Diffusion in a class of exactly solvable non-harmonic potentials. Intrinsic effects induced by non-linearities

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