J. Chem. Phys. 110, 2159 (1999); h ps://doi.o g/10.1063/1.477826 110, 2159
© 1999 Ame ican Ins i u e o Physics.
The mally d i en escape o e a ba ie o
a bi a y shape
Ci e as: J. Chem. Phys. 110, 2159 (1999); h ps://doi.o g/10.1063/1.477826
Submi ed: 13 Ma ch 1998 . Accep ed: 21 Oc obe 1998 . Published Online: 12 Janua y 1999
A. N. D ozdo , and J. J. B ey
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The mally d i en escape o e a ba ie o a bi a y shape
A. N. D ozdo a) and J. J. B ey
Fı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain
~Recei ed 13 Ma ch 1998; accep ed 21 Oc obe 1998!
The K ame s heo y o he he mally ac i a ed a e o escape o a B ownian pa icle om a
po en ial well is ex ended o a ba ie o a bi a y shape. The ex ension is based on an app oxima e
solu ion o he unde lying Fokke –Planck equa ion in he spa ial di usion egime. Wi h he use o
he Mel’niko –Meshko esul o he unde damped B ownian mo ion an o e all a e exp ession is
cons uc ed, which in e pola es he co ec limi ing beha io o bo h weak and s ong ic ion. I
gene alizes in a na u al way a ious di e en a e exp essions ha a e al eady a ailable in he
li e a u e o pa abolic, cusped, and qua ic ba ie s. Applica ions o symme ic pa abolic and
cusped double-well po en ials show good ag eemen be ween he heo y and es ima es o he a es
om nume ical calcula ions. © 1999 Ame ican Ins i u e o Physics. @S0021-9606~99!01404-X#
I. INTRODUCTION
E e since he pionee ing con ibu ion o S an e A hen-
ius, he p oblem o he mally d i en escape om a me a-
s able s a e has become one o he mos undamen al p ob-
lems in physics and chemis y.1The mode n heo y o
ac i a ed a e p ocesses is essen ially due o K ame s,2who
p o ided a dynamical amewo k o he o iginal concep s o
A henius. The unde lying idea o he K ame s heo y is o
model he escape p ocess by he mo ion o a B ownian pa -
icle wi h mass weigh ed coo dina e xin a po en ial o mean
o ce V(x). The dynamics is go e ned by he ollowing
Fokke –Planck equa ion o he p obabili y densi y P(x, , )
o inding he pa icle a ime a posi ion xwi h eloci y :2
]
P~x, , !5@2
]
x1V8~x!
]
1
g
]
~ 1
b
21
]
!#P~x, , !.
~1.1!
He e he p ime deno es he de i a i e wi h espec o x,
g
is
he ic ion coe icien , and
b
he in e se ene gy a ailable
om he he mal ba h,
b
215kBT. The po en ial is assumed
o ha e a well wi h minimum a xw,0, sepa a ed om he
con inuum by a ba ie a x50 o heigh E52V(xw).
He eby we se o con enience V(0)50. The quan i y o
in e es is he escape a e Go he pa icle om he well. The
la e can always be w i en in he o m
G5
m
GTST ,~1.2!
whe e GTST is he ansi ion s a e heo y ~TST! esul
GTST5
H
A
2
p
b
E
2`
0dx e2
b
V~x!
J
21,~1.3!
and
m
is a ansmission coe icien desc ibing he de ia ion
o he a e om GTST .
K ame s s udied he dependence o he escape a e on
he ic ional damping in wo egimes, namely, o small and
in e media e o la ge ic ion
g
. In he o me egime, he
coupling be ween he sys em and he ba h is assumed o be
anishingly weak so ha he a e limi ing s ep is he ans e
o ene gy om he ba h o he pa icle. The ansmission
coe icien akes in his case he o m
m
~
g
→0!5D~
g
→0!52
gb
E
xp
0dx
A
22V~x!,~1.4!
whe e Dis he dimensionless loss o ene gy pe oscilla ion o
a pa icle wi h ene gy close o he ba ie heigh , and xp he
le -hand side u ning poin o he asymp o ic unde damped
ajec o y, V(xp)50. In he in e media e o la ge ic ion e-
gime, when he ans e o ene gy becomes as enough o
main ain he mal equilib ium o escaping pa icles, he a e
limi ing s ep is spa ial di usion ac oss he ba ie egion.
One o he basic assump ions o he K ame s heo y2in his
egime is a pa abolic ba ie app oxima ion. I consis s in
di iding he ull po en ial in o a pa abolic ba ie pa
U~x!52 1
2
2x2,~1.5!
wi h
252V9(0), and an anha monic co ec ion eading
V~x!5U~x!1O~x3!.~1.6!
In he immedia e icini y o he ba ie op which domina es
he dynamics, he nonlinea i y o V(x) anishes as e han
he pa abolic pa 21
2
2x2and he e o e can be neglec ed.
This yields he ollowing exp ession o he ansmission co-
e icien :
m
pb5
A
11
g
2
4
22
g
2
.~1.7!
I should be no ed ha Eq. ~1.7!is alid o
g
*
/(2
p
b
E). Consequen ly, in he ex eme high ba ie
~low empe a u e!limi ,
b
E→`, one will ul ima ely almos
always be in he spa ial di usion egime.
K ame s’ model, al hough simple, is o wide- anging
signi icance o a de ailed unde s anding and e alua ing he
in luence o he medium on eac ion a es. I has ound a i-
ous gene aliza ions o he ull ic ion ange,3non-Ma ko ian
ac i a ed a e p ocesses,4,5 mul idimensional sys ems,6and
cases wi hou de ailed balance7~ o a e iew see Re . 1!.In
a!
Pe manen add ess: Ins i u e o High Tempe a u es, 13/19 Izho skaya
S ee , 127412 Moscow, Russia.
JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 4 22 JANUARY 1999
21590021-9606/99/110(4)/2159/5/$15.00 © 1999 Ame ican Ins i u e o Physics
all hese in es iga ions he ba ie is assumed o be pa abolic,
hough his assump ion is no always me in eal physical and
chemical ba ie c ossing p ocesses. Fo example, he ba ie
o cha ge ans e eac ions is o en o a cusp-shaped o m.8
K ame s also de i ed he ansmission coe icien o a cusp-
shaped ba ie ,2U(x)52a
u
x
u
,
m
cusp~
g
→`!5a
g
A
1
2
p
b
,~1.8!
bu his exp ession is alid only in he asymp o ic limi o
la ge ic ion whe e Eq. ~1.1!can be app oxima ed by a
Smoluchowski equa ion. The e a e a ious a emp s in he
li e a u e o b idge he s ong ic ion limi esul o a cusp-
shaped ba ie wi h he TST alue,
m
TST51, a ze o
damping.9,10 An analogous in e pola ing o mula is known
o a qua ic ba ie .1Only e y ecen ly, Be ezhko skii
e al.11 ha e ex ended his o mula o an a bi a y nonpa a-
bolic ba ie o he o m
U~x!52~a/
a
!
u
x
u
a
.~1.9!
Thei gene aliza ion eads11
m
a
5
H
E
2`
`dx exp@
b
U~x!#
J
21
3
E
2`
`dx exp
H
b
F
U~x!21
2
g
2x2
G
J
.~1.10!
The e is, howe e , a ce ain i ony he e; he abo e o mula
ag ees wi h he known escape a es o nonpa abolic ba ie s,
bu ails o ep oduce he exac esul o a pa abolic ba ie .
In he la e case, i yields ins ead o Eq. ~1.7!an app oxi-
ma e exp ession
m
25~11
g
2/
2!21/2.~1.11!
The aim o his pape is wo old. Fi s , we wan o
p esen an app oxima e a e o mula, which indeed is alid
o a bi a ily shaped ba ie s and in e pola es be ween he
limi s o small and la ge ic ion. And second, we wish o
compa e his o mula wi h exac nume ical a es in di e en
ypes o po en ials.
II. INTERPOLATING FORMULA
To begin wi h we conside he spa ial di usion egime.
Ou pu pose is o de i e an app oxima e solu ion o he
Fokke –Planck equa ion which would allow one o eco e
Eqs. ~1.7!and ~1.10!. This goal can be achie ed in many
di e en ways.12 He e we employ he lux o e popula ion
me hod de eloped by K ame s.2Wi hin i s scope, he escape
a e is de ined as he a io o a s a iona y di usion cu en a
he op o he ba ie o he popula ion o he well. Acco d-
ingly, we ha e o look o a cu en ca ying s a iona y p ob-
abili y densi y P(x, ), ha smoo hly ma ches he equilib-
ium dis ibu ion
Peq~x, !5exp@2
b
V~x!21
2
b
2#~2.1!
in he well and anishes beyond he ba ie . The wo s a ion-
a y densi ies a e ela ed by a o m unc ion
j
(x, ),
P~x, !5
j
~x, !Peq~x, !,~2.2!
which is de e mined om
$
2
]
x1@V8~x!2
g
#
]
1
gb
21
]
2
%
j
~x, !50. ~2.3!
Once he o m unc ion is known, he eac i e lux o mula
yields o he ansmission coe icien
m
5
b
E
2`
`d
j
~
0, !exp
S
21
2
b
2
D
.~2.4!
Following K ame s, we app oxima e he po en ial V(x)
en e ing Eq. ~2.3!by i s ba ie pa U(x). The la e is no
necessa ily pa abolic, i may be a sum o a bi a y ~pa abolic
and nonpa abolic! e ms
U~x!52 1
2
2x22a
a
u
x
u
a
2¯.~2.5!
Mo eo e , we assume ha
j
(x, ) is a unc ion o some lin-
ea combina ion o xand ,
j
~x, !5
j
~%!,%5cx1b .~2.6!
Then, i is no di icul o check by di ec subs i u ion ha in
leading o de in %and (
b
E)21an app oxima e solu ion o
Eq. ~2.3! eads
j
~x, !5Z21
E
%
`dy e
b
U~y!,~2.7!
wi h
%5
A
/~
gm
pb!@x2~
m
pb /
! #.~2.8!
In he abo e
m
pb is gi en by Eq. ~1.7!, while he no maliza-
ion cons an Zis de ined by he equi emen ha he o m
unc ion
j
(x, ) app oaches uni y in he ini ial well and ze o
in he p oduc side. This immedia ely yields
Z5
E
2`
`dy e
b
U~y!.~2.9!
I will be ecalled he e ha he ba ie ~ empe a u e!is as-
sumed o be high ~low!enough so ha he po en ial can be
well app oxima ed by i s local beha io in he icini y o he
ba ie op. O he wise one can use in Eqs. ~2.7!and ~2.9!
ins ead o he ba ie pa U(x) he ull po en ial V(x) i sel .
In such a case, he in eg a ion has o be es ic ed o he
ba ie egion wi h a lowe limi a , say, xwand he uppe
limi a a alue beyond he ba ie om whe e he ec ossing
p obabili y o a pa icle wi h ze o ini ial eloci y can sa ely
be neglec ed.
Inse ing Eq. ~2.7!in o Eq. ~2.4!, we ob ain he ollow-
ing exp ession o he ansmission coe icien :
m
ab5Z21
E
2`
`dx exp
H
b
F
U~x!21
2~
g
/
m
pb!x2
G
J
.
~2.10!
I is a simple ma e o check ha o a pa abolic ba ie he
abo e o mula coincides wi h he exac K ame s esul , Eq.
~1.7!, while o a pu ely nonpa abolic ba ie (
50) i e-
p oduces Eq. ~1.10!. One may also no e ha i ag ees in he
limi ing case o high ic ion wi h he ansmission ac o o
an a bi a ily shaped ba ie ollowing om he co espond-
ing Smoluchowski equa ion13
2160 J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey
m
~
g
→`!5
H
g
A
b
2
p
E
2`
`dx e
b
U~x!
J
21
,~2.11!
and educes o uni y a ze o damping.
A a e exp ession alid in he ull damping ange can be
ob ained by making use o an elegan app oach de eloped by
Mel’niko and Meshko .3This gi es in a s aigh o wa d
way
m
5
m
abA~D!,~2.12!
wi h
A~D!5exp
S
1
p
E
0
`dx ln
$
12exp@2D~x211
4!#
%
x211
4
D
,
~2.13!
whe e Dis gi en by Eq. ~1.4!. I should be no ed ha he
ansa z o w i ing a uni o m o mula o nonpa abolic ba ie s
as a p oduc o a spa ial di usion exp ession and he depopu-
la ion ac o Ais ad hoc. I ollows nei he om Mel’niko
and Meshko no om Pollak, G abe , and Ha
¨nggi u no e
heo ies. I is ou aim he e o p o e he u ili y o Eq. ~2.12!
by compa ing wi h exac nume ical a es. The la e is no so
ob ious as one migh hink. Speci ically, Mel’niko and
Meshko de i ed he depopula ion ac o ~2.13!unde he
assump ion ha he escape dynamics can be desc ibed by a
p obabilis ic in eg al equa ion in ene gy-ac ion a iables,
whose G een unc ion co esponds o he ba ie ajec o y.
Fo a smoo h po en ial he ajec o y ha lea es he ba ie
wi h he en i e ene gy close o ze o e u ns o i a e ime
T→`. This in ini e ime, howe e , is no longe ue o a
cusped ba ie whe e he ime is o he o de o he pe iod o
pa icle oscilla ion in he well. Thus he in e es ing issue we
shall add ess in ou nume ical applica ions is as ollows:
Does he ini e pe iod o he ba ie ajec o y spoil he ap-
plicabili y o Eq. ~2.12!?
III. NUMERICAL RESULTS
The aim o his sec ion is o p esen exac nume ical
a es o di e en ypes o po en ial ba ie s ha would allow
one o es analy ical p edic ions. One migh , a i s , belie e
ha his issue should ha e been se led long ago, mainly
because o i s con inuous impo ance in many p oblems o
chemical physics. To he bes o ou knowledge, howe e ,
he e a e no nume ical solu ions o such a ype, o he han
hose ob ained in Re s. 10 and 11 unde he assump ion ha
he po en ial consis s only o a ba ie pa . This assump ion
esul s in a mono onic dependence o he ansmission coe -
icien on
g
; he coe icien inc eases wi h dec easing
g
and
eaches i s maximal alue a ze o damping, when he e is no
coupling be ween he sys em and he ba h. I is clea ha he
da a so ob ained a e no sui ed o es ing analy ical p edic-
ions in he mos p oblema ic in e media e and weak damp-
ing egimes.
He e we deal wi h ac i a ed a e p ocesses in a symme -
ic double-well po en ial o he o m
V~x!5E
112a@x424a
u
x
u
22~12a!x2#,a.2 1
2.
~3.1!
I s ba ie pa a ies wi h he pa ame e a om a pu ely
pa abolic (a50) o a pu ely cusped (a>1) ba ie , see Fig.
1. Acco dingly, he equency
en e ing ou a e exp ession
eads
25
H
4~12a!E/~112a!21
2,a<1,
0a.1. ~3.2!
The me hod used o nume ically sol e Eq. ~1.1!will be de-
sc ibed elsewhe e.14,15 Table I shows a lis o he i s non-
ze o eigen alue in he conside ed po en ial o
b
E510 and
a50, 0.5, and 1. The calcula ion is pe o med o e a la ge
ange o
g
which co e s all egimes o chemical in e es ,
om he unde damped B ownian mo ion o he spa ial di u-
sion egime.
Be o e es ing he alidi y o he p esen a e exp ession,
we no e ha Eq. ~2.12!gi es he ansmission coe icien o
he escape om a me as able s a e. Using he app oach sug-
ges ed by Mel’niko and Meshko ,3 he coe icien o a
symme ic double well can be w i en as
m
5
m
abA2~D!/A~2D!.~3.3!
FIG. 1. Di e en shapes o he po en ial V(x), Eq. ~3.1!, o a50~ he
dashed line!and a51~ he solid line!.
TABLE I. Fi s nonze o eigen alue o symme ic double-well po en ials,
Eq. ~3.1!wi h
b
E510 and a50, 0.5, and 1. Exponen ial no a ion 2k
means ha he numbe p eceding is o be mul iplied by 102k.
g
a50a50.5 a51
0.05 0.17124 0.14624 0.14424
0.1 0.30424 0.25924 0.24724
0.25 0.59324 0.49424 0.45624
0.5 0.86824 0.71324 0.64024
1 0.10623 0.88924 0.79124
2 0.10623 0.95224 0.85624
5 0.85824 0.91424 0.85024
10 0.60724 0.78624 0.77024
20 0.36124 0.55724 0.56424
50 0.15424 0.26824 0.28624
100 0.78025 0.13324 0.14524
1000 0.78326a0.13425a0.14725a
aExac es ima e o he eigen alue calcula ed om he espec i e Smolu-
chowski equa ion.
2161J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey
The leas non anishing eigen alue o he co esponding
Fokke –Planck ope a o is hen gi en by wice he a e de-
ined by Eq. ~1.2!. The nume ical alues o he ansmission
coe icien ex ac ed in his way a e exhibi ed in Fig. 2, o-
ge he wi h he analy ical p edic ions ob ained in e ms o
Eq. ~3.3!. As e idenced by Fig. 2, he app oxima e a e ex-
p ession gi es an uppe bound o he exac esul o he a e
in he pa abolic double-well po en ial. Fo he cusped po en-
ials he heo y o e es ima es he a e in bo h limi s o weak
and s ong ic ion and unde es ima es i in he in e media e
ic ion egion. I is also seen ha o all alues o a he bes
ag eemen is achie ed in he s ong damping limi (
g
*100). Wi h dec easing
g
he e o made by he ansa z ~3.3!
inc eases and eaches maximal alues in he weak damping
egion (
g
&0.1). The heo e ical exp ession o e es ima es
he a e in his egion by 14% o a pa abolic ba ie (a
50) and by 18% o a pu ely cusped ba ie . I should be
poin ed ou ha he same is ue o he u no e heo y o
Pollak, G abe , and Ha
¨nggi.5As we ha e shown in ecen
pape s,15,18 hei heo y also conside ably o e es ima es he
a e in he weak ic ion egime.
Finally, o conclude his sec ion we no e ha he ba ie
equency
appea ing in Eq. ~2.10!may s ill be le e en i
he ba ie is pu ely nonpa abolic. In such a case, i should be
ea ed as a a ia ional pa ame e .16 Ye ano he way o im-
p o e he a e o mula is o ake in o accoun ini e-ba ie
co ec ions. These a e ob ainable sys ema ically in bo h e-
gimes o weak17 and in e media e o s ong ic ion.12 A u -
he imp o emen o he o e all a e exp ession can be
achie ed by using in Eq. ~2.10!a p ope ly de e mined ene gy
loss o he de e minis ic pa icle dynamics. In con as o he
weak ic ion exp ession o Dp oposed by Mel’niko and
Meshko , Eq. ~1.4!,3as well ha sugges ed by Pollak, G ab-
e , and Ha
¨nggi5in hei u no e heo y, he de e minis ic
app oach o his quan i y yields an app oxima ion which e-
mains co ec in he ull damping ange, ega dless o he
pa icula shape o he po en ial ba ie .15,18
IV. CONCLUDING REMARKS
In his pape , an app oxima e o mula o he a e o
escape o e an a bi a ily shaped ba ie has been con-
s uc ed by means o he lux o e popula ion me hod and he
app oach by Mel’niko and Meshko . The esul ing exp es-
sion ag ees in he limi ing case o high ic ion wi h he a e
ollowing om he co esponding Smoluchowski equa ion
and, in he ex emely unde damped egime wi h he a e ob-
ained by K ame s om a di usion equa ion in ene gy ~ac-
ion! a iables. I gene alizes in a na u al way he known a e
o mulas o pa abolic and nonpa abolic ba ie s.
Besides, we ha e p esen ed o he i s ime nume ically
exac a e cons an s o po en ials wi h di e en ba ie
shapes in all egimes o chemical in e es , om unde -
damped o o e damped B ownian mo ion. These esul s p o-
FIG. 2. T ansmission coe icien and pe cen age e o ,
1003~app oxima e2exac !/exac , made in
m
by using
Eq. ~3.3!. Exac nume ical esul s a e shown by ci cles.
~a!a50; ~b!a50.5; ~c!a51.
2162 J. Chem. Phys., Vol. 110, No. 4, 22 Janua y 1999 A. N. D ozdo and J. J. B ey

ides he necessa y ounda ion o es ing a ious di e en
a e exp essions ha al eady exis in he li e a u e. Compa i-
son wi h he nume ical da a shows ha he p esen o e all
a e exp ession is a he accu a e in he s ong damping limi ,
unde es ima es he a e by ;0%–18% in he in e media e
ic ion egion and o e es ima es he a e by ;14%–23% in
he weak damping egime.
ACKNOWLEDGMENTS
One o us ~A.N.D.!is g a e ul o A. M. Be ezhko skii,
P. Talkne , and V. Yu. Zi se man o many help ul discus-
sions. We acknowledge he suppo o he Di eccio
´n Gene al
de In es igacio
´n Cien ı
´ icayTe
´
cnica o Spain o inancial
suppo ~A.N.D.!and o P ojec No. PB95-534 ~J.J.B.!.
1P. Ha
¨nggi, P. Talkne , and M. Bo ko ec, Re . Mod. Phys. 62, 251 ~1990!.
2H. K ame s, Physica ~U ech !7, 284 ~1940!.
3V. I. Mel’niko and S. V. Meshko , J. Chem. Phys. 85, 1018 ~1986!.
4R. F. G o e and J. T. Hynes, J. Chem. Phys. 73, 2715 ~1980!;P.Ha
¨
nggi
and F. Moj abai, Phys. Re . A 26, 1168 ~1982!.
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