Volume75A, number3 PHYSICSLETTERS 7 January1980

BROWNIAN MOTION WITH ABSORBING BOUNDARIES

S. HARRISDepartmentof Physics,Universityof Surrey, Guildford, Surrey, UKand Collegeof Engineering,SUNY,StonyBrook,NY11794, USA

Received31 October1979

We considerthe I -D problemof brownianmotion with two absorbingboundaries.A recentlyformulatedintegralequa-tion approach(for the oneboundaryproblem) is used,in suitablymodified form, to obtain resultsfor the full position,velocity distributionfunction.

In this letterwe will be concernedwith the l-D the first absorbingeventoccurringat O,L respectively.brownianmotionbetweentwo absorbingboundaries, Thesefunctionsdescribethe first absorbingevent,notspecificallythedeterniinationof theposition,velocity thefirst eventwithvelocity v given a prior eventwith(q,v) distribution function for this problem.Some velocity v’ ~ v, andtakeinto accountthepresenceofyearsagoWangandUhlenbeck[I] notedthe difficul- bothboundaries,i.e. anabsorptionat q = 0 at time t

tiesof the singleboundarycase,whichhasremained precludesanearlierabsorptionatL andvice versa.an openproblem.Theyconsideredthis problemusing TheLaplacetransformsof thesefunctionssatisfy thethe Fokker—Planckequation(FPE),but wereunable following set of coupledintegralequationswhich gen-to solveit becauseof the lackof a boundarycondition eralizetheresultsof ref. [2]:for incidentparticles.We haverecentlyre-formulated

f0(v, s;x,~)= F(0, v, s;x0)this problem[2] usingan integralequationapproachand obtainedan approximatesolutionin the high fric- 0tion limit (HFL; seebelow for a precisedescriptionof — J” dv’ Iv’lfd(v’, s;x0)F(0,v,s;0, v’)this limit). At longtimesour result reproducesthe dif- —~

fusion equationresult for thenumberdensity,but ingeneralwe find a morecomplicatedbehavior.Two r ‘ ‘

— I dVIVIfr(V s;x~)F(0,u,s;L,v), v<0,specific integralequationapproacheswere consideredin our earlierwork. The first, basedon first turning (la)points [2], is lessphysicalbut leadsto aWiener—Hopf 1

fL(u,s;xA)F(L,v,s;x )equationwhichin principleis solvableby standardmethods.Thesecondis basedon first passagetimes, 0

andis bestsuitedfor generatingapproximateresults — f dv’ u’I f~(v’,s;x0)F(L,v,s;0, v’)[3]. In what follows we usethelatterapproachto —~

formulatethetwo-boundaryproblemand obtainanapproximatesolutionto the resultingequationsin r ‘ ‘ 1 ‘

theHFL. —J dvjvlfL(v,s;xo)F(L,v,s;L,v), v>0,We denoteq, u byx,q0,vo = x0 arethe initial 0 (ib)

data,andwe considerabsorbingbarriersatq = 0, Lsothat 0 <q0<L. Definef~(v, t; x0),v < 0, and whereF is the distribution function for thecaseoff~(v,t; x0), v >0 asthedistribution functionsfor no boundaries(infinite-space)ands is the Laplace

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Volume75A, number3 PHYSICSLETTERS 7 January1980

variable.Thedistribution functionf(x, s;x0) for the andIL(S,x0) f~°dv vf)~(v,s;x0) givenby an iden-two-boundaryproblemis thengiven as tical expressionwithq = 0 replacedby q L every-

whereand viceversa;theconstantA = (kT/2irm)L’2.

f(x, s;x0)= F(x, s;x0) Substitutinginto eq.(2) and making useof someob-

0 viousidentitiesthen gives— f dv’lv’if~(v’,s;x0)F(x,s;0,v’) f(x,t;x0)=F

0(v)[n(q,s;q0)

—~{(l+a)n(q,s;—q0)—an(2L +q,s;q0) (7)

— f dv’lv’l f~(v’, s;x0) F(x, s; L, v’), + (1 + a) n(2L — q, s; q0) — an(2L — q, s; —q0)}]0 wherecs=A(4Ds)—

1/2and /3=a[{1 +a}2 — {An(0,0~<q~<L. (2) s;L)}2]~.

- In the limit L -~ oo eq.(7) reducesto our previousWecanshowthat if f is the solution to theFPE subject result [3] for thesingleboundaryproblemwhich atto theboundaryconditionsfor two absorbingbounda- long timesreproducesthe diffusion equationresultries,f(0, v, t; x

0) = 0, v >0 andf(L, v, t; x0) = 0, for 5 dv f. For finite L the longtime limit of eq. (7)v< 0 then doesnot,however,appearto reduceto the diffusion

f= [U’ ~— O(L~1f (3) equationresult,which is given asan infinite series[4].‘ ‘~ ‘ It may well be that in thetwo-boundaryproblemthere

whereU is the Heavysidefunction,is compatiblewith is no hydrodynamiclimit, i.e. no timescalefor whicheq.(2). Operatingon eq.(3) with si~wheresr~f= 0 is the diffusion equationis valid, and only theFPE,ortheFPEgives eq.(2), providesanadequatedescription.Thiswould

- notaffectthe validity of the HFLwhich pertainsonly

= —lvi ~(q)f(0, v, t;x0) to thepropertiesof the infinite spacesolution.In the— v~(q— L)f(L, v, t; x0) (4) limit L -÷oo thediffusion equationis certainlyvalid,

at longtimes;aswe haveremarkedabove,in the limitandthe sameoperationon eq.(2) gives t -÷ oo, L -÷oo our resultsare consistentwith thediffu-

1 sionequation(note,L —~oo mustbe takenfirst here).~f= —lvi ~(q)f0(v, r;x0)

— v~(q— L)f)~(v,t; x0). (5) Thiswork was supportedby the UniversityofSurreythroughthe awardof a Visiting Fellowship.

Becauseof the deltafunctions,it follows from thedefinitionsoff~andf~that eqs.(4) and(5) are iden- Referencestical. It thenfollows that ther.h.s.of eqs.(2) and(3)areidentical sincetheycanonly differ by g where [11M. Wang andG. Uhlenbeck,Rev. Mod. Phys.17 (1945)

9lg = 0 andtheinitial conditioninsuresthatg = 0. 323.The HFLapproximationconsistsof takingF(x, t; [2] S. Harris,J. Chem.Phys.(January15, 1980),to bepublished.

0 0 . [3] S. Harris in preparation.x0) F (v) n(q,t;q0)whereF is themaxwelhan [4] W. Feller,An introduction to probability theoryanditsdistribution and n is thesolutionto thediffusion applicationsII (Wiley, New York, 1966).

equation.Substitutingfor Fin eq.(1), multiplyingby lvi and integratingover _oo~v~0 andO ~v~oo,

respectively,allows usto find, e.g.

I0(s;x0)~f dv lvi f~(v,s;x0)A[n(0,s;q0)

+An(0,s;q0)n(0,s;0)—An(0,s;L)n(L,s;q0)]

X[{1 +An(0,s;0)}2_{An(0,s;L)}2]~ (6)

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