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.