The depinning transition in presence of disorder: a toy model

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by  Bernard Derrida

JStatPhysmanuscriptNo.
(willbeinsertedbytheeditor)
BernardDerrida · MartinRetaux
The depinning transition in presence of
disorder: a toy model
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n Received:date/Accepted:date
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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
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d
n January28,2014
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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,
: E´coleNormaleSupe´rieure,Universite´PierreetMarieCurie,Universite´DenisDiderot,CNRS
v
24,rueLhomond,75231ParisCedex05-France
i
X E-mail:[email protected]·E-mail:[email protected]
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 1<i2<i3···<ik−1<L
3
where,inthesum,kisthenumberofcontacts,i ,···i arethepositionsofthe
1 k
contacts(wehavechosenheretoimposecontactsatpositions1andLsothat
i =1andi =L).Inthedisorderedversionofthemodel,theε’sarequenched
1 k i
i.i.d.randomvariables.
InthePolandScheragamodelitiswellknown[19,35,48,49]thatthenature
ofthetransitiondependsonthelargel bevaviorofω(l).Usuallyonechooses
alargel dependenceoftheform
sl
ω(l)∼ . (1)
lc
wheresandcaretwoparameters(llogsistheextensivepartoftheentropyof
alargeloopofsizel whilethecriticalbehavioratthetransitiondepends[19,
35,48,49]ontheparameterc).
Depending on the large L behavior of logZ , the system is either in the un-
L
pinnedorinthepinnedphase
(cid:104)logZ (cid:105)
L
lim =logs intheunpinnedphase
L→∞ L
(cid:104)logZ (cid:105)
L
lim >logs 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.For1<c<2thetran-
sition is second order with an exponent ν which varies continuously with c
whileforc>2,itbecomesfirstorder
(cid:26)
lim logZL −logs∼(Tpure−T)ν with ν =1/(c−1) for 1<c<2
L→∞ L c ν =1 for c>2.
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.
For1<c<3/2thesepredictionshavebeenconfirmedrigorously:ithaseven
been shown that for a weak enough disorder (i.e. if the distribution of the
ε’s is narrow enough) the quenched and the annealed models have the same
i
transition temperature [2,40,51] (and (2) becomes an equality) and the same
criticalbehaviornearthetransition[28,40].Whenthedistributionoftheε’s
i
is broader, one expects (2) to become a strict inequality [11,51,52] and the
natureofthetransitionremainsdebated[6,7,36,38,50].
For 3/2<c , the relevance of disorder has also been confirmed [17,27,29].
Then,evenforanarrowdistributionoftheε’s,theinequality(2)isstrictand
i
thenatureofthetransitionisstilldebated(asforabroadenoughdistribution
whenc<3/2).OnehoweverhasboundsforthedifferenceTannealed−Tquenched
c c
when disorder is small [3,4,52]. One also knows that for all values of c>1
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 1<b<2
Zcritical=
0 b−1 2<b
andthecriticalbehaviorisgivenby
logZ (cid:26)ν =log2/log(2/b) for 1<b<2
lim n ∼(Z −Zcritical)ν with
n→∞ 2n 0 0 ν =log2/log(2(b−1)) for b>2.
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 2<b<2+ 2whileitisrelevantfor1<b< 2andforb>2+ 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
1<b<2.
(1) (2)
It is easy to check that if Z and Z are both larger than b−1, then the
n−1 n−1
recursion(3)givesZ >b−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.