JStatPhysmanuscriptNo. (willbeinsertedbytheeditor) BernardDerrida · MartinRetaux The depinning transition in presence of disorder: a toy model 4 1 0 2 n Received:date/Accepted:date a J 7 Abstract Weintroduceatoymodel,whichrepresentsasimplifiedversionofthe 2 problemofthedepinningtransitioninthelimitofstrongdisorder.Thistoymodel canbeformulatedasasimplerenormalizationtransformationfortheprobability ] h distributionofasinglerealvariable.Forthistoymodel,thecriticallineisknown c exactlyinoneparticularcaseanditcanbecalculatedperturbativelyinthegeneral e case.Onecanalsoshowthat,atthetransition,thereisnostrongdisorderfixeddis- m tributionaccessiblebyrenormalization.Instead,bothournumericalandanalytic - approaches indicate a transition of infinite order (of the Berezinskii-Kosterlitz- t a Thoulesstype).Wegivenumericalevidencethatthisinfiniteordertransitionper- t sistsfortheproblemofthedepinningtransitionwithdisorderonthehierarchical s . lattice. t a m PACS 05,05.10Cc,05.70Jh - d n January28,2014 o c [ Thedepinningtransitioninpresenceofimpurities[2,18,21,25,31,38,46,47, 1 50] (in the version of the Poland Scheraga (PS) model with uncorrelated disor- v der) is one of the simplest problems for which the effect of disorder, at a phase 9 transition,isnontrivial. 1 Though the problem is very simple to formulate and despite all the progress 9 doneoverthelast30years[26],manybasicquestionsonthepreciselocationof 6 the critical surface or on the nature of the depinning transition when disorder is . 1 relevantarestilldebated. 0 4 B.Derrida·M.Retaux 1 LaboratoiredePhysiqueStatistique, : EćoleNormaleSupe´rieure,Universite´PierreetMarieCurie,Universite´DenisDiderot,CNRS v 24,rueLhomond,75231ParisCedex05-France i X E-mail:derrida@lps.ens.fr·E-mail:martin.retaux@ens.fr r a 2 Severalauthorshavestudiedasimplifiedversionoftheproblem,byconsider- ingthedepinningproblemonahierarchicallattice[18,30,47,50],butinthiscase too,thesamebasicquestionsremainhardtoanswer. Herewetrytolookatanevensimplerproblem,atoymodel,whichresembles the depinning problem on the hierarchical lattice in the limit of strong disorder. Ourtoymodelcanbeformulatedasaverysimplerenormalizationtransformation foraprobabilitydistributionofasinglevariable.Ourmainresultisthat,incon- trast to usual critical phenomena, the transition is not characterized by a critical fixed distribution. Instead, the transition is of infinite order (of the Berezinskii- Kosterlitz-Thoulesstype[37]). Thisarticleisorganizedasfollows.First,inordertoshowtheconnectionbe- tween our toy model and the depinning problem, we make in section 1 a short reviewonpastresultsonthePoland-Schergamodelwithdisorderandonitssim- plified version on the hierarchical lattice. Then, in section 2, we present several numericalresultsonthelocationofthetransitionandonthenatureofthecritical behaviourofourtoymodel.Insection3,weexplainhowtheinfiniteordertran- sition can be understood analytically. This is confirmed by the absence of fixed criticaldistributions.Wealsoshowhowtocharacterizethecriticalmanifoldper- turbatively. Lastly in section 4, we present some numerical results which indi- catethatthehierachicalmodelandourtoymodelhavesimilarcriticalbehaviors, namelyaninfiniteordertransition. 1 FromthePolandScheragamodelwithdisordertoourtoymodel 1. ThePolandScheragamodelinpresenceofdisorder ThePolandScheragamodel[48]isamodelforthedenaturationoftheDNA molecule(thetransitionfromadoublestrandmoleculeintotwosinglestrands) orforthedepinningtransitionofalinefromasubstrate.InthePolandScher- aga model, one represents the two strands of DNA as in figure 1. There is a bindingenergyε whenthetwostrandsareincontactatpositioni.Inaddition i thereisanentropyfactorω foreachloopoflengthlbetweentwoconsecutive l contacts(aloopoflengthl correspondstol−1consecutiveunpairedbases). ThepartitionfunctionZ ofamoleculeoflengthListhengivenby L Fig.1 InthePolandScheragamodelthereisanenergyforeachpairofbasesincontactandan entropyfactor(1)foreachloopofunpairedbases. (cid:20) (cid:21) ε +ε +···ε Z = ∑ ∑ ω(i −i )···ω(i −i )exp − i1 i2 ik L 2 1 k k−1 T k≥2 1logs inthepinnedphase L→∞ L where(cid:104).(cid:105)denotesanaverageoverthedisorder(i.e.overtherandomenergies ε)andthesimplestquestionsonemayaskaboutofthedenaturationtransition i are: – WhereisthepreciselocationofthetransitiontemperatureT whichsepa- c ratesthesetwophases? – HowdoesthedifferencelogZ /L−logsvanishasT →T ? L c Inthepurecase,i.e.whenalltheε areequal,thesequestionshavewellknown i answers[19,35,48,49]anditisknownthatthereisaphasetransition(forat- tractiveenergies,i.e.fornegativeε)wheneverc>1.For12,itbecomesfirstorder (cid:26) lim logZL −logs∼(Tpure−T)ν with ν =1/(c−1) for 12. Inthispurecase,letusdefineupure by c (cid:20) (cid:21) ε upure=exp − c Tpure c whichwillbeusefulbelow. In the random case, that is when the ε are i.i.d. random variables, there is i stillatransitionforc>1but,forageneraldistributionoftheenergiesε,the i 4 transition temperature Tquench is not known and the nature of the transition c is still debated. In this random case one can however calculate the annealed partition function (cid:104)Z (cid:105) (where as above (cid:104).(cid:105) is an average over the ε’s) and L i showthatthatitundergoesatransitionatatemperatureTannealed givenby c (cid:28) (cid:20) (cid:21)(cid:29) ε exp − i =upure. Tannealed c c UsingthenJensen’sinequalityonehas[13]that(cid:104)logZ (cid:105)≤log(cid:104)Z (cid:105)andthere- L L fore Tquenched≤Tannealed . (2) c c Since the mid seventies, one knows after the work of Harris [33] under what conditionthecriticalbehaviorofapuresystemismodifiedbyaweakamount ofdisorder.ForthedepinningtransitiontheHarriscriterionpredictsthatdisor- derisirrevelantwhenc<3/2,(meaningthataweakenoughdisordershould not change the critical behavior at the transition) while it is relevant for c> 3/2. For11 thetransitionissmooth[31,32,1]. The case c = 3/2 has been the most difficult to analyse, as it is the case where, according to the Harris criterion, a weak disorder is marginal and it has been debated for years whether disorder was marginally relevant with Tannealed (cid:54)=Tquenched or irrelevant with Tannealed =Tquenched [9,10,18,21,22, c c c c 23,45]. The problem has finally been settled [27,29] and the weak disorder behaviorofthedifferenceTannealed−Tquenchedhasbeenestimated[18,27,29]. c c In conclusion for the Poland Scheraga model, with strong enough disorder whenc<3/2orwitharbitrarydisorderwhenc≥3/2,thepreciselocationof TquenchedandthecriticalbehaviorasT →Tquenchedremaindebatedquestions. c c In particular one does not know how the critical behavior depends on c if it doesatall. 5 2. TheHierarchicallattice In order to gain some insight on the previous problem, several authors have studiedasimplerversionoftheproblem:thedepinningtransitiononahierar- chicallattice.Onsuchalatticetheproblemcanbeformulatedasfollows:the partitionZ ofaninterfaceoflengthL =2n canbecalculated(uptoatrivial n n normalization factor) by the following recursion relation [8,18,30,39,41,47, 50] (1) (2) Z Z +b−1 Z = n−1 n−1 . (3) n b (1) (2) In(3)Z andZ aretwoindependentrealizationsofthepartitionfunction n−1 n−1 of an interface of length 2n−1 and b is a parameter which characterizes the lattice. As for the Poland Scheraga model, the pinned and the unpinned phases are definedby (cid:104)logZ (cid:105) n lim =0 intheunpinnedphase n→∞ 2n (cid:104)logZ (cid:105) n lim >0 inthepinnedphase. n→∞ 2n TomaketheconnectionwiththePolandScheragamodel,eachpartitionfunc- tionZ (whichcorrespondstoastrandoflength20=1)israndomlydistributed 0 accordingtoagivendistributionP(Z)orequivalentlyonecanwrite 0 (cid:16) ε(cid:17) Z =exp − 0 T whereeachenergyε ischosenaccordingtoagivendistributionρ(ε). Inthepurecase,i.e.whenP(Z)isdeltadistributed,thecriticalvalueofZ is 0 0 givenbytheunstablefixedpointofthemapZ→(Z2+b−1)/b (cid:26) 1 for 12. b (4) So in the pure case the transition is always second order, but the exponent varies with b. Thus b plays a role similar to the parameter c in the Poland Scheragamodel. Inth√erandomcas√e,theHarriscriteriontellsus[18]t√hatdisorderisirrele√vant for 22+ 2; When disorder is irrelevant, very much like in the PS model, the quenched andtheannealedmodelshavethesametransitionpointandthesamecritical behaviorwhenP(Z)isnarrowenough. 0 6 When disorder is relevant, or when disorder is irrelevant but strong enough, themainresultsestablishedsofararesimilartothoseofthePolandScheraga model:theannealedandthequenchedmodelshavedifferenttransitiontemper- √ √ atures,thetransitionissmooth[42]andforb= 2andb=2+ 2,disorder ismarginallyrelevant[30,39,41].Theprecisepositionofthetransitioninthe quenched case is not known (only bounds are known [42]) and the nature of thetransitionisstilldebated[18,47,50]. One can remark that the recursion (3) is invariant under the transformation {b,Z}→{b(cid:48)=b/(b−1),Z(cid:48)=Z/(b−1)}.Itisthereforesufficienttoconsider therange 1b−1. n IntermsofthefreeenergyX =logZ,therecursion(3)becomes   (1) (2) 1+(b−1)e−Xn(−1)1−Xn(−2)1 Xn=Xn−1+Xn−1+log b  (5) andtherangeX >log(b−1)isstableundertherecursion.Forbcloseto1we n seethatthethirdterminther.h.s.of(5)isessentially0exceptwhenthesum (1) (2) X +X isclosetoorlessthanlog(b−1). n−1 n−1 3. Thetoymodelstudiedinthepresentpaper Ourtoymodelisasimplifiedversionofrecursion(5): (cid:104) (cid:105) (1) (2) X =max X +X ,−a (6) n n−1 n−1 where a is a fixed positive number (which plays the role of −log(b−1) in (5)).Asin(5),therangeX>−aisstableandateachsteponeessentiallyadds (1) (2) twoindependentvariablesX andX exceptwhenthesumisclosetoor n−1 n−1 lessthantheboundaryvalue−a(seefigure2). Thequestionisasbefore:givenaninitialdistributionP(X)ofX ,whatisthe 0 0 largenlimitofthefreeenergy (cid:104)X (cid:105) n F = lim . ∞ n→∞ 2n Inthistoymodelthetwophasescanbeidentifiedby F =0 intheunpinnedphase ∞ F >0 inthepinnedphase (7) ∞ and by varying the initial distribution P(X) one can observe a transition be- 0 tweenthesetwophases. Inthepurecase,thatiswhentheinitialdistributionisadeltafunction P(X)=δ(X−µ), 0 7 4 2 0 f(X(1) +X(2)) n−1 n−1 -2 -4 -6 -8 -6 -4 -2 0 2 X(1) +X(2) n−1 n−1 Fig.2 Therighthandsideof(5)andof(6)areplottedversusX(1) +X(2) inthecaseb=1.01 n−1 n−1 whena=−log(b−1). itiseasytoseethatthetransitionisfirstorder: F =0 for µ ≤0 ∞ =µ µ ≥0. (thetransitionisfirstorderatµ =0becausedF /dµ isdiscontinuous). ∞ In the random case imagine that (for a>1) the initial distribution depends onaparameterλ asinthefollowingexample P(X)=(1−λ)δ(X+1)+λδ(X−1). (8) 0 Byvaryingtheparameterλ onecanobserveatransitionfromthepinnedphase totheunpinnedphase. Intheexample(8),forλ =1,itisobviousthat(cid:104)X (cid:105)=2nandsoλ =1belongs n tothepinnedphase(7).Forλ =0itisalsoobviousthat−a≤(cid:104)X (cid:105)<0andso n λ =0belongstotheunpinnedphase(7).As(cid:104)X (cid:105)increaseswithλ,thephase n transitionshouldoccuratsomecriticalvalueλ . c Onecanobtainasequenceofupperboundsλ forλ bylookingatthevalue n c λ suchthat n (cid:104)X (cid:105) =0. (9) n λn Toseethatλ definedby(9)isanupperboundofλ onecanusethefactthat n c (cid:68) (cid:104) (cid:105)(cid:69) (1) (2) (cid:104)X (cid:105) = max X +X ,−a ≥2(cid:104)X (cid:105) . n λ n−1 n−1 n−1 λ λ Since(cid:104)X (cid:105) isacontinuousfunctionofλ,andassoonas(cid:104)X (cid:105) >0,onehas n λ n λ (cid:104)X (cid:105) F ≥ n λ >0. ∞ 2n 8 Wearegoingtoseethatonesignatureoftheinfiniteordertransitionisthatthe upperboundsλ definedin(9)satisfyforlargen n (cid:18) (cid:19) 1 λ −λ =O . (10) n c n2 To relate (10)to the infinite order transition,one can use the followingargu- ment:from(6)onecaneasilyshowthat 2(cid:104)X (cid:105) ≤(cid:104)X (cid:105) ≤2(cid:104)X (cid:105) +a n−1 λ n λ n−1 λ (wehaveseenthattherangeX≥−aisstableandthesecondinequalityfollows fromthefactthatX ≥−a).Thereforeif(cid:104)X (cid:105) =y,forsomepositivey,one n−1 n λ has 2my≤(cid:104)X (cid:105) ≤2my+(2m−1)a n+m λ andonecanbesurethat y y+a (cid:104)X (cid:105) =y ⇒ ≤F ≤ . (11) n λ 2n ∞ 2n Ifonedefinesµ (y)asthevalueofλ suchthat n (cid:104)X (cid:105) =y (12) n µn(y) andif,asin(10),onehas A µ (y)−λ ∼ , (13) n c n2 andfrom(11)onegets (cid:32) √ (cid:33) Alog2 F (µ (y))∼exp − . ∞ n (cid:112) µ (y)−λ n c InprincipletheamplitudeAin(13)coulddependony.Onecanhoweverargue thatitdoesnot:forexample,forlargey,changingybyafactor2hastheeffect ofchangingninton+1andthisdoesnotchangetheamplitudeA.Soforlarge noneexpects(13)toholdwithaconstantAindependentofy(fory≥0). Thisisconfirmedinfigure3whereweplotµ (y)definedby(12)versus1/n2 n fory=0,1,10inthecasea=1andweseethatforlargenthedataarecon- sistentwith(13)andanamplitudeAindependentofy. Thereforeweexpectthatasλ →λ c (cid:32) √ (cid:33) Alog2 F (λ)∼exp −√ . (14) ∞ λ−λ c Remark:Onecanalsofindlowerboundsforλ bynoticingthataconsequence c of(6)isthatforanyα (cid:104)eαXn+1(cid:105)≤(cid:104)eαXn(cid:105)2+e−αa . Th√erefore λ ≤ λc whenever one can find some α > 0 for which (cid:104)eαXn(cid:105) ≤ 1+ 1−4e−αa. This kind of lower bound is in the spirit of those obtained from 2 estimatesofnon-integermomentsofthepartitionfunctionindisorderedsys- tems[17]. 9 0.204 µn(0) + ∗ µn(1) × 0.203 µn(10) ∗ +× ∗ × 0.202 + ∗ +× ∗ +× 0.201 +×∗ +×+×+×∗∗+×∗+×∗+×∗+×∗∗+×∗+×∗+×∗+×∗+×∗+×∗+×∗+×∗+×∗ 0.2 0 0.0001 0.0002 0.0003 0.0004 1/n2 Fig.3 Valuesofµn(y)solutionsof(12)fory=0,1and10.Theconvergencetoλc isaspre- dictedin(13)withanamplitudeAwhichseemstobeindependentofy(inthefigure,Aisthe slopeattheorigin). 2 Numericalevidenceoftheinfiniteordertransition Wesawintheprevioussectionthatonesignature(10)oftheinfiniteordertransi- tion(14)isthattheupperboundsλ (solutionsof(9))convergetoλ as1/n2.In n c figure4weseeclearlythis1/n2convergencewhenweplotλ versus1/n2forthe n initialdistribution(8). a=1 a=5 a=10 0.203 + 0.4288 + 0.4644 + 0.202 + + 0.4642 0.4284 + λn + + + + + 0.464 0.201 ++++++++++++ 0.428 ++++++++++++ 0.4638 +++++++++++++ 0.2 0 0.00020.0004 0 0.0002 0.0004 0 0.0002 0.0004 1/n2 1/n2 1/n2 Fig.4 Theupperboundsλn obtainedbysolving(9)for10≤n≤200.Oneseesclearlythe 1/n2convergencewhichisexpectedforaninfiniteordertransitionoftheform(14). 10 Intable1wegiveestimatesofthecriticalvaluesλ andoftheamplitudeAin c (10)forseveralchoicesofa. a= 1 2 5 10 20 50 λc(cid:39) .2000 .3333 .4278 .4638 .48182 .4927 A(cid:39) 7 5.2 2.3 1.1 .7 a(λc−1/2)(cid:39) .3 .333 .361 .362 .364 .364 Table1 Thevaluesλc areestimatedbyextrapolatingtheboundsλn obtainedbysolving(9). TheamplitudeAisestimatedastheslopeattheoriginofthedataoffigure4. Onecannoticefromthelastlineoftable1that,forlargea, (cid:18) (cid:19) 1 .36 (cid:104)X (cid:105) =2 λ − ∼ . (15) 0 λc c 2 a IntheappendixAwegiveageneralargumentforthis1/adependenceof(cid:104)X (cid:105) . 0 λc Another way of visualizing the infinite order transition (14) is to try to plot 1/log(F )2 asafunctionofλ.If(14)isvalidoneshouldobservealinearcross- ∞ ing with the real axis. In figure 5 we plot 1/log[(cid:104)X (cid:105)/2n]2 versus λ for n = n 10,15,···60. The envelope appears to cross the positive real axis with a non- zero slope, at values of λ consistent with the estimates of table 1 and the slope c (1/A/log(2)2)(estimatedwiththevalueAoftable1)shownasathinlineseems tobetangenttotheenvelopeasexpectedfrom(14). 0.035 0.03 0.025 (cid:18)(cid:104)X (cid:105)(cid:19)−2 0.02 log n 2n 0.015 0.01 0.005 0 0.2 0.22 0.24 0.26 0.28 λ Fig.5 [log((cid:104)Xn(cid:105)/2n)]−2 versusλ.Theenvelopeseemstovanishlinearlyasλ →λc.Thisis consistent with the infinite order transition (14). The thin dashed line is the linear behaviour expectedfrom(14)withthevalueAgivenbythenumericalestimatesoftable1.

The depinning transition in presence of disorder: a toy model

Similar The depinning transition in presence of disorder: a toy model