PhysicsLettersA181 (1993) 119—122 PHYSICSLETTERS ANorth-Holland

Renormalizationgroupapproachfor randomwalkson a disorderedbond-dilutedlattice

MichaelSchulzAbteilungfurTheoretischePhysik,Universität Ulm, W-7900 Ulm, Germany

Received9 June1993; revisedmanuscriptreceived29July 1993;acceptedfor publication30 July 1993Communicatedby V.M. Agranovich

Randomwalks in a d-dimensionalstochasticenvironmentwith a staticstructurefactorS(q)—. constandS(q)— q —2, respec-tively, areanalyzedby renormalizationgrouptechniques.In dependenceon thestrengthof thedisordernormal, anomalousorlimiteddiffusion (finite diffusionradius)for t-.co wasdetected.

It is well knownthatthebehaviorofrandomwalks timeevolutionisgivenby theone-stepmasterequa-on a diluted lattice with static disorderdiffers from tion [5]the situation in an external static stochasticforcefield. In the last case,ananomalousdiffusion is ex- -~-P(i,j,t)=w ~ A~,AJkP(J,k, t)

dt kpectedfor dimensionsd<2 andan arbitrarysmallbut nonzerostatic randomforcefield [1] (for d=2 — w ~ A

1kA~P(i,j,t) (1)follows adiffusionwith logarithmiccorrections[2]), k

in thefirst casewegetananomalousdiffusion, if the Note,thatA0~1 only if i and]are neighboringlat-latticehasa fractal structure[3,41.Therefore,for a tice sitesand the bond betweeni andj is occupied.randomdiluted lattice an anomalousdiffusion can This formulation guaranteesthat a transitionoverbe expectedonly at the percolationthreshold.The nonoccupiedbonds is strictly forbidden. A well-causefor thisis thedifferentbehaviorof therandom knowngeneralizationofeq.(1) isthebackwardjumpwalker indifferentenvirons.Fora randomwalkerin model [6—81on a bond-dilutedlattice [9]. Here,a randomforce field thereexists a nonzeropossi- threetypesof jumpsare possible:jumpsto thepre-bility to crossa barrier,whereasfor a walker in the viously occupiedsite with transitionrateWb, jumpssecondcasethe transitionoveran obstacleis strictly from a site ito itself with ratew(z—z1) (z is theco-forbidden. ordinationnumberof theoriginal regularlatticeand

In principle, a reasonablestartingpoint is a for- z the coordinationnumberof site i on the dilutedmulation like to the Fokker—Planckequation.The lattice)andjumps in all otherdirections(transitionbond-dilutedlatticeis givenby present(A~=1) and rate w). From ref. [9] the time evolution of theabsent(A,~=0) bonds (i, ~ are the numbersof the probabilitysites,which are neighboringthebond(if)) andonlyoccupiedbondscanbecrossedby thewalker.There- P( i, t) = ~ P( i, j, t)

Il ifore, one can obtain a Markovian descriptionbyconsideringthe historyof therandomwalker.In the (thesumincludesall neighboringlatticesites]ofthepreviouscasethe knowledgeof the time evolution point i) is givenbyfor the probability P(i, j, I) to find a walkerat sitei at time t, which hasoccupiedthesitejimmediatelyprior the lastjump, is necessaryandsufficient. This

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Volume181, number2 PHYSICSLETrERSA 4 October1993

(d2 d p p+2(Wb—w)+Zw

~ +[2(Wb—W)+WZ,]~ m0(p)= ~ p+Wb+W(Z1)

+ [Wb + w(z— 1)]~~1) andthe dimensionlessdiffusion constantD= <z> /P(:, t) z. Contraryto (3), this equationreflectsno longer

all detailsof the diluted lattice.While the informa-tion for the longrangescalesof the dilution is con-

= w ~ A0 P~,I) se~edcomplete,thedetailsofthe local structurearej

reducedto anirrelevantmeanfield description.The+ [wb+w(z— 1)1w ~ AIJP(J, 1). (2) diffusion limit needsmainlyinformation aboutthe

3

largescalebehaviorof thedistribution of the latticeSettingz1=<z> + 8z1andA(1 D,~<Z> /Z+ 6A~,(Dii is defects,i.e. the influence of the large scalelatticethenearestneighborfunction,D~,=1 if the latticesites structurebecomesimportantfor thebehaviorof thei and] are nearestneighborsandD~=0otherwise, particlefor t—*oo. In the longtimelimit t—~oo,which<z> is the averagedcoordinationnumberof the di- is equivalentto p—~0, we get m0 ~ p and(4) cor-luted lattice) with the connection&z~= >~6A~,,the respondsto the usualFokker—Planckequationtype.Laplacetransformationwithrespectto timeandthe In comparisonto the Fokker—PlanckequationwithFourier transformationwith anexternalrandomdrift forceF,

SA~=D~~aqexp[~iq(rj+rj)] (p+Dk2)+igoJdd1qk.F~_qP(q,p)=I*, (5)bfq

and eq. (4) hasonly a different structurein the integralterm.In particular,this differenceis evident,if the

P(i, t) = ~ P(q,t) exp(iq~r,) forceFhasa potentialV. In this casethe termk~qa~

in eq. (4) is replacedby ik (k—q)Vk_q. Notethat(q a first Brillouin zoneof thelattice), give g0 is an unessentialprefactor,which resultsfrom the

transitionto thecontinuumlimit in bothcases(eqs.{p

2+2[(wb — w)+zw]p (4) and (5)). In thefollowingweconsidertwo typesof bond-dilutedlattices.

+W[p+Wb + w(z—1)] <z> [I —a(k)]}P(k,P) (i) Thecorrelationbetweentheoccupiedbondsisrandom.Therefore,the correlationof the dilution

+z ~ a4[a(~q)—c(k—~q)]P(k—q,P)=Ik. becomes<aqaq’>=4~5(q+q’).

(3) (ii) The bond dilution showsa long rangecorre-lation.Hence,we get <aqaq’>=A/q

2o~5(q+q’).In thisa(q)= ~ exp(iq~g.) particularcasethecorrelationof the lattice structure

has the same form as the potential correlationis the well-knownlatticefunction (g are thevectors <v~Vq~>=4/q2ô(q+q’) (which correspondsto afrom a site to the nearestneighborson a regularlat- short range random potential force fieldtice) andI,. representsthe initial conditions.Forthe <F~F.>=Aqaqfl/q2o(q + q’)).determinationof the longtime behaviorof theran- Thecorrelationfunctioncanbeidentifiedwith thedomwalker, it is reasonableto usethe continuum staticstructurefactorS(q),which is in thefirst caselimit of eq. (3) for small k.Then,eq. (3) reduces S(q)=constand in the secondcaseS(q)—~q2.Toto investigatethe long time behaviorof the random

walker for small ~= d~— d, it is naturalto try to per-[m

0(p) +Dk2]P(k, ~o) form a dynamicrenormalizationgroupexpansionin

+g0 Jddq a~_4kqP(q,p) = ‘A, (4) powersof the parameterA for sufficientlyweak dis-

order [1,2,10,11]. The resultsof the renormaliza-

with the mass lion groupapproachare restrictedto the lowestnon-

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trivial order of . In principle, the dimensionless collapseregime, e.g. the meansquaredisplacementinteraction constantu0 =A Ag~SdD

2(Sd is the radiushasa finite limit <x2> ‘~ ~ const for t—~oo.unit surfaceof a d-dimensionalsphere,A is theusual This is the typical situationfor a diffusion on a di-cutoff) behaves under the rescaling procedure luted latticewith occupationprobabilityp, e.g. adi-k—~e~klike lution rate q= 1 —p. Forp<p~,the allowedbonds

form finite clustersand1im1.,~.<x

2> is determinedd/ — (u

0) by the finite meansquareclusterradius.Forp > p~,

the latticecontainsoneinfinite cluster,whichhason

with the well-known /3 function. At least, the mean large scalesa homogeneousstructure,e.g. the dif-squaredisplacementof the randomwalker is deter- fusionbecomesnormal <x

2> t. Only for adilutionminedby the scaling law rateq,= 1 —pa, which canbe identifiedwith the un-

stablefixpoint ur, we get an anomalousdiffusion(6) with

with the exponentC=2—y,,,y~=2+AaIn(Z0)/8A <x2>~tI_[d(~2)]/(22+8~). (7)

(Z0 is the field renormalizationconstant).The comparisonof the anomalousdiffusion coeffi-(i) Using the one-loopapproximation(fig. 1) of

perturbationtheorywe get in the first case cient /1 in <x2> t’~with the numericalresults [3]

allows the determinationof the valuesshown in ta-l+4d ) U

0 blel./3(uo)= _~0(~+(2+d)d U0 ~ c’ Clearly,the resultsareonly from the lowest order

of the e expansion.Therefore,onecanexpectthatwith thecritical dimensiond~= 0. Thatmeanse= — d the contributionsof thefollowing termswith higherisalwaysa negativevalue.The fixpoint u~=0 is sta- ordersof c realizea convergenceof the c-powerSe-bleforalld>0,whereasu~=IeId(d+2)/(ll +4d) nesanda stabilizationof ji at thevalue ~. Notethatisunstable,e.g. fora sufficientsmalluo<u~thedis- as a resultof the presentone-loopapproximation,jtorder increasesandtendsto the stablefixpoint U0 becomesnegativefor d~9, while accordingthestan-underrescaling.The diffusion exponentC becomes dardpercolationtheory~ = ~ for de6. However,weC= 2 andwe havethe normaldiffusion case.The in- havethe remarkablepoint that the renormalizationversesituationpresentsa disorder,U0> u~.Forlarge group analysisof eq. (4) determinesthe exponentu0> u~ the disorderbecomesmore and more im- 2/C for a random diluted lattice with occupationportantandtendsto infinity. As a result of this be- probabilityp andshowsan anomalousdiffusion forhaviorthe exponent2/C tendsto zeroandwe get a the unstablefixpoint. On the otherhand, this ex-

ponentis alsocalculableusingthePottsmodel [12]r (2) ____________ + for the percolationcase [111. Here,from the well-

known critical exponents/3, v and t follows C=2 + (1—/3)/v [4]. Consequently,thesameexponentsresult from different approaches,which are hardly

_______ relatedto eachother.

r - + I I + (ii) In thecaseof longrangedisorderwe get with

___________ ____________ Table I

d MRG l~rn,n,~rio

2 0.79 0.71+.“ 3 0.67 0.56

____________ 4 0.56 0.425 0.44 0.37

Fig. 1. Graphs ofthe one-loopapproximation for the vertex func- 6 0.31 0.33tions [‘(2) and [‘(4)~ ____________________________________________________

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Volume 181, number2 PHYSICSLETTERSA 4 October1993

thesamemethodacritical dimensiond~=2. Ford< 2 differentbehaviorof therandomwalker for differentthereexistsonly theunstablefixpoint U~= 0, e.g.for environs.eachdisorderstrengthA follows for t-+cc a finite Characteristicisthebehaviorofthewalker in d= 1:meandiffusion radius,contraryto the situationin a the exactsolution of (3) givesa finite diffusion ra-randompotentialforce field [2], wherean anoma- dius for t—* ~ in environsof randomdistributedob-busdiffusion is availablefor d< 2 andan arbitrary staclesand on the otherhanda scaling law for theA. Ford> 2 wehavea situationanalogoustothe first diffusion in a randomforce field [13] <x2> ln4i.case,e.g.a stablefixpoint U~= 0 andan unstablefix- Consequently,the differencebetweenthe two dif-point ur = I ci d, from which follows a normal dif- fusion processesis very strongand we canexpectafusionfora smalldisorderU

0<ur andafinite mean continuationof thisbehaviorin higherdimensions,diffusion radiusfor the oppositecase u0> U~. An as shown in the presentpaper.anomalousdiffusion follows also only for onedis- Whereasthe diffusion in a static stochasticforceorder strength.Only here,we get field leadsto ananomalousdiffusion belowa critical

dimensiond =2 anda normal diffusion aboved~/ 2\ ,,~,tI_(d_2)/8 ‘8~ C\ X / . ‘ / onegets in thecaseof static randomdistributedob-

Note that for d> 2 in a randompotentialforce field staclesbelow d~only a limited diffusion in a finitefor eachA follows a normal diffusion. A lastspecial region,whereasthreeregimes(dependingonthedis-caseis the behaviorat the critical dimensiond= 2. order strengthA) exist for d> d~:normal diffusionHerewe get for weakly disorder, limited diffusion in a finite re-

gionfor strongdisorderandanomalousdiffusion for

—2U2 onevalueA, which canbeinterpretedasthe strength

8! — of the disorderat the percolationthreshold.andconsequently,usingtherenormalizationrelationfor thecharacteristicfrequency(mass)scale0m(1)/81= zm(1), it follows that References

lfl(m(l)’~=2l_~lfl[l_2U0(O)l]. (9)

\m(0)J [I] D.S.Fisher,Phys.Rev.A 30 (1984)960.

[2] D.S. Fisher, D. Friedan, Z. Qui, S.J. Shenker and S.H.At sufficiently long times t, with correspondingbar Shenker,Phys.Rev.A 31(1985)384.

massm(0) l/t, themeansquaredisplacementres- [3] D. Stauffer,Phys.Rep.54 (1979) 1.

cales as e21. Thereforewe get from eq. (9) with [41S.AlexanderandR.Orbach,J. Phys.(Paris)Lett. 43 (1982)

m(1) const [5] C.W.Gardiner,Handbookof stochasticmethods(Springer,

/ 2~ Berlin, 1983).\X / (10) [6]R. Fuerth,Z. Phys.2 (1920)244.

(1—u0ln<x

2> ) ~ [7] J.W. HausandK.W. Kehr,Phys.Rep. 150 (1987)265.[8] M.H. Ernst,J.Stat.Phys.53 (1988) 191.

From this equationwe concludethat the meandif- [9] R. Hilfer, Phys.Rev.B 44 (1991)638.

fusion radiusat longtimesis Rma,, e’,‘2~~• [10] E. Medina, T. Hwa, M. KardarandY.C. Zhang,Phys.Rev.In principle, we find a remarkabledifferencebe- A 39 (1989)3053.

tween the diffusion in randomforce fields (or ran- [11] D.J.Amit, J. Phys.A 9 (1976) 1441.[12]R.B. Potts,Proc.CambridgePhilos.Soc.48(1952)106.

dompotential fields) on theone handandthe dif- [13] G. Sinai, in: Proc. Berlin Conferenceon Mathematicalfusion in a systemwith randomdistributedobstacles problemsin theoreticalphysics,eds.R. Schrader,R. Seiler

on the otherhand.Thecauseis the abovementioned andD.A. Ohlenbrock(Springer,Berlin, 1982).

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