We consider the dynamics of a diluted mean-field spin glass model in the
aging regime. The model presents a particularly rich heterogeneous behavior. In
order to catch this behavior, we perform a **spin-by-spin analysis** for a
**given disorder realization**. The results compare well with the outcome of a
static calculation which uses the ``survey propagation'' algorithm of Mezard,
Parisi, and Zecchina [Sciencexpress 10.1126/science.1073287 (2002)]. We thus
confirm the connection between statics and dynamics at the level of single
degrees of freedom. Moreover, working with single-site quantities, we can
introduce a new response-vs-correlation plot, which clearly shows how
heterogeneous degrees of freedom undergo coherent structural rearrangements.
Finally we discuss the general scenario which emerges from our work and
(possibly) applies to more realistic glassy models. Interestingly enough, some
features of this scenario can be understood recurring to thermometric
considerations.Comment: 4 pages, 5 figures (7 eps files

A. Barrat

A. Montanari

C. A. Angell

E. Vincent

F. Ricci-Tersenghi

J.-P. Bouchaud

L. C. E. Struick

English

arXiv

Crossref

Microscopic Description of Aging Dynamics: Fluctuation-Dissipation Relations, Effective Temperature, and Heterogeneities

MONTANARI A.

RICCI TERSENGHI, Federico

Archivio della ricerca- Università di Roma La Sapienza

P H Y S I C A L R E V I E W L E T T E R S week ending10 JANUARY 2003VOLUME 90, NUMBER 1Microscopic Description of Aging Dynamics: Fluctuation-Dissipation Relations,Effective Temperature, and HeterogeneitiesA. Montanari1,* and F. Ricci-Tersenghi2,†1CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris, France2Dipartimento di Fisica and INFM, Università di Roma ‘‘La Sapienza,’’ Piazzale Aldo Moro 2, I-00185 Roma, Italy(Received 18 July 2002; published 8 January 2003)017203-1We consider the dynamics of a diluted mean-field spin glass model in the aging regime. The modelpresents a particularly rich heterogeneous behavior. In order to catch this behavior, we perform a spin-by-spin analysis for a given disorder realization. We confirm the connection between statics anddynamics at the level of single degrees of freedom. Moreover, working with single-site quantities,we can introduce a new response-vs-correlation plot, which clearly shows how heterogeneous degrees offreedom undergo coherent structural rearrangements. We discuss the general scenario which emergesfrom our work and (possibly) applies to more realistic glassy models. Interestingly enough, somefeatures of this scenario can be understood recurring to thermometric considerations.DOI: 10.1103/PhysRevLett.90.017203 PACS numbers: 75.50.Lkmicroscopic, but site-independent, effective temperature. in the zero temperature limit, mthT  thT, withThe description of the off-equilibrium dynamics inaging systems is one of the major challenges in contem-porary statistical mechanics. Aging systems, such as spin,structural, and polymeric glasses [1], are slowly evolving,heterogeneous systems which do not reach thermal equi-librium, at low enough temperatures, on any experimentaltime scale. In the last 20 years a great effort has beendevoted to study the dynamics of some prototypical mod-els, namely, mean-field spin glasses [2]. These modelsexhibit an extremely rich behavior: slow relaxation,memory effects, aging. Important tools which havebeen introduced in this context are the off-equilibriumfluctuation-dissipation relation (OFDR) [3] and the effec-tive temperature [4] one can derive from that relation.Such an effective temperature is, roughly speaking, whatwould be measured by a thermometer responding on thetime scale on which the system ages.One of the weak points of the results obtained so far isthat they focus on global quantities, e.g., correlation andresponse functions averaged over the spins. On the otherhand, we expect one of the peculiar features of glassydynamics to be its heterogeneity [5]. For instance, corre-lation and response functions of a particular spin dependupon its local environment [6]. Two simple remarks are inorder here: (i) These spin-to-spin fluctuations are nonzeroeven in the thermodynamic limit; (ii) they disappearwhen average over quenched disorder is taken [7].Moreover the present definition [4] of effective tem-perature has some problems. Indeed it corresponds towhat would be measured by a specific, properly tuned,slow thermometer, while generalizations to more genericthermometers give disagreeing results [8], still to beclarified.In this Letter, inspired by the new approach of [9], westudy the aging dynamics focusing on single-site corre-lation and response functions. In this way we are able toconsider the heterogeneities in the system and to define a0031-9007=03=90(1)=017203(4)$20.00 An interesting context for addressing these issues isprovided by diluted mean-field models [10]. In thesemodels each spin interacts with a finite number of otherspins, just as in finite-dimensional models. On the otherhand, the absence of a finite-dimensional geometricalstructure makes them tractable from an analytic pointof view.We consider a ferromagnetic Ising model with three-spin interactions, defined by the HamiltonianH  XMm1imjmkm; (1)where the M triples im; jm; km are chosen randomlyamong the N3 possible ones. Although ferromagnetic,this model is thought to have a glassy behavior for M >0:818N, due to self-induced frustration [11].We work on a single sample withM  N  100, whoselargest connected component contains 96 sites. We lim-ited ourselves to such a small sample because single-spinmeasures require huge statistics.We use the survey propagation (SP) algorithm ofRef. [9], or better its generalization at finite temperatures[12], to compute the free energy density Fm; for ourspecific sample at the one-step replica symmetry break-ing (1RSB) level. In Fig. 1 we report the complexityT  @mFm;jm1. The dynamic and static tem-peratures are defined, respectively, as the points where anontrivial (1RSB) solution to the cavity equations firstappears, and where its complexity vanishes. From theresults of Fig. 1 we get the estimates Td  0:5572 andTc  0:4672. In the standard picture, the aging dynam-ics for discontinuous spin glasses is dominated by‘‘threshold’’ metastable states [2]. The corresponding1RSB parameter mthT can be computed by imposingthe condition @2mmFm;  0. We computed mthTon our sample for some temperatures below Td, and2003 The American Physical Society 017203-100.20.40.60.811 10 102 103 104 105 106C0(t w+t,tw)ttw = 10  tw = 102tw = 103tw = 10400.20.40.60.810 0.2 0.4 0.6 0.8 1T χ0(t w+t,tw)C0(tw+t,tw)tw = 10  tw = 102tw = 103tw = 104FIG. 2. The correlation function and the -vs-C plot for thespin i  0 at T  0:5, which has q0th  0:7167 (dashed line,top). The bold line (bottom) is the static 0C curve.00.010.020.030.040.050.4 0.45 0.5 0.55 0.6Σ(T)TTc Td00.20.40.60.810 0.2 0.4 0.6mth(T)TdFIG. 1. The complexity  and the 1RSB parameter forthreshold states mth (inset) versus the temperature for thesample studied in this Letter. The continuous line in the insetis the polynomial fit mthT  1:08T  0:038T2  2:17T3.Finite size corrections are negligible [12].P H Y S I C A L R E V I E W L E T T E R S week ending10 JANUARY 2003VOLUME 90, NUMBER 1th  1:081. These results are summarized in the insetof Fig. 1.Another important outcome from the SP algorithm isthe value of the local Edwards-Anderson order parameterqiEAm, which depends on the 1RSB parameter m. Onthreshold states, the local order parameter is connected tothe single-site correlation function (defined below):qith  qiEAmth  limt!1limtw!1Citw  t; tw: (2)We consider Metropolis dynamics starting from ran-dom initial conditions, it  0  1. After a time twwe turn on a small random magnetic field, hi  h0, andwe measure single-spin correlation and integrated re-sponse functionsCitw  t; tw 1tX2t1t0thitw  t0itwi; (3)itw  t; tw 1th0X2t1t0thitw  t0signhii; (4)where hi denotes the average over the Metropolis trajec-tories and the random perturbing field. The sums over t0are introduced in order to reduce the statistical error. This‘‘experiment’’ is repeated Nruns times, each time with adifferent thermal noise and perturbing field. Typical val-ues for Nruns range between 106 and 5 106.The first remark on the numerical data is that the spinscan be clearly classified in two groups. Type-I spinsbehave as if the system were in equilibrium: the corre-sponding correlation and response functions satisfy time-translation invariance and the fluctuation-dissipationtheorem (FDT). Type-II spins are out of equilibriumspins: their correlation and response functions are non-homogeneous on long time scales and violate FDT.017203-2Type I includes isolated sites, but also 12 nonisolatedsites. Remarkably these sites are the ones for which the SPalgorithm returns qith  0; i.e., they are paramagneticfrom the static point of view. These sites can also beidentified via a simple algorithm [13].Let us hereafter focus on type-II spins, that is, onglassy degrees of freedom. In Fig. 2 we show the corre-lation function for a generic type-II spin (i  0 here) andthe corresponding -vs-C plot. For this spin we haveqth  0:7167, shown with a dashed line in Fig. 2 (top).In the limit of very large times we expect (in analogywith [3]) the OFDR it; t0  iCt; t0 to hold. More-over the function iC is related to static quantities [14].Numerical data, for this and for all the other spins, seemto converge to the static curve iC. This is an evidencefor a strong link between static and dynamic observableseven at the level of single degrees of freedom.The 1RSB static calculation yieldsTiC  1 CC qith  1 qith mthC qith qith  C: (5)The OFDR changes from site to site because the qithchanges. Note, however, that the iC curves are parallelin the aging regime, since mth depends only on the tem-perature (cf. Fig. 3).017203-20.51.0P H Y S I C A L R E V I E W L E T T E R S week ending10 JANUARY 2003VOLUME 90, NUMBER 1System heterogeneities manifest themselves in thelarge variability of the qith local order parameters. The-vs-C plot for seven generic sites, see Fig. 3 (top), clearlyshows this variability. In order to check that single-sitedata can be well described by Eq. (5) we rescale data ofFig. 3 using the scaling variables Cresi  1 Ai1 Ciand resi  Aii, where Ai  1 q=1 qith , q being areference overlap which can be chosen freely. Rescaleddata are shown in Fig. 3 (bottom).A last important question, in order to complete thedescription of the time evolution of single-site quantities,regards the time law with which spin i runs along thecurve iC. The answer to this question is very surpris-ing and it is shown in Fig. 4, where we plot the timeevolution of all the 100 Ci’s and i’s.Amazingly, site-II data leave the FDT line coherently(when the system undergoes a global structural re-arrangement) and they remain very well aligned for latertimes. Fits to a functionitw  t; tw 1 Citw  t; twTfittw; t(6)give very accurate results with Tfit  0:459 (for t  212),0:536 (t  215), 0:564 (t  216), 0:590 (t  217).In the following discussion we outline a few generalproperties which can be extrapolated from our numeri-cal results, and, possibly, applied to a wider variety ofsystems. Our basic objects are the local correlation andresponse functions Cit; t0 and Rit; t0  @t0it; t0.Following Ref. [3], we guess that @tCit; t0;@tRit; t0  0 and @t0Cit; t0; @t0Rit; t0  0. Moreover00.050.10.150.20.250.30.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1T χiCi00.050.10.150.7 0.75 0.8 0.85 0.9 0.95 1T χresiCresiFIG. 3. In the -vs-C plot for seven generic sites at T  0:3system heterogeneities become apparent. The seven data setscan be nicely collapsed with no fitting parameters (bottom).017203-3Cit; t0; Rit; t0 ! 0 as t! 1 for any fixed t0. All theseproperties are well realized within our model.The first nontrivial property [15] is that, for any ‘‘non-exceptional’’ pair of spins i and j there exist two continu-ous functions fij and fji such thatCit; t0  fijCjt; t0; Cjt; t0  fjiCit; t0; (7)asymptotically for t; t0  1. We do not specify whatnonexceptional means [16], but the reader is urged tobear in mind the example of type-I (paramagnetic) spinsin our model: if i is type I, and j is type II, then relation(7) clearly does not hold.It is easy to show that transition functions ffijg can bewritten in the form fij  f1i  fj. Of course the func-tions fi are not unique: in particular, they can be modifiedby a global reparametrization fi ! g  fi.Moreover Eq. (7) implies a one-to-one correspondencebetween the correlation scales (in the sense of [3]) of sitesi and j. Notice that, for our model, this is unavoidable ifwe want the connection between statics and dynamics tobe satisfied both at the level of global and local (single-spin) observables. Physically this first property meansthat structural rearrangements occur coherently in thewhole system. Notice that this is not unphysical, because0.0 0.5 1.0Ci(t+tw,tw)0.00.51.0T χi(t+t w,t w)0.51.00.5 1.0FIG. 4. The ‘‘movie’’ plot: evolution of the single-spin corre-lation and response functions in the C; plane. Here we useh  0:1, T  0:4, and tw  105. Different frames correspond to(from left to right and top to bottom) t  24, 29, 212, 215, 216,and 217. Black and white circles refer, respectively, to type-Iand type-II sites. Black circles are not exactly on the FDT linebecause of finite-h0 effects, to be discussed in Ref. [12]. Forthis reason, the system has to be considered in a quasiequili-brium regime as long as type-I and type-II data are aligned,even if on a line which is slightly below the FDT one (see, e.g.,the third plot). The dotted lines are fits to type-II data.017203-3C(t,t’)χ (t,t’)FIG. 5. A pictorial view of the heterogeneous aging dynamicsin a diluted p-spin model. It is sufficient to know the dynamicsof a single spin in the system (a full curve) in order toreconstruct the behavior of any other one (once the staticparameters qith are known). Arrows are the points velocities.P H Y S I C A L R E V I E W L E T T E R S week ending10 JANUARY 2003VOLUME 90, NUMBER 1far apart degrees of freedom are coherent only on a coarsetime resolution (diverging with tw).The second property has been illustrated at lengthabove: For large times t; t0, a local OFDR of the formit; t0  iCit; t0 exists (the connection with thestatic result is not a crucial point here).Our third property determines the form of the transi-tion functions fij  f1i  fj in the aging regime. In fact,the results in Fig. 4 suggest thatit; t01 Cit; t0jt; t01 Cjt; t08 i; j: (8)Combined with the OFDR, this means that we can takefiC  iC=1 C for C< qith . This relation cannotbe extended to the quasiequilibrium regime Ci > qith ,because we would obtain fiC   identically, whichis not invertible, and to very small values of C. In moregeneral physical systems [12] this formula must be re-placed by fiC  iC; C, with a site-independent; .Finally the fourth property is0iCit; t0  0jCjt; t0; (9)or, in other words 0iCi  0jCj when Ci  fijCj.Notice that, for our model, this property is satisfied bythe prediction we draw from the statics. However, a directnumerical verification is quite hard.Suppose now we measure the temperature of the spin i,by weakly coupling it to a thermometer [17]. If thethermometer responds quickly, it measures the bath tem-perature T for any site i, independently of the details ofthe thermometer itself. However, if the thermometer is‘‘slow,’’ it measures an effective temperature Tmeas > T,which depends upon the specific thermometer used [8].Nevertheless, the above scenarios imply [12] that Tmeas issite independent. The converse is also true: if one of theabove four properties ceases to hold, one can construct athermometer which distinguishes colder sites from hotterones and use it to transfer heat.017203-4Our conclusions are summarized in Fig. 5. Dynamicalheterogeneities turn out to be strongly constrained in theaging regime. These constraints can be derived from thehypothesis of thermometric indistinguishability of differ-ent sites [12].We are deeply indebted to Riccardo Zecchina andMarc Mézard, who discussed with us their results [9]before publication. A. M. thanks Leticia Cugliandoloand Jorge Kurchan for their interest in this work, andthe ESF for financial support.*Electronic address: Andrea.Montanari@lpt.ens.fr†Electronic address: Federico.Ricci@roma1.infn.it[1] E. Vincent et al., in Proceedings of the Sitges Con-ference, edited by M. Rubi (Springer-Verlag, New York,1997); C. A. Angell, Science 267, 1924 (1995); L. C. E.Struick, Physical Aging in Amorphous Polymers andOther Materials (Elsevier, Amsterdam, 1978).[2] J.-P. Bouchaud et al., in Spin Glasses and Random Fields,edited by A. P. Young (World Scientific, Singapore, 1997).[3] L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. 71,173 (1993); J. Phys. A 27, 5749 (1994).[4] L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys.Rev. E 55, 3898 (1997).[5] M. D. Ediger, Annu. Rev. Phys. Chem. 51, 99 (2000);W. K. Kegel and A. van Blaaderen, Science 287, 290(2000); E. R. Weeks et al., Science 287, 627 (2000).[6] P. H. Poole et al., Phys. Rev. Lett. 78, 3394 (1997);A. Barrat and R. Zecchina, Phys. Rev. E 59, R1299(1999); F. Ricci-Tersenghi and R. Zecchina, Phys. Rev.E 62, R7567 (2000); H. E. Castillo et al., Phys. Rev. Lett.88, 237201 (2002).[7] We are considering here the simplest disorder average.More sophisticated averages, as in A. K. Hartmannand M. Weigt, Phys. Rev. E 63, 056127 (2001), maygive nontrivial results, which depend on the localconnectivity.[8] R. Exartier and L. Peliti, Eur. Phys. J. B 16, 119 (2000).[9] M. Mézard, G. Parisi, and R. Zecchina, Science 297, 812(2002); M. Mézard and R. Zecchina, cond-mat/0207194.[10] G. Semerjian and L. F. Cugliandolo, cond-mat/0204613.[11] F. Ricci-Tersenghi, M. Weigt, and R. Zecchina, Phys. Rev.E 63, 026702 (2001).[12] A. Montanari and F. Ricci-Tersenghi (to be published).[13] S. Cocco et al., cond-mat/0206239; M. Mézard, F. Ricci-Tersenghi, and R. Zecchina, cond-mat/0207140.[14] G. Parisi, cond-mat/0208070.[15] A somewhat similar statement appears in L. F.Cugliandolo, J. Kurchan, and P. Le Doussal, Phys. Rev.Lett. 76, 2390 (1996).[16] One possibility is to take Eqs. (7) as the definition ofdynamically connected degrees of freedom. Our discus-sion is therefore valid within each dynamically con-nected component.[17] Here we consider a thermometer based on the zeroth lawof thermodynamics [4].017203-4

Microscopic description of the aging dynamics: fluctuation-dissipation relations, effective temperature and heterogeneities

A microscopic description of the aging dynamics: fluctuation-dissipation
  relations, effective temperature and heterogeneities

A microscopic description of the aging dynamics: fluctuation-dissipation relations, effective temperature and heterogeneities

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