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epl draft Patch-repetition correlation length in glassy systems Chiara Cammarota and Giulio Biroli Institut de Physique Théorique (IPhT), CEA, and CNRS URA 2306, F-91191 Gif-sur-Yvette, France 2 1 0 2 PACS 64.70.P – Glass Transition PACS 05.20.-y–Statistical Mechanics classical n a Abstract–Weobtainthepatch-repetitionentropyΣwithintheRandomFirstOrderTransition J theory(RFOT)andforthesquareplaquettesystem,amodelrelatedtothedynamicalfacilitation 0 theoryofglassydynamics. Wefindthatinbothcasestheentropyofpatchesoflinearsize(cid:96),Σ((cid:96)), 1 scales as s (cid:96)d+A(cid:96)d−1 down to length-scales of the order of one, where A is a positive constant, c s is the configurational entropy density and d the spatial dimension. In consequence, the only ] c h meaningfullengththatcanbedefinedfrompatch-repetitionisthecross-overlengthξ=A/s . We c c relate ξ to the typical length-scales already discussed in the literature and show that it is always e of the order of the largest static length. Our results provide new insights, which are particularly m relevant for RFOT theory, on the possible real space structure of super-cooled liquids. They - suggestthatthisstructurediffersfromamosaicofdifferentpatcheshavingroughlythesamesize. t a t s . t a m Thesearchforstaticamorphousorderanditscharacter- severallengthsareencodedinthe(cid:96)dependentbehaviorof - d ization in glass-forming liquids has become a very active the entropy, Σ((cid:96)), measuring the repetition of patterns of n research topic in the last few years. Indeed, after that the linear size (cid:96): a cross-over length ξ determining the regime o understandingandthemeasurementofdynamicalhetero- in which Σ((cid:96)) scales extensively, i.e. proportionally to (cid:96)d c geneity and dynamical correlation lengths reached a ma- (d the spatial dimension) and a cooperativity length ξ [ c turestage[1],thefocuspartiallyshiftedtounveiling,mea- at which Σ((cid:96)) becomes larger than one. It was suggested 1 suring and explaining static correlation lengths. This is a that a proxy for the latter is provided by the configura- v newandverypromisingwaytounderstandtheglasstran- tional entropy s , which is the intensive value of the PR- 4 c sition and prune down several theoretical explanations of entropy, to the power −1/d. In this work we find that 6 1 the glassy behavior of super-cooled liquids. In a nutshell, within both frameworks Σ((cid:96)) scales as sc(cid:96)d+A(cid:96)d−1 down 2 one would like to understand whether (and why) super- tolength-scales(cid:96)oftheorderofone,whereAisapositive 1. cooled liquids display increasing amorphous order and to constant. Within RFOT we assumed a surface tension 0 what extent this phenomenon is related to the very fast exponent θ = d − 1, were this not the case the second 2 growth of the relaxation time. term should be replaced by A(cid:96)θ. The main consequence 1 Many different static lengths have been proposed in the of the scaling we find for Σ((cid:96)) is that the only meaningful : v literature. Here we only focus on the ones aimed at mea- andgrowingstaticlengththatcanbedefinedfrompatch- i suring growing static order—whatever this order is. This repetition is the cross-over length: ξ =A/s . By compar- X c restricts the lengths available to the point-to-set [2,3] and ing ξ to the typical length-scales already discussed in the r a the patch-repetition [4] ones. Several numerical and ana- literature,wefindthatξ alwayscoincideswiththelargest lyticalstudiesoftheformerhave beenalreadyperformed. staticcorrelationlengthoftheproblem—aresultpossibly Instead,verylittleisknownonthelatter,apartfromwhat valid in general. The point-to-set length and ξ are only discussed in the original papers of Levine and Kurchan equalinthecaseofRFOT.Forthesquareplaquettemodel [4,5]andinanumericalworkonamonodisperseLennard- ξ instead increases faster than the point-to-set length. A Jones liquid on the hyperbolic plane [6]. final finding worth mentioning is that our results suggest Here we shall analytically study patch-repetition within that that the real space structure of super-cooled liquids two frameworks: the Random First Order Transition differs from a mosaic of different patches having roughly (RFOT) theory [7,8] and the square plaquette system, the same size, a result particular relevant for RFOT the- a model that is related to the dynamical facilitation theory, as we shall discuss in detail at the end of this letter. ory of glassy dynamics [9]. In [4,5] it was proposed that p-1 Chiara Cammarota Giulio Biroli Patch-repetition entropy. – The patch repetition Thus, ξ would be the largest scale over which Σ((cid:96)) be- c entropy was defined to probe amorphous order present in haves as in the ideal glass phase; ξ was thought to corre- c a given snapshot of a glassy liquid1. In order to obtain it, spond to the spatial extent of medium range amorphous onehastocountthefrequencywithwhichagivenpattern order. Itwassuggested[5,6]thatanestimateofξ canbe c repeatsinaconfiguration. Actually, oneshouldnotcount obtained by assuming that the extensive behavior is valid as different, patterns which only change because of short- until Σ((cid:96)) is of order one; this leads to ξ =s−1/d. As we c c time thermal fluctuations. This is a tricky issue that was shallsee,inthecaseswehaveanalyzedtheseassumptions solved in [5]. For simplicity, we will neglect it below and do not work. The only meaningful length is ξ. discuss it later when needed. Before concluding this introduction on PR lengths we The frequency with which a given pattern P repeats in a would like to discuss another motivation to focus on ξ. configuration C reads: In the works by Kurchan and Levine the study of pattern repetition was motivated by asking the question: Suppose 1 (cid:90) f = drΩ(C,P ) . (1) we have a region of volume V with a configuration A. To P V r what extent does A determine the configuration (say B) of a neighboring region, also of size V? This issue is more whereV isthevolume. ThenotationΩ(C,P )representsa r general than the one often addressed in analysis of point- deltafunctionmeasuringwhetherthepatternP ispresent to-setcorrelations,inwhichitisstudiedhowmuchaset(a inC aroundthepositionr. Inthethermodynamiclimitf P neighboring configuration, a boundary condition, etc) de- coincides with its ensemble average since intensive quan- termines the average density profile. It goes beyond that, tities do not fluctuate. Thus, because it asks to what extent A determines B without referring to any specific correlation function: for example f =(cid:104)f (cid:105)=P(P) , (2) P P A could determine the three point correlation functions in B but not the average density profile. From this per- where P(P) is the probability to observe the pattern P spective, the answer to the question posed by Kurchan around a given point. As usual for entropies, one prefers and Levine should lead to a static length always larger or to focus on the logarithm of P(P). Moreover, instead of equal to the point-to-set. In order to directly show that focusing on each single pattern, a quite huge task, it is this length is actually ξ let us rephrase the question in a better to compute an average quantity that measures the more formal way: given the configuration C in A, how repetitionoftypicalpatterns2. Followingtheserecipesone A much the entropy of the configurations in B is reduced? ends up with the expression of the PR entropy of [4,6]: This can be obtained by subtracting to the entropy of B (cid:88) the one obtained by constraining the configuration in A Σ((cid:96))=− P(P)logP(P) . (3) tobeequaltoatypicalconfigurationC . Thisquantityis A P well known in information theory, it is called the mutual ThesumoverP isrestrictedtopatternsappearingwithin information and reads: a given region of linear size (cid:96). We shall focus on (hyper)- (cid:88) I(A,B) = Σ(B)−Σ(B|A)=− P(C )logP(C ) B B cubes or spheres, as done in [6]. This should not be an CB important restriction except if relevant patterns are very (cid:88) (cid:88) + P(C ) P(C |C )logP(C |C ) (4) much diluted and fractal. A B A B A Levine and Kurchan proposed that at least two length- CA CB scales can be obtained in the super-cooled regime by ItisveryeasytoshowthatI(A,B)=Σ(A)+Σ(B)−Σ(A+ studying the dependence on (cid:96) of Σ((cid:96)). Since amorphous B). IntheextremecasewhereAdeterminesBcompletely, order (if present) does not have an infinite range, one ex- one obtains Σ(B|A) = 0 and, hence, Σ(A+B) = Σ(A). pects Σ((cid:96)) to be extensive at large (cid:96), i.e. Σ((cid:96)) (cid:39) sc(cid:96)d. Instead,ifAdoesnotdetermineB atall,Σ(B|A)=Σ(B) Thus, one can define a cross-over length ξ for which the and Σ(A+B) = Σ(A)+Σ(B). In consequence, in the extensive behavior, Σ((cid:96))/(cid:96)d (cid:39) sc, is attained. A second ordered phase, one expects a PR-entropy that grows very lengthξc (calledcooperativein[4])possiblydifferentfrom slowly with (cid:96), whereas in the super-cooled phase, which thefirstone, canbedefinedasthelargestlengthatwhich is characterized (at best) only by medium range amor- Σ((cid:96))isstilloforderone. Theideaisthatforasystemchar- phous order, the PR-entropy scales extensively for large acterizedbyinfiniterangeamorphousorderΣ((cid:96))neverbe- enough (cid:96). When (cid:96) reaches the value ξ at which extensiv- comes extensive and remains of the order of one for any ityisattainedtherelativevalueofthemutualinformation, (cid:96) or just very slowly dependent on (cid:96), e.g. logarithmically. [Σ(B)−Σ(B|A)]/Σ(B), vanishes. This means, recalling Kurchan and Levine question, that ξ can be interpreted 1Actually,alsothepoint-to-setlengthcouldbeinprinciplemea- as the length beyond which A does not determine B any- sured from a given snapshot but in reality this is impractical; it is more. bettertouseensembleaverages. 2Patterns identical by reflection or rotation will be consider as Square Plaquette model. – This model consists in different for simplicity. This double counting leads to an error at mostoftheorderofln(cid:96),whichisirrelevantforourdiscussion. Ising spins interacting with their neighbours through pla- p-2 quette interactions. We consider the two dimensional pla- + quette model on the square lattice [9] whose Hamiltonian + reads: (cid:88) H =−J σiσjσkσl (5) + − ijkl∈(cid:3) + + where the sum runs over all plaquettes of the lattice. The dynamics of this system can be shown to be effectively + + described by a kinetically constrained dynamics leading y+ + + − + to an Arrhenius behavior. Its triangular counterpart is characterized by super-Arrhenius dynamics at low tem- + perature [9,10]. + Bychangingvariablesanddefiningτ =σ σ σ σ thesys- i i j k l tem becomes non-interacting. At low temperature almost + + + + + + + + + x all τs are up and very few are down. The concentration of the latter is given by c = (1 − tanhβJ)/2 (cid:39) e−2βJ Fig.1: Squareplaquettelatticemodel. Thegrayregiondenotes Themappingfromσstoτsisverynon-linear. Thus, even all the plaquettes determining the value of σ . The pattern x,y though its thermodynamics is trivial, the model displays region corresponds to the dotted square. An example of spin interesting features such as diverging static correlations configuration B of the dotted square is shown explicitly. atlowtemperature. The2-pointspincorrelationfunction remains featureless but, instead, the 4 point correlation function remaining spins a typical given B: G (l)=(cid:104)σ σ σ σ (cid:105)=tanh(βJ)l (cid:39)e−2cl Σ((cid:96))=Σ(B)+Σ(I|B) . (9) 4 x,y x,y+1 x+l,y x+l,y+1 (6) It is easy to show that in the thermodynamic limit all Bs displays a growing correlation length diverging as 1/c are equiprobable, i.e. P(B) = 1/22(cid:96)−1. Once the spins when T → 0. This is related to the fact that the sys- B are fixed, the mapping between the spins I and the tem orders at low temperature. The number of ground plaquette variables is one to one (just a generalization of states with open boundary conditions can be shown to be eq. 7). In consequence the PR-entropy can be computed equal to 22L−1 where L is the linear size of the system. easily: In the following we shall consider a very special boundary condition that simplify the computation: all the spins on Σ((cid:96)) = −((cid:96)−1)2[clnc+(1−c)ln(1−c)]+(2(cid:96)−1)ln2 the left and bottom edges of the square are fixed in the (cid:39) −(cid:96)2clnc+2(cid:96)ln2−ln2 (10) up state and the remaining ones are free. This is by no meansrestrictivesinceweareinterestedinbulkproperties where the last expression is valid at low temperature (in thatareindependentonthechoiceofboundaryconditions. the last expression for each power of (cid:96) we have only re- The mapping between σs and τs is particularly simple in tained the leading contribution in c). The two terms con- this case: tributingtotheentropy,theboundaryandthebulkterm, (cid:89) σx,y = τl (7) have a clear physical interpretation: the former counts l∈(x,y) thenumberof(equiprobable)patternsonthebottomand left edge of the square. The latter counts the number of wheretheproductrunsoverallplaquettesbelongingtothe possible patterns for a fixed boundary configurations: it square with vertices (0,0);(x,0);(y,0);(x,y), see Fig.1. corresponds to the entropy of an ideal gas of defects, cor- The patch-repetition entropy in this case is responding to down plaquette variables. (cid:88) Letusnowinvestigatewhichlengthsonecanextractfrom Σ((cid:96))=− P(P)logP(P) . (8) the(cid:96)behaviorofthePR-entropy. Fromthecross-overbe- P tween extensive and sub-extensive terms one obtains the where P is a spin configuration of a (cid:96)×(cid:96) square in the length ξ (cid:39) 2ln2/|clnc|, see Fig.2. The PR-entropy in- bulk of the system, see Fig.1. We shall denote (x,y) its creases starting from a value of the order of one with a bottom left vertex. Note that we are interested in the finite slope, hence ξc remains microscopic and does not low temperature limit, in which the approximation of ne- grow lowering the temperature. Moreover, the length ob- glecting thermal fluctuations in the definition of the PR- tained by dimensional analysis from the extensive contri- entropy is correct. In order to simplify the computation, bution s−1/2 where s = −clnc also does not play any c c wedecomposethePR-entropyinaboundarycontribution, role, see Fig.2 for a visual summary. Thus in this model corresponding to the configuration B of the spins on the the only meaningful length that can be extracted from bottom and left edges, and in a bulk contribution, cor- the PR-entropy is ξ. It is interesting to compare it to the responding to the entropy of the configuration I of the other lengths already discussed for this system [10,11]. p-3 Chiara Cammarota Giulio Biroli thepatternregion(weskippedforsimplicitythesub-index Σ(ℓ) denoting that the pattern is taken around a given point). c Introducing F(m)(q ), which is the Legendre transform3 ab ofthefreeenergyofthesystemofmreplicaswithrespect to q , we find: ab β (cid:16) (cid:17) Σ((cid:96))= lim Extr F(m)(q )−F(m)(0) m→1m−1 qab:qab|P=1 ab (12) where we have used that the solution of the extremiza- tionconditioninabsenceofconstraintandabovetheideal glass transition temperature is q =0. We shall now ad- O(1) ab dress the problem of thermal fluctuations. Taking them ξc ξc(proxy) ξ ℓ intoaccountmeansthatoneshouldnotcountasdifferent, patternsthatareconnecteddynamicallybyshort-timedy- namical fluctuations. In order to avoid this double count- Fig.2: Sketchofthelog-logplotofaPR-entropyhavingalin- ing, one can lump together patterns that have an over- ear and quadratic contribution (we have also added the loga- laplargerorequalthantheEdwards-Andersonparameter rithmiccontributiondiscussedinthesection”patch-repetition q . The value of q can be determined from the dy- entropy”): ξ is the cross-over length at which Σ((cid:96)) becomes EA EA extensive, ξ(proxy) = s−1/d is the proxy for the cooperative namical correlation function. Of course, this procedure c c makesonlysensewhenthereisaclearseparationoftime- length ξ . c scales. The final replica expression for the PR entropy is therefore4: The point-to-set length, ξ , was shown to increase as √ PS d (cid:16) (cid:17)(cid:12) 1/ c, hence much slowly than ξ. Instead, the dynamical Σ((cid:96))=β Extr F(m)(q )−F(m)(0) (cid:12) correlation scales as 1/c, i.e. essentially as ξ. dm qab:qab|P=qEA ab (cid:12)m=1 (13) In conclusion, the analysis of the square plaquette model PR entropy and RFOT theory. In order to perform shows that, at least in this case, only ξ can be defined the computation of Σ((cid:96)) we need βF(m)(q ). This is from the (cid:96) behavior of the PR entropy. This length is ab not known in general since its computation would re- much larger than ξ and scales approximatively as the PS quire to solve the full theory by integrating over al- dynamicalcorrelationlengthandthelargeststaticcorrela- most all fluctuations. However, in the Kac-limit [14] tionlength,whichisobtainedfromG . Asimilarbehavior 4 or on general grounds, one can argue that β∆F = is also expected for the triangular plaquette model [12]. β(cid:0)F(m)(q )−F(m)(0)(cid:1) should have a form like: ab RFOT, patch-repetition entropy and replicas. – In this section we shall derive the PR-entropy in the β∆F =(cid:90) ddx(cid:26)a20 (cid:88)m (∂q (x))2+V˜(q (x))(cid:27) (14) super-cooled regime within RFOT theory. Before doing ad 2 ab ab 0 a,b=1 that, we develop a general formalism based on the replica method to obtain Σ((cid:96)). Our derivation is related to the Thefirsttermrepresentsthetendencyofreplicatoremain Renyi complexities introduced in [5]. coupled. We used a squared gradient to mimic this effect. Replica and PR-entropy. The PR-entropy can be written Although this is just an approximation, other forms, e.g. using the following trick: non-local but still elastic ones, would lead to the same (cid:32) (cid:33) result as long as they have a finite range a0, which physi- 1 (cid:88) Σ((cid:96))=− lim ln P(P)m (11) callycorrespondstothecharacteristicmicroscopiclength, m→1m−1 i.e. the length corresponding to the first peak in the ra- P dial distribution function (henceforth we shall put a =1 0 Forsimplicitywecontinuetoneglectthermalfluctuations. and measure lengths in unit of a ). Instead, in the case 0 We shall take into account their effect at the end of the of a more general elastic term leading to an interface cost derivation. The expression inside the logarithm can be scaling with an exponent θ < d−1 the results below are rewritten as: modified: d−1hastobereplacedbyθ inthefinalexpres- ln(cid:32)(cid:80)P,C1,...,Cme−βH(C1)...−βH(Cm)(cid:81)mi=1Ω(Ci,P)(cid:33) sion for Σ((cid:96)). The precise expression of the second term, (cid:80)C1,...,Cme−βH(C1)...−βH(Cm) 3Actually, strictly speaking, we do not focus on the complete = ln(cid:32)(cid:80)C1,...,Cme−βH(C1)...−βH(Cm)Ω(qab|P =1)(cid:33) Ldeeggreenedsreoftfrraenesdfoomrmabreutinotengrtahteedavoeurat,geasacintiounsuoanlcneuacllelasthioonrt-psrcoable- (cid:80) e−βH(C1)...−βH(Cm) lems[13]. C1,...,Cm 4Inthisexpressionweconstrainedallreplicastohaveanoverlap The numerator of the above expression is the sum over m qEA inside the pattern region. This gives the leading contribution sincelargeroverlapsaresuppressedexponentially(inthesizeofthe replicasconstrainedtohaveanoverlapequaltooneinside patternregion). p-4 A part from irrelevant numerical constants, the (cid:96) depen- V(q) dence of Σ((cid:96)) is the three dimensional analog of the one found for the square plaquette model in the previous section. In consequence, as previously, the only meaningful length that one can extract from the PR-entropy is ξ = 3Y/s . Remarkably, this scaling with s , which is c c validinanydimension, isthesameoneofthepoint-toset V(qEA)=sc length ξPS (even in the case θ < d−1). Thus, within RFOT, these two length-scale coincide. The fuzzy mosaic state. The description of the super- cooled liquid state resulting from RFOT is often repre- q q sented or envisioned as a mosaic of patches, each one lo- EA calized (temporarily) in a given amorphous configuration [7,8]. Showing whether such a state does exist and what is its real space structure remains a pressing open prob- Fig. 3: Sketch of the shape of V(q). lem in the field. The PR-entropy should be a good probe to verify the mosaic representation. Unexpectedly, the explicit computation of Σ((cid:96)) within RFOT leads to an the potential, actually is not important. The only thing expression which is not compatible with such real space which matters is that in the m → 1 limit and within the structure. In order to clarify this point, let us obtain our hypothesis of replica symmetry, q (x)=q(x) ∀a(cid:54)=b, the ab result again, but in a different way by rewriting Σ((cid:96)) as potentialV˜(q (x))simplifiesto(m−1)V(q(x))withV(q) ab the sum of the PR entropy for the boundary, Σ ((cid:96)), and B havingtheshapeshowninFig.3. Thisisindeedwhathap- the PR entropy of the interior of the sphere given a fixed pens in the Kac-limit investigated by Franz [14] and it is typicalboundary, Σ ((cid:96)). Forasystemwithshortrange C|B what one would expect if RFOT theory retains a validity interactions once the boundary of a closed region is fixed infinitedimensions. Previousstudieshaveshownthatthe any information with the exterior is lost, thus fixing a value of q at the local secondary minimum of V(q) corre- boundary is like fixing all the exterior. Hence, Σ ((cid:96)) C|B sponds to q and that V(q ) is the intensive value of EA EA coincides with the configurational entropy of the particles the configurational entropy s , which goes to zero linearly c inside a cavity when all particles outside are blocked in at the ideal glass transition temperature T [15,16]. K a typical equilibrium configuration. This, actually, is the Byassumingthatreplicasymmetryisnotbrokenandcon- protocol used to compute point-to-set lengths. The anal- sidering, without loss of generality, a spherical pattern re- ysis of [18] shows that close to the ideal glass transition gion we find that the solution of the variational problem Σ ((cid:96)) is zero for (cid:96)<ξ and equal to 4π(cid:96)3s −4π(cid:96)2Y (13) satisfies the equation: C|B PS 3 c for (cid:96) > ξ . This is exactly what one expects from the PS definition of the point-to-set length: boundary conditions d2q d−1dq − − +V(cid:48)(q)=0 (15) determines the bulk amorphous configuration below, but dr2 r dr notabove,ξ . ThebehaviorofΣ ((cid:96))canbeobtainedby PS B Close to T the local minimum is almost degenerate with generalizing our previous computation: one has to solve K theglobaloneandqisfoundtovaryonlargelength-scales. a variational problem in which the profile q(x) is con- Thus, making the thin-wall approximation [17], i.e. by strained to be equal to q in a small annular region. EA dropping the first derivative, one obtains a one dimen- The result is that Σ ((cid:96)) is equal to 4π(cid:96)3s +4π(cid:96)2Y be- B 3 c sional Newton equation for a particle moving in the po- low the point-to set length and equal to 8π(cid:96)2Y above it. tential −V(q) during an effective time r. For the optimal Summing the two contributions, one finds back (16), as solution V(q) must vanish for r →∞. Thus, the solution it has to be. Remarkably, the single contributions dis- q(r)correspondstothe trajectoryof aparticlethatstarts play a transition at the point-to-set length but their sum, at q = q at r = 0, with a ”kinetic energy” equal to the PR entropy, has no singularity but just a cross-over. EA V(q ), and reaches q =0 at infinite time with vanishing The behavior of Σ ((cid:96)) is quite surprising. Close to T , EA B K kinetic energy. Using this result and following standard for (cid:96) = ξ , the number of typical boundary patterns PS manipulations [17] one finds (in three dimension): is very large. Actually, it starts to be much larger than one as soon as (cid:96) > 1. This suggests that given a snap- (cid:18) (cid:19) 4 βF(m)(q )−βF(m)(0)=(m−1) π(cid:96)3s +4π(cid:96)2Y shot (even one coarse grained in time, as discussed in ab 3 c [5]) no clear boundaries between local amorphous states do exist5. Were they present, one would expect Σ ((cid:96)) where Y = (cid:82)qEA(cid:112)2V(q)dq. Using eq. (13) we finally B 0 obtain the RFOT prediction for the PR-entropy: 5The non-existence of boundaries was already proposed by S. Franz [19] on the basis of the result that the average energy (or of 4 anyotherlocalobservable)insideacavitywithamorphousboundary Σ((cid:96))= π(cid:96)3s +4π(cid:96)2Y (16) 3 c conditions coincides with the average energy for the free system. p-5 Chiara Cammarota and Giulio Biroli to be related to the number of possible different amor- where colors change rapidly and in an apparently random phous states on scale (cid:96) for (cid:96)(cid:28)ξ . What is this number fashion instead than to a mosaic of patches with differ- PS N ((cid:96))? Naively, one could think that there is a one to entcolors—itwouldresemblemoretoaPollock’sabstract S onecorrespondencebetweenthetilesofthemosaic(amor- paintingthantoaMondrian’sone. Thepresenceofamor- phousstatesonscaleξ )andthepossiblestatesonscale phousorderinthisdrawingwouldbeunveiledfocusingon PS (cid:96). This would lead to logN ((cid:96)) = s ξd , which clearly allregionswherethesame(apparentlyrandom)boundary S c PS does not hold. A more refined reasoning takes into ac- appears: one would typically find the same pattern inside countthatdifferenttilescancontainthesamepatternson the boundary if this has a linear size smaller than ξ . PS length-scales smaller than ξ . The simplest expectation PS Conclusion. – We studied the behavior of the PR- would then be that NS((cid:96)) = (NS(ξPS))((cid:96)/ξPS)d = esc(cid:96)d entropy in the square plaquette model and within RFOT [4,5]. However, this also does not hold. The only way theory. Inbothcaseswefindthatthe(cid:96)dependenceofΣ((cid:96)) to get our result for 1 (cid:28) (cid:96) (cid:28) ξ is that N ((cid:96)) (cid:39) PS S isgivenbyanextensivetermfollowedbyasub-leadingone (NS(ξPS))12((cid:96)/ξPS)d−1 = e4πY(cid:96)d−1, a relation that seems scaling as the area of the pattern region. We found that hard to justify starting from a picture in terms of a mo- the only meaningful length one can define, ξ, is the one saic with well-defined boundaries. corresponding to the cross-over between leading and sub- Theabsenceofboundariesbetweenlocalamorphousstates leading behavior in (cid:96) of Σ((cid:96)). This length, however, is can be backed up by an independent argument based on neither ξ , at which the PR-entropy becomes of order one c RFOT and our recent results on random pinning glass (thisremainsfeatureless, microscopicanddoesnotgrow), transitions [20]. We have recently shown that within nor its proxy s−1/d. Comparing ξ to the other lengths RFOT by pinning at random particles of an equilibrated c involved, we discovered that it coincides with the point- configurationonecaninduceidealglasstransitionsforthe to-set length in the RFOT case only and it always corre- remaining free particles at temperatures higher than T . K sponds to the largest static length, a fact that is possibly Now, imagine that one is able to show the existence of valid in general. Our results provide new insights, which boundaries for the unconstrained liquid. The only way areparticularlyrelevantforRFOTtheory,onthepossible to do that consists in finding local fluctuations of some real space structure of super-cooled liquids. They clarify sort (of energy, entropy, density, etc.) which pinpoint the that its illustration in terms of a mosaic of different pat- presence of the boundary in real space. A paradox arises terns is not qualitative correct. For lack of a better name when considering the effect of random pinning. By in- wedubthecorrectone,holdingatleastwithinRFOT,the creasing the fraction of pinned particles one can approach fuzzy mosaic state. (atfixedtemperature)theglasstransitionarbitrarilyclose and, correspondingly, obtain an arbitrarily large point- ∗∗∗ to-set length and very large boundaries. The problem is that this has to be true also for the initial configuration, We thank J.-P. Bouchaud, R. Jack, J. Kurchan and P. sincethisisanequilibratedconfigurationforthefreeparti- Wolynesforhelpfuldiscussions. GBacknowledgessupport cles, independently on the fraction of pinned ones. Thus, from ERC grant NPRG-GLASS. the initial configuration must contain boundaries on all length-scales. This seems—and actually is—impossible. 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