 +ks) +apk2ks +akim k2ks 7'7 k2k3 +Pklk2k3 (6) The beha io o ( ) in he wo limi ing alues o pis easily unde s ood. When pis e y la ge, he abo e esul becomes ( ) ( ) 2(kg +k2 +ks) 23 which does no depend on he alue o he ini ial condi- ion a. In ac , his is he exp ession one ge s by com- pu ing he MFPT o he p ocess and subs i u ing he ansi ion a es by hei equilib ium alues, which a e kq/2 o he ansi ion om B o A, k2/2 o he ansi ion om A o B, and ks/2 o he ansi ion &om B o C. On he o he hand, in he s a ic diso de limi p~0, ( ) goes o in ini y (excep in he pa icula case a=0) because he ansi ion om A o Cis no possible when Bis in he s a e B+. The ques ion now is whe he he MFPT p esen s aminimum be ween he abo e wo limi - ing alues. F om Eq. (6) i ollows ha such aminimum exis s i k, +k, +k, +(k,k,)& The e o e, a esonance e ec occu s when he ini ial p obabili y dis ibu ion o he Buc ua ing s a es o B, cha ac e ized by he pa ame e a, and he a es kq, k2, and ks e i y he ela ion gi en in Eq. (9). The ele an ole played by he ini ial condi ion o de e mine whe he he e is esonance mus be no iced. In pa icula , o gi en alues o he ansi ion a es, he e will always be esonance o small enough alues o a. As an example, we ha e plo ed in Fig. 2 he MFPT as a unc ion o p o kq — — k2 — — ks — —1. The ini ial condi ion is a=1/2, which 6.8 6.6 6.4- (~l 62- 2k& &ak2k3. (9) Mo eo e , he minim»m is a ained o aswi ching a e 5.4010 20 and i is gi en by kg (ak2ks) ~~2 2~~ ky — (ak2ks) ~(10) FIG. 2. Mean S s passage ime as a unc ion o he swi ch- ing a e o he model ske ched in Fig. 1. The alues o he pa ame e s a e kq — —kq — —k3 — —1and he ini ial condi ion is a=1 2. 118 J.J.BRRY AND J.CASADO-PASCUAL co esponds o he s eady dis ibu ion o he swi ching mechanism i i we e isola ed. As p edic ed by Eq. (10), he esonance akes place o po — —l. The o igin o he esonance shows up clea ly i we ew i e Eq. (6) as A 8+ jI ~l l il 8 a2k' p '1 1) (T) =—++2/ — +—/. k2k, q+k, ~k, k, y(12) The lipping a e appea s in wo summands on he igh hand side o his exp ession. The 6 s one is adec easing unc ion o pand educes o ze o o p~oo o a= 0, while he second one is an inc easing unc ion o p and ends o he asymp o ic alue 2k'/k2ks. Besides, he la e does no depend on he ini ial condi ion a. The e o e, he Bipping plays a wo old ole. On he one hand, i acili a es he decay o he ini ial p obabili y o 6nding he sys em in he conlgu a ion in which he ansi ion o he p oduc Cis impossible. On he o he hand, i makes i ha no all he ansi ions om A o Bp oceed up o C. Some o he subs ance eaching Bis e u ned o A. The compe i ion o hese wo oles leads in some cases o esonan ac i a ion. The condi ion is ha he second endency domina es o la ge enough Bipping a es. I we in oduce ameasu e Bo he esonance e ec as B jl II F 8 (b) FIG. 3. Two a ian s o he kine ic model discussed in Sec. a ian o he model ha is ske ched in Fig. 3(a). The in e p e a ion o he scheme is simila o he one in Fig. 1and he only di e ence wi h he p e ious model is ha now he decay o he p oduc Coccu s when he in e - media e subs ance Bis in s a e B+. Using he same p ocedu e as abo e and he ini ial condi ions gi en by Eqs. (3) and (4), he MFPT o his model is ob ained as 2p(kg +k2 +ks) +2kgk2 +(1+a)k2ks ( pk2ks i is seen om Eqs. (7) and (11) ha , o agi en alue o a, he esonance is s onge he la ge he alue o kq and he smalle he alues o k2 and k3. In he limi kq moo, (14) In he limi p~oo, his exp ession educes o Eq. (7). Thus, bo h models ha e he same beha io in he high equency limi . This is he expec ed esul since he discussion gi en below Eq. (7) also applies in his case. Ne e heless, om Eq. (15) one ge s and Bdi e ges as aconsequence o he di e gence o ( ) . B(7.)2k' +(u+ 1)ks Op pk3 (16) III. VARIANTS AND GENERALIZATIONS OF THE MODEL In o de o iden i y he basic mechanism ha is e- sponsible o he esonance beha io , we now u n o he I which indica es ha he MFPT is amono onously de- c easing unc ion o p o all alues o k~, k~, k3, and a and he e o e his model does no p esen he esonance phenomenon. Le us s il1 conside ano he a ian , namely, ha de- sc ibed by Fig. 3(b). The exp ession o he MFPT o i eads (~) =2p (kq +k2 +ks) +p(2k' k2 +(2 — a)kqks +ak2ks) +akqk2ks pks(kg +k2) +pkgk2ks Taking in o accoun ha 0&a&1, i is easily seen ha we again ha e amono onous decay. Le us poin ou ha he high equency limi o his model is 2(k, +k, +k&) ks(kg +k2) which is di e en om he esul ob ained in he same limi o he wo p e ious models, Eq. (?). Wha is he conc1usion eme ging om he compa i- l son o he h ee models we ha e analyzed'? Conside he gene al model wi h aBuc ua ing in e media e s a e ep- esen ed in Fig. 4. Fo kq, k2, and k3 di e en &om ze o, he MFPT is acon inuous unc ion o all he ansi ion a es. The e o e, ou esul s sugges ha he possibil- i y o ge ing a esonan beha io equi es ha k~ &k2, k2 )k», and k3 &k4. The Buc ua ions o Bmus be such ha when he ansi ion a e om B o Cinc eases, he ansi ion a e om A o Balso inc eases, while ha as- socia ed wi h he ans o ma ion oxn B o Abecomes 50 RESONANT ACTIVATION IN ASIMPLE KINETIC MODEL 119 B gl II II B FIG. 4. Ske ch o he gene al kine ic model wi h an in e - media e 6uc ua ing s a e. 0 — P+(1, ) =— (1 — A+ +p)Pi(1, ) +(1+A+)P+(2, ) +pP (l, ), (19a) — P(l, ) =— (1 — A+q)P (l, ) +(1+A)P (2, ) +pP+(l, ), (19b) 8 — P+(» ) =— (2+7)P+(» ) +(1— A+)P+(l, ) +pP (2, ), (19c) 19 — P(2, ) =— (2+q)P (2, ) +(1— A)P (l, ) +pP+(2, ), (19d) whe e we ha e in oduced a eBec ing bounda y a n=0 and an abso bing one a n=2. Since he model e i ies he condi ions o mula ed abo e, i is expec ed o p esen esonan ac i a ion in some egion o i s space o pa am- e e s. As o he o he models, i is s aigh o wa d o ob ain he MFPT using he ini ial condi ions P+(1,0) = a, P(1,0) =1— a, and P+(2,0) =P(2, 0) =0. Ne e - heless, he exp ession is e y la ge and will be omi ed he e. In he limi pmoo, i educes o smalle . As an example, le us assume ha he p ocess o going &om A o Cco esponds o he di usion o apa icle on aone dimensional la ice &om he si e n=0 o he si e n=2, in he p esence o an ex e nal po en ial ield. The p obabili y o jumping pe uni o ime is u(1 —A) o jumps o he igh and (d(1 +A) o jumps o he le , wi h 0(A&1. The pa ame e Ais p opo ional o he slope o he ex e nal po en ial [8] and (a is ana u al &equency o he la ice, which will be used o ix he ime scale and he e o e i will be aken equal o uni y in he ollowing. Now, suppose ha ABuc ua es, wi h aBipping a e p, be ween wo alues A+ and Aas aconsequence o he Buc ua ion o he slope o he ex e nal po en ial. The pa ially a e aged p obabili ies o inding he pa icle a si e n, o agi en alue o A, obey he mas e equa ion The exis ence o esonance can be analyzed by s udy- ing he i s o de co ec ion in pi o ( ) .When his co ec ion is nega i e, ( ) mus p esen aminimum o some ini e alue o p. Fo a=1j2, he sys em always exhibi s esonan ac i a ion, independen o he alues o A+ and A.Ne e heless, his is no ue o o he alues o a. Fo ins ance, o a=0.8, A+ — — 0.2, and A=0.1, ( ) decays mono onously o ( ) I he numbe o in e media e s a es Bis inc eased, he ansi ion p obabili ies a e scaled wi h he densi y o si es, and he con inuous limi is aken, he sys em educes [8] o he B ownian mo ion in aHuc ua ing lin- ea ba ie s udied by Doe ing and Gadona [3]. Le us ema k ha hei calcula ions co espond o he ini ial condi ion a=1/2 and he e o e i is no su p ising ha hey ound esonance o all pai s o alues o he slope o he ba ie . In iew o he analysis p esen ed he e, we specula e ha he esul could be di e en o o he ini ial condi ions. IV. SUMMARY AND CONCLUSIONS In his pape we ha e shown ha he esonance phe- nomenon e e ed o as esonan ac i a ion can be un- de s ood in e ms o asimple kine ic model. The main ea u e o he model is he p esence o an in e media e Buc ua ing s a e. The echnical simplici y o he mas- e equa ion go e ning he ime e olu ion o he sys em allows us o ob ain an explici exac exp ession o he mean i s -passage ime in e ms o he pa ame e s de in- ing he model and he ini ial condi ions. In his way, a de ailed analysis o he condi ions equi ed o he dis- play o esonan beha io has been possible. Anecessa y condi ion seems o be ha he Buc ua ions o he in- e media e s a e ha e di e en quali a i e e ec s on he ansi ion a es co esponding o p ocesses poin ing in he di ec ion o he inal s a e and on hose associa ed wi h p ocesses in he opposi e di ec ion. An in e es ing ea u e shown he e is he e ec o he ini ial condi ions on he esonance. They a e no only inBuencing he ampli ude o he phenomenon, bu ac u- ally de e mining whe he i will ake place o no . This is some hing o be aken in o accoun upon designing expe - imen s looking o esonan ac i a ion. In his con ex , al hough we ha e no ied o ela e ou model o any eal sys em, i is clea om i s s uc u e ha i can be use ul o he desc ip ion o some chemical ansi ions. Besides, gene aliza ions o he model, like he one desc ibed in Sec. III, can be applied o a a ie y o physicaI p oblems. ACKNOWLEDGMENTS 2(6 — A+ — A) "--(A. +A 2)' (20) Pa ial suppo om he Di eccion Gene al de In es- igacion Cien i ica yTecnica (Spain) h ough G an No. PB92-0683 is g a e ully acknowledged. 120 J.J.SREY AND J.CASADO-PASCUAL SG [1] B.McNama a and W. Wiesen eld, Phys. Re . A39, 4854 (1989). [2] A. Bulsa a and F.Moss, 3.S a . Phys. Special Issues (1,2) 70 (1993). [3] C. R. Doe ing and 3. C. Gadona, Phys. Re . Le . 69, 2318 (1992);U. Zii che and C. R. Doe ing, Phys. Re . E 47, 3862 (1993). [4] M. Bie and R. D. As umian, Phys. Re . Le . 71, 1649 (1993). [5] C. Van den B oeck, Phys. Re . E47, 4579 (1993). [6] R. Zwanzig, Acc. Chem. Res. 23, 148 (1990). P ocesses o he same kind a e ea ed in Re . [7], Sec. VII.7, whe e hey a e called "composi e Ma ko p ocesses. " [7] N. G. an Kampen, S ochas ic P ocesses in Physics and Chemis y(No h-Holland, Ams e dam, 1992). [8] See, o ins ance, S ochas ic P ocesses in Physics and Chemis y (Re . [7]), Sec. XI.2.