Physica D 41 (1990) 79-88
No h-Holland
DIFFUSION IN A CLASS OF EXACTLY SOLVABLE NON-HARMONIC POTENTIALS.
INTRINSIC EFFECTS INDUCED BY NON-LINEARITIES
J.L.
ROMERO and J. RAMIREZ
Depa amen o de Ma em~ icas, Unioe sidad de Cddiz, P.O. Box 40, Pue o Real (Cadiz), Spain
F. ROMERO and J.F.R. ARCHILLA
Depa amen o de Fisica Te ica, Facul ad de Fisica, P.O. Box 1065, Seoilla, Spain
Recei ed 1 Ma ch 1988
Re ised manusc ip ecei ed 11 Decembe 1988
Accep ed 17 June 1989
Communica ed by A.V. Holden
This pape deals wi h he p oblem o a pa icle ha di uses in a po en ial wi h a e lec ing ba ie and has a poin o s able
equilib ium and a poin o uns able equilib ium. Based on he exac solu ions ob ained ea lie o he Fokke -Planck equa ion
o a class o hese models, we analyze he beha io o he p obabili y densi y, he mean pa h and he onse ime which
de e mines he ansi ion om unimodal o bimodal p obabili y densi ies. The s udy is made o e di e en ini ial posi ions,
wo o hem e y close o he uns able poin , which pe mi s a clea compa ison among he subsequen e olu ions, and he
obse a ion o some in insic e ec s induced by non-linea i ies.
1. In oduc ion
The p oblem o di usion in one-dimensional
po en ials, and pa icula ly in non-ha monic po-
en ials, has been he subjec o many ecen s ud-
ies. One o he mo i es o he impo a ~ce o his
p oblem lies in i s connec ion wi h Lange in's
desc ip ion o non-equilib ium phase ansi ions.
When he whi e noise cons i u es a good idealiza-
ion o he luc ua ions conce ning a mac o a i-
able, he p obabili y densi y o his mac o a iable
obeys a Fokke -Planck equa ion (FPE), which
mus be sol ed unde app op ia e bounda y con-
di ions.
In he heo y o eac ion kine ics, many s udies
use a model o a pa icle mo ing in a one-dimen-
sional po en ial unde he e ec o he mal noise
and damping. Following K ame s [1], many use a
model o a po en ial wi h a single well and one
ba ie . Ini ially he pa icle is placed in he po en-
ial well and can only escape by passing o e he
po en ial ba ie . The objec i e is o calcula e he
a e o escape ou o he well, and he depend-
ence o his a e on empe a u e, ic ion, and
he pa ame e s o he po en ial. In he high-
ic ion egime one is led o analyze a FPE o he
Smoluchowski ype [2, 3].
The model o K ame s has played a cen al ole
in many a eas, o he han chemical eac ions in
condensed phases [4], such as su ace deso p ion
[5] and su ace ca alysis [6]. Since he o iginal
analysis by K ame s a numbe o au ho s ha e
conside ed pa ame ic o ms o he po en ial in an
a emp bo h o desc ibe he e olu ion o di e en
p ocesses and o check he alidi y o K ame s'
esul s. One o he i s in es iga ions is due o an
0167-2789/90/$03.50 © Else ie Science Publishe s B.V.
(No h-Holland)
80
J.L. Rome o e aL / Di usion in a class o exac ly sol able non-ha monic po en ials
Kampen [7], who showed ha K ame s' calcula-
ion leads o a co ec alue o he case o a
sol able double-well symme ic po en ial.
Owing o i s impo ance he e has been consid-
e able esea ch bo h in he de elopmen o ap-
p oxima ion schemes [8-10] and in ob aining
exac ly sol able models [4, 11] which co e a ich
a ie y o beha iou s. Al hough he ma hema ical
exp essions o he po en ials ob ained o hese
exac ly sol able models a e ela i ely complica ed,
hei quali a i e o ms a e e y close o some
in e es ing po en ials o which no exac solu ions
exis . We hink ha he s udy o hese models is
impo an o wo easons. Fi s , o lea n abou
in insic e ec s induced by non-linea i ies, such as
ime-con olled ansi ions om unimodal o bi-
modal p obabili y densi ies, b anching imes and
boome ang beha iou o he mean pa h. Second,
exac solu ions can se e o es he alidi y o he
di e en app oxima ion me hods.
In e . [12] we showed he exis ence o a la ge
class o non-linea s ochas ic p ocesses, wi h con-
s an di usion, which admi s exac solu ions. This
class is de ined by a wo-pa ame e amily o
non-ha monic po en ials, and includes po en ials
wi h ei he e lec ing o abso bing ba ie s. This
pape conce ns wi h a pa icula model o ha
amily: he model o he po en ial has a e lec ing
ba ie a he o igin o coo dina es, a s able equi-
lib ium poin and an uns able equilib ium poin .
I is analogous o he po en ial used by K ame s
[1] in he con ex o eac ion kine ics, al hough in
ou case we can only ea he high- ic ion egime,
as we only know he exac solu ions co esponding
o he FPE o he Smoluchowski ype.
This pape is o ganized as ollows. Fi s , we
expose a summa y o some p e ious esul s, show-
ing a pa icula class o models which has exac
solu ions, i s explici solu ions, and he h ee qual-
i a i ely di e en kinds o po en ials ha a e in-
cluded. Second, we analyze a model o he ype
men ioned abo e by adequa ely selec ing wo al-
ues o he pa ame e s ha de ine he amily o
po en ials. Ha ing cen e ed on a speci ic model,
we expose de ails on he e olu ion o he p obabil-
i y densi y, he beha iou o he mean pa h and
he spli ing o he p obabili y densi y, o di e -
en ypes o ini ial condi ions. We conclude wi h
some commen s abou he obse ed phenomena
and some u he s udies ha a e unde way.
2. Smmna y o p e ious
esul s
We conside he s ochas ic di e en ial equa ion
(Lange in equa ion) o a d i en a iable x( ):
dx _ D(x)
+ 71( ) = dU(x)
d d------x~ + T/( ), (1)
whe e
U(x)
is a po en ial ield om which he
d i
D(x)
is de i ed, and 7/( ) is a Gaussian
whi e noise, cha ac e ized by
(,/( )) =0, (2)
(~/( ) ~i( ')) = 28( - '), (3)
wi h he ini ial condi ion being
x(0) =x o. (4)
Unde hese condi ions, he co esponding FPE
o he dis ibu ion unc ion
P(x, )
is
OP(x, )~ =-~[[~]a [[~U(X) Ip¢ , )]
a2p(x, )
+
~X 2
wi h he p ope ies
P(x, )>_O, Vxen,
eR+=[0, oo), (Sa)
P(x, )dx=l, V ~a +,
(5b)
P(x,o) -- a(x - go). (5c)
Fo some cases i may be equi ed ha x ~ R +.
J. L Rome o e al./ Di usion in a class o exac ly sol able non-ha monic po en ials 81
The e exis exac solu ions o his FPE o he
special choice o he po en ial U(x):
U(x)
= -21n V(x), (6)
wi h
V(x) = (½x 2)"+ 1/4 exp (- ¼x 2)
X ,Fx(A, B,
½x2), (7)
whe e
,FI(A, B, z)
deno es Kumme 's hype geo-
me ic unc ion o he i s kind [13], and
A=½+a-~ l,
B=l+Za, (8)
wi h
li ,)
o] A=O B:6
cl I : .18/5 I~ : 4.5
Fig. 1. Shapes o
U(x)
wi h a = 0.25 and (a) l = 6, (b) /8 =
5.99, (c) l = 4.5.
a>-}, A>0, (9) wi h
When a > - ¼, he po en ial may adop some
physically in e es ing beha iou s. The h ee quali-
a i ely di e en o ms o
U(x)
when a > - ¼ a e
ep esen ed in ig. 1. All he cu es end asymp o -
ically o + oo when x ends o ze o, i.e. he poin
x = 0 ep esen s a e lec ing ba ie . When A = 0
he po en ials end o + m o x ending o + o0,
i.e. he po en ials a e a ac i e Vx ~ [i +, like ha
ep esen ed in ig. la. When A q. 0, he po en ials
end o - c~ o x ending o +
oo.
Fo a gi en alue o a >- ¼, he e always
exis s a alue
A(a)
> 0, such ha i A ~ (0,
A(a))
he po en ial adop s a o m like ha ep esen ed
in ig. lb, i.e. wi h one s able poin and one
uns able poin . The alue
A(a)
may be ound by
sol ing a anscenden al equa ion• In o he cases,
wi h A 4: 0, he po en ials ha e no dep h, i.e. hey
a e like ig. lc.
Fo he pa icula case a = ¼,
U(x)
adop s he
o m
O(x) = - 2 ln[(x/¢~)exp(- ¼x 2)
XxFl( A,a2, ½x2)],
(10)
a=l-i#.
Wi h he p esc ip ion ha x ~ R +, he solu ion
o
P(x, ),
wi h
P(x,O) = 8(x - Xo),
is
P(x, ) = Pl(X, ; Xo) + Pl(X, ; - go) ,
x ~ R +, (11)
whe e
Px( x, ; Xo)
= (4~) - '/2(sinh ) -'/2(X/Xo)
iFI(A,3 _; ½x 2)
• 1 2
X ,Fx(A, 3, _~Xo)
X exp(¼ l )exp(-¼(X-Xoe- )2e')
sinh
(12)
3. Analysis o he model
We shall now show ha ou class o models
co e s a g ea a ie y o beha iou s induced by
non-linea i y. Ne e heless, i is no possible o
ob ain an exac explici calcula ion o all he ele-
82
J.L. Rome o e aL / Di usion in a class o exac ly sol able non-ha monic po en ials
an physical quan i ies. The p incipal eason is
ha he e is no possibili y o choosing an ini ial
alue x 0 ha makes (11) a symme ical unc ion.
Thus, o example, o he calcula ion o he posi-
ions and he na u e o he ex ema o
P(x, ),
which a e de e mined by means o he equa ion
P'(x,
)---0, we a e led o he esolu ion o a
anscenden al equa ion, which canno gene ally
be sol ed analy ically.
To o e come his di icul y pa ially we ha e
made exhaus i e compu e de e mina ions, using
(11) and (12), o he mos ele an quan i ies and
o he ime e olu ion o
P(x, )
o se e al di e -
en alues o he ini ial posi ion x 0. In wha
ollows we will expose de ails o some esul s
conce ning hese compu a ions.
In ou s udy we ha e ixed he pa ame e s a
and l o he alues a = 0.25 and l --- 5.99. These
gi e a po en ial
U(x)
like ha o ig. lb, wi h one
s able poin x s and one uns able poin x~. The
compu a ion o x s and o x u gi es he alues
x s = 1.416 and x~ = 4.537. These pa ame e s co -
espond o a po en ial wi h enough dep h ha one
can expec a ich a ie y o phenomena o appea
easily. Fo his case U(xu) -
U(xs)
--- 5.39.
4. Time e olu ion o he p obabili y densi y
Fo he s udy o he beha iou o he p obabil-
i y densi y
P(x,
), and o some ele an physical
quan i ies, we may selec di e en ini ial posi ions
x0, which co espond o quali a i ely di e en ini-
ial physical si ua ions. We shall ske ch and com-
men on he ime e olu ion o he p obabili y
densi y
P(x, )
o cases which clea ly show phe-
nomena induced by nonlinea i y.
P(xA)
0.0 1.6
3.~
/
x,= 4.50
4.B 6.4 8.0 9.6 II.2 12.8
1(
Fig. 2. P obabili y densi y
P(x,
); x o = 4.50, = 0.1 ~ (0.1)
0.6.
e olu ion, o = 0.1 --* (0.1) ~ 0.6. A = 0.1 he
cu e is p ac ically Gaussian, he abscissa o he
maximum Xu is such ha x M < xu. Ne e heless
by = 0.2 one may app ecia e he exis ence o wo
peaks o he p obabili y densi y. The e exis s a
c i ical ime a, de ined as he ime o his spli -
ing o he p obabili y densi y such ha a _< 0.2.
We obse e ha he o dina e o he i s peak is
g ea e han he o dina e o he second one, and
ha he abscissa o his second peak is g ea e
han xu. The o ma ion o his second peak in he
egion (x~, oo) may be iewed as a " unneling
p ocess" h ough he po en ial ba ie d awn in
ig. lb. Ini ially he abscissa o he i s peak
mo es owa ds x s, and he abscissa o he second
peak mo es owa ds oo.
4.1.
Case
x 0 = 4.50
As he uns able poin co esponds o xu=
4.5379 , he ini ial si ua ion is nea his uns able
poin . One may expec phenomena such as he
o ma ion o wo maxima o he p obabili y den-
si y, i.e. he occu ence o a ansi ion om uni-
modal o bimodal s a es. Fig. 2 shows us his
4.2. Case x o = 4.60
Now x 0 >__ x u. The e is also a ansi ion o bi-
modal s a es a e a ime g ea e han 0.1, as we
can see in ig. 3. The " unneling p ocess" leads o
he o ma ion o a maximum in he egion [x s, xu],
i.e. i is p oduced in backwa d di ec ion. I we
J.L. Rome o e aL / Di usion in a class o exac ly sol able non-ha monic po en ials
83
P x,L)
6o
0 2
V
6 8 lO 12 14 16 x
Fig. 3.
P obabili y densi y
P(x,
); x o = 4.60, =
0.1 --*
(0.1)
--, 1.1.
compa e he o ms o
P(x, )
o --0.3 co e-
sponding o he cases x 0 = 4.5 and x 0 = 4.6, we
can see ha o he second case he cu e is mo e
symme ical, which is a consequence o he asym-
me ical o m o he po en ial in he egion a ound
x,. La e on we shall commen on his asymme y.
Fo
he imes conside ed in ig. 3, i.e. = 0.1
(0.1) ---, 1.1, he e exis s a mono onic mo emen o
he abscissa o he i s peak owa ds x,. A ime
= 1.1 an inc ease in he o dina e o he i s peak
wi h espec o he o dina e o = 0.9 is obse ed.
The exis ence o his inc ease is much be e ap-
p ecia ed in ig. 4, whe e o = 60 he o dina e is
g ea e han 0.2. A successi e ins an s o ime he
o dina es dec ease mono onically and abscissas
a e s abilized nea x s. The exis ence o a e lec ing
wall a x = 0 p e en s he p obabili y densi y om
passing h ough i . Re lec ion is esponsible o he
ansien g owing o he i s peak and o he
exis ence o a second " unneling p ocess" which
causes a u he dec ease in he o dina es.
P(x,LI
6
8-
,e~_ _
0.0
x, : LGO
I. i 3.2 4.8 6.4 8.0 x
Fig. 4. P obabili y densi y
P(x,
); x o = 4.60, = 60 --, (60)
--, 110.
4. 3. Case Xo = 5
This is he las case we conside . The e exis s
also a ansi ion o bimodal s a es as a conse-
quence o a " unneling p ocess" h ough he po-
en ial ba ie . Now he o m o he cu es o
P(x, )
a e his ansi ion a e much mo e asym-
me ical (see ig. 5).
5. Beha iou o he mean pa h
Le us analyze he beha iou o he mean pa h
(x( )>
co esponding o he p ocess (1) when he
p obabili y densi y is gi en by (11) and (12). This
magni ude is de e mined by calcula ing he in e-
g al
(x( )> = o°°Xe(x, ) dx.
(13)
The analy ical ope a ions equi ed o calcula e
his in eg al a e qui e edious, so we ha e p e-
e ed o de e mine his in eg al ia compu e
calcula ion only o some special ini ial alues x 0.
No phenomena o in e es appea o he cases

84
J.L. Rome o e al./ Di usion in a class o exac ly sol able non-ha monic po en ials
PIx,[I
X =
5
0 4 8 12 16 20 24
X
Fig. 5. P obabili y densi y
P(x,
); x 0 = 5, = 0.1 ~ (0.1) --*
0.6.
x 0 < Xs, o x 0 > x u. We ha e selec ed wo cases
co esponding o x s < x 0 < x u.
In ig. 6 we ep esen
(x( ))
e sus o x 0 =
1.5. We obse e ha ini ially he alues o
(x( ))
dec ease un il a ime * ---0.1, and hen he mean
pa h inc eases mono onically, i.e. he mean pa h is
s abilized o < * and su e s a des abiliza ion
o > *. The minimum alue o
(x( ))
does no
a ain x s, (X>n~n ~ 1.491. This si ua ion is called
boome ang beha iou o he mean pa h [14].
Fig. 7 co esponds o he case x o = 3. We ob-
se e also he exis ence o an ini ial pe iod o
s abiliza ion ollowed by a des abiliza ion a *
1.5. As be o e he minimum alue o (x( )> does
no a ain xs, o his case (x)~n --- 1.81.
Fo he case x0=4.50 we de ec also a
boome ang beha iou , bu in his case * is o he
o de o 0.06, i.e. he ini ial pe iod o s abiliza ion
is negligible.
The nume ical s udies ha we ha e made
demons a e he exis ence o an in e al o ini ial
alues x o o which he p ocess p esen s a pe iod
<x>~
I0-'
,d-,
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Fig. 6. Boome ang beha iou o he mean pa h o x o = 1.5.
o s abiliza ion ollowed by a des abiliza ion o
he mean pa h. Al hough no demons a ed ma h-
ema ically, i could be in e ed ha he condi ion
o he exis ence o he so-called "boome ang
e ec " is ha x s < x 0 < x u.
6. T ansi ion o bimodal s a es. Times o
bi u ca ion
As we ha e poin ed ou in sec ion 4, o a gi en
alue x 0' he p obabili y densi y
P(x, )
p esen s
wo peaks a e a c i ical ime m called he ime
o bi u ca ion o o ansi ion o bimodal s a es,
which depends ob iously on he ini ial posi ion
x 0. Bi u ca ion such ha he wo peaks ha e sig-
ni ican alues, occu s only o a limi ed in e al
o alues x o a ound x u.
We can conside he space (x o, ) as ha ing he
ole o a pa ame e space C, and (x) he ole o a
beha iou space X. In his way he p obabili y
densi y is ep esen ed by a smoo h map:
P: C x X ~ R. (14)
J.L. Rome o e aL / Di usion in a class o exac ly sol able non-ha monic po en ials
85
o
8
,-J.
...... , , , , , ,
0.0 0.4 0.8 1.2 1.6 2.0
Fig. 7. Boome ang beha iou o he mean pa h o x 0 = 3.
, , , , , , , , , , , , , , , ,
1.2 2.0 2.8 3.6 4.4 5.2 6.0 6.8 7.6 x,
The ime-dependen posi ions o he ex ema o
P(x, )
a e de e mined by he equa ion
3P(x, ) _ 0. (15)
P'(x, ) - Ox
The se M = ((x 0,
, x)/P'(x,
) = 0} cons i u es
a mani old, o dimension wo, ha can be consid-
e ed as a ca as ophe mani old. Each poin o M
co esponds o a minimum o a maximum o
P(x, ).
In pla es I and II we ep esen wo pe -
spec i e d awings o he mani old M. Axes a e
ep esen ed in blue, ed and black, and co e-
spond o x o, , x, espec i ely. Pla e I co esponds
o a wo-dimensional p ojec ion de ined by he
ec o ( -0.5, 5,4) o he space (x 0, , x), and pla e
II is a p ojec ion de ined by he ec o (0.6,
-0.6,5). In his way, i is possible o gi e he
same imp ession ha one would ob ain om look-
ing h ough he ac ual h ee-dimensional su ace,
along he di ec ion o p ojec ion. In bo h igu es,
he colo black, o he p ojec ed su aces, co e-
sponds o he second maxima o
P(x,
), ed
co esponds o he minima o
P(x,
), and blue
co esponds well o he i s maxima, o well o
~=,
,=;
8
Fig. 8. G aphical ep esen a ion o he beha iou o he ime
o bi u ca ion,
~,
o di e en alues o x 0.
he unique maxima o imes p io o bi u ca ion.
This mani old is like ha co esponding o he
so-called "dual cusp ca as ophe" o ca as ophe
heo y. In bo h pic u es he old line is ela i ely
well dis inguished, his being he smoo h cu e:
M 3 ((Xo, ,x)/P"(x, )
=0}. (16)
P ojec ion o he old line, on o he plane (x0, ),
p oduces a cu e, ep esen ed in ig. 8, ha cons i-
u es he bi u ca ion se , i.e. i gi es us o each
ini ial alue x0, he co esponding ime o bi u -
ca ion B. This bi u ca ion se has a singula i y o
X0m
~
4.577,
Bm =
0.137, and we obse e ha his
alue o Xom is sligh ly g ea e han he abscissa o
he uns able poin x u - 4.537.
Ou po en ial
U(x)
has an asymme ical o m
o a neigbou hood o x u, i.e. he uns able egion
has no a symme ical o m. This can be seen by
s udying he beha iou o he second de i a e o
U(x)
wi h espec o
x, U"(x),
in ha uns able
egion. The cu e, ep esen ed in ig. 9, has a
minimum a x m --- 4.620, U"(Xm) -~ -7.481, and
86 J.L. Rome o e al./ Di usion in a class o exac ly sol able non-ha monic po en ials
Pla e I. Two-dimensional p ojec ion o he ca as ophe mani old M, de ined by he ec o ( - 0.5, 5, 4).
hese alues pe mi us o de e mina e he coe i-
cien s o he ollowing app oxima e o m o U'(x)
o a neigbou hood o Xu:
U'( x ) ~ a( x -
xu) 3 +
b( x -
Xu) 2
+c(x-Xu),
(17)
esul ing in a -~ 7.756, b -~ - 1.931, c ~- - 7.295.
The ~ac ha b #: 0 indica es a non-symme ical
o m o U(x) in he uns able egion.
The minimum ime o bi u ca ion Bm co e-
sponds o an ini ial posi ion x 0 = X0m, such ha
x u < XOm < Xn We obse e a ela ion o Bm wi h
U"(Xom ) gi en by
1
/Bm ~
IU,,(XOm)[. (18)
J.L. Rome o e al. / Di usion in a class o exac ly sol able non-ha monic po en ials
87
Pla e II. Two-dimensional p ojec ion o he mani old, de ined by he ec o (0.6, - 0.6, 5).
By s udying he same ques ion wi h an exac ly
sol able model wi h a symme ical uns able e-
gion, as one o he models p oposed by Hongle
and Zheng [11], one ob ains ha he minimum
ime o bi u ca ion co esponds exac ly o an
ini ial posi ion loca ed a he uns able poin . Fo
an asymme ical uns able egion he e is a dis-
placemen o x0m owa ds he poin whe e he
cu a u e is g ea e , i.e. he poin whe e he abso-
lu e alue o he second de i a i e o U(x) is
g ea e .
7. Final commen
In his pape we ha e limi ed ou sel es o some
aspec s o he beha iou o he solu ions o an
exac ly sol able model o he K ame s ype. By
conside ing di e en ini ial condi ions i has been
possible o obse e phenomena induced by non-
linea i y, such as boome ang beha iou o he
mean pa h and spli ing o he p obabili y densi y.
We ha e quan i a i e es ima es bo h o he onse
ime B o his spli ing and he ime * which