Dynamical Mean-Field Theory and Aging Dynamics
DDynamical Mean-Field Theory and Aging Dynamics
Ada Altieri, Giulio Biroli, and Chiara Cammarota Laboratoire de Physique de l’École Normale Supérieure, Université PSL,CNRS, Sorbonne Université, Université Paris-Diderot, Paris, France Department of Mathematics, King’s College London, Strand London WC2R 2LS, UK
Dynamical Mean-Field Theory (DMFT) replaces the many-body dynamical problem with onefor a single degree of freedom in a thermal bath whose features are determined self-consistently.By focusing on models with soft disordered p -spin interactions, we show how to incorporate themean-field theory of aging within dynamical mean-field theory. We study cases with only oneslow time-scale, corresponding statically to the one-step replica symmetry breaking (1RSB) phase,and cases with an infinite number of slow time-scales, corresponding statically to the full replicasymmetry breaking (FRSB) phase. For the former, we show that the effective temperature of theslow degrees of freedom is fixed by requiring critical dynamical behavior on short time-scales, i.e. marginality. For the latter, we find that aging on an infinite number of slow time-scales is governedby a stochastic equation where the clock for dynamical evolution is fixed by the change of effectivetemperature, hence obtaining a dynamical derivation of the stochastic equation at the basis of theFRSB phase. Our results extend the realm of the mean-field theory of aging to all situations whereDMFT holds. a r X i v : . [ c ond - m a t . d i s - nn ] M a y CONTENTS
I. Introduction 3II. Dynamical mean-field equations 3A. Models with p -spin interactions 3B. Dynamical Mean-Field Theory formalism 4III. Slow Time-Scales: Aging and dynamical phases 5A. Two classes of landscapes 5B. Aging and its two dynamical regimes 6C. Fast and slow noises 7IV. Fast and Slow Time-Scales: Analysis of the TTI and Aging Regimes 7A. TTI regime 8B. Aging Regime 9C. Relationship with the statics 9D. DMFT for Ising and spherical p -spin models 111. Ising p -spin model 112. Spherical p -spin model 12V. Effective temperature in the RSB case 12A. A diagrammatic approach 12B. Applications to Ising and spherical p -spin models with p > p -spin model 142. Spherical p -spin model 14VI. Aging dynamics in the Full RSB regime 15A. Relationship and analogies with Sompolinsky’s dynamical approach 16VII. Conclusions and Perspectives 17Acknowledgments 18A. Alternative analysis of the spherical p-spin in terms of the virial equation 181. 1st approach 182. 2nd approach 193. 3rd approach 20B. Ising spin model: failure of the previously proposed approach 211. Double-well potential: perturbative expansion in the limit of a infinitely narrow double well 22C. Connection between effective temperature and breaking parameter of the static solution 25D. Connection with previous formalisms and identification of the anomaly I. INTRODUCTION
Many metastable states, slow dynamics, and aging are hallmarks of glassiness. The study of mean-field models hasbeen instrumental in revealing these features and understanding such phenomena. The first analysis of the dynamicsof mean-field glassy systems was pioneered by Sompolinsky and Zippelius [1], who were the first to obtain dynamicalmean-field equations for glassy systems. At that time, the interest was mainly on the equilibrium properties. Later,the focus shifted on off-equilibrium , and an exact analysis of the aging dynamics was worked out for a disparate setof models [2–5]. The peculiarity of these models is that their dynamical mean-field equations simplify considerably.In fact, instead of dealing with a self-consistent stochastic process, representing the dynamics of a single degree offreedom in the self-consistent bath formed by the rest of the system, their dynamics can be studied via a closed setof integro-differential equations on correlation and response functions, a fact that played an important role in theirexact analysis.The picture resulting from these works goes beyond the exact solution of these simplified models and provides ageneral scenario for aging dynamics for all mean-field glassy systems (see [6, 7] for very recent surprises). Yet, acomplete dynamical mean-field theory of aging that applies to generic cases where the dynamics can be studied onlythrough the analysis of the self-consistent stochastic process is still missing. This is not a mere technical curiosity, itis actually relevant for the study of topics as diverse as ecosystem dynamics, the glass transition, and optimizationdynamics of neural networks [8–10].The aim of our work is to extend the mean-field theory of aging to generic dynamical mean-field theories (DMFT).We take the mean-field picture of aging [11, 12] as a starting point, and work out its main implications for DMFT. Wemake use of many results obtained along the years. In particular, we combine the ideas put forward by Sompolinskyand Zippelius [1, 13] on dynamics on very large time-scales with the ones developed by Cugliandolo and Kurchan oneffective temperatures and slow thermal baths [14].We shall show how to obtain explicit equations on the correlation and the response of the systems on divergingtime-scales. In particular, in cases (called RSB-like) where the slow dynamics is described by only one divergingtime-scale we find that the effective temperature of the slow degrees of freedom is determined by the condition thatthe dynamics on fast time scales is marginal, i.e. the correlation function decreases as a power law in time. In caseswhere the slow dynamics is described by an infinite set of diverging time-scales (so-called Full RSB-like) we find thatthe slow dynamics contribution is given by a stochastic equation where the clock for dynamical evolution is fixed bythe change of the effective temperature. These results generalise the ones found in simplified models. The latter oneprovides a dynamical derivation of the stochastic equation at the basis of full-replica symmetry breaking [15].We will comment in the Conclusions on possible extensions, and applications of our results to theoretical ecology[8], out-of-equilibrium dynamics of hard spheres in the limit of infinite dimensions [10, 16], and gradient-descent basedalgorithms for non-convex optimization problems [17, 18].The paper is organised as follows: in Section II we will present a class of disordered models defined by p -spininteractions for which the DMFT formalism applies; in Section III the aging hypothesis is described in full generalityalong with the discussion on two different kinds of dynamically broken phases, according to a RSB and a Full RSBAnsatz respectively. We will then present our formalism based on a sharp separation of time scales, focusing first onthe (fast) TTI regime in Sec.(IV A) and then on the slow dynamical phase corresponding to aging, in Sec. (IV B).Next Sec. (V A) will be then devoted to the definition of the effective temperature for models that display a one-stepreplica symmetry breaking solution: the key result relies on the determination of the marginal stability conditionfor two different classes of p -spin models, with continuous and discrete variables respectively. In Section VI we willextend our predictions to a pairwise interaction model, namely the classical Sherrington-Kirkpatrick model. We willprove that also in this case we can write a dynamical effective stochastic process for the effective fields, which exactlymaps into the equation for ultrametricity as it was obtained in the past in a static formalism. Finally, in Section VIIwe will present our conclusive remarks and some perspectives for future investigations in related fields. II. DYNAMICAL MEAN-FIELD EQUATIONSA. Models with p -spin interactions In order to develop the theoretical framework we focus on a simple class of mean-field models, but our results canbe generalized to more complex cases. The elementary degrees of freedom of the models are real variables, that weshall call spins and denote as s i ( i = 1 , . . . , N ). Each spin is subjected to an external potential V ( s i ) . The interactionpart of the Hamiltonian is given by random p -spin interactions: H I = − (cid:88) i <...
The dynamics we are going to focus on is induced by Langevin equations that read: ds i ( t ) dt = − ∂V∂s i + 1( p − (cid:88) i ...i p J i,i ...i p s i ...s i p + η i ( t ) . (4)The first and second term appearing on the right hand side (RHS) correspond to the derivative of the Hamiltonian withrespect to the given spin s i , whereas the noise is (for simplicity) δ -correlated, (cid:104) η i ( t ) η j ( t (cid:48) ) (cid:105) = 2 T δ ij δ ( t − t (cid:48) ) ( T is thetemperature). We shall consider a high-temperature like initial condition at t = 0 given by a non-interacting productmeasure on the spins: P ( s i , t = 0) = (cid:81) Ni =1 P ( s i ) . One can then write the corresponding generating functional interms of a bare contribution and a J -dependent term, which has to be eventually averaged over the disorder [1, 19, 26],and from it obtain the dynamical mean-field equations. Alternatively, one can use the dynamical cavity method [8, 15].These derivations are standard. Hence, we directly state the final result, i.e. the DMFT equation that reads: ˙ s ( t ) = − ∂V ( s ( t )) ∂s + p ( p − (cid:90) t dt (cid:48)(cid:48) R ( t, t (cid:48)(cid:48) ) C p − ( t, t (cid:48)(cid:48) ) s ( t (cid:48)(cid:48) ) + ξ ( t ) , (5)where the noise is such that (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = 2 T δ ( t − t (cid:48) ) + p C p − ( t, t (cid:48) ) . (6)The first contribution corresponds to the usual noise, whereas the second one accounts for the interaction with therest of the system. The correlation and the response functions, C ( t, t (cid:48) ) and R ( t, t (cid:48) ) , are defined respectively as C ( t, t (cid:48) ) = 1 N (cid:88) i s i ( t ) s i ( t (cid:48) ) ,R ( t, t (cid:48) ) = 1 N (cid:88) i δs i ( t ) δh i ( t (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) h i =0 , (7)where h i is an external field linearly coupled to s i . As N → ∞ , these quantities converge to a non-fluctuating value.The DMFT equation has to be solved self-consistently, i.e. one has to find C ( t, t (cid:48) ) and R ( t, t (cid:48) ) such that the stochasticprocess in Eq. (5), with initial condition given by P ( s ) , leads to correlation and response functions equal to C ( t, t (cid:48) ) and R ( t, t (cid:48) ) . In very specific instances, e.g. in the so-called spherical limit, the problem simplifies and C ( t, t (cid:48) ) and R ( t, t (cid:48) ) can be shown to satisfy closed-form integro-differential equations. Note that those closed equations haveformally the same structure as Mode-Coupling Theory equations for structural glasses [27].In general, this does not happen and one has to deal with the self-consistent process defined above. The aim of ourwork is to show how its solution can be handled for aging dynamics. III. SLOW TIME-SCALES: AGING AND DYNAMICAL PHASES
As we discussed in the Introduction, one key feature of glassy systems is that they display slow and aging dynamicsafter a quench from high to low temperature. The behavior at a large time t w after the quench is characterized by:(i) power law (or even slower) relaxation of one-time quantities, and (ii) a decorrelation time that grows with the time t w .The theoretical analysis based on mean-field models performed in the 90s has unveiled that there are at least twodifferent classes of aging dynamics, correspondingly to two different classes of free-energy landscapes [12]. We nowrecall their main salient features. A. Two classes of landscapes
In the case of mean-field glassy models one can give a precise meaning to the free-energy landscape, which isobtained from the TAP free-energy [28]. The number of minima, and more generally of critical points of given index,has been computed and analysed thoroughly [29–32], even rigorously in recent years [33, 34]. These works establishedthe existence of two very different classes of landscapes, which are related to different thermodynamical and dynamicalproperties: • Spin-glass landscapes : In this case, the number of free-energy minima is not exponential in the system size.The free-energy barriers are expected to be sub-extensive, and despite quenches to low temperature induce aging,one-time observables converge to their equilibrium thermodynamic limit. For instance, the asymptotic valueof the energy density coincides with the equilibrium value obtained within the static approach, E ∞ = E eq .Moreover, there is a strong connection between the asymptotic aging dynamics and the thermodynamics [35].In fact, the system asymptotically approaches the marginal free-energy minima that are relevant for equilibriumproperties. Finally, the dynamical transition at which the system falls out of equilibrium takes place at thesame critical temperature of the thermodynamic spin-glass transition. • Simple structural glass landscapes : In this case, the number of free-energy minima is exponential in thesystem size. The free-energy barriers are extensive, and one-time observables do not converge to their equilibriumvalue. A connection with free-energy minima still holds. In fact, long-time aging dynamics approaches free-energy minima with the largest basin of attraction, which are generally not the ones relevant for equilibriumthermodynamic properties. Starting from random initial conditions, i.e. quenches from infinite temperature,these have been identified as the typical most numerous minima that are marginally stable (called thresholdstates ). In this case, the dynamical transition at which the system falls out of equilibrium takes place at highertemperature, T d , than the thermodynamic glass transition T s .As we have recalled above, marginally stable free-energy minima play a key role in aging dynamics. Marginalitymeans that the free-energy Hessian matrix at the minima is characterised by arbitrary small eigenvalues. There aremodels characterised by more complicated free-energy landscapes that combine features of spin-glass and structuralglass landscapes, as for instance when a Gardner phase transition takes place (a likely general feature for structuralglass models at low enough temperature) [36–38]. These cases are left for future works, and hence not described indetail here.The relationship recalled above between free-energy minima and aging dynamics is at the core of the weak ergodicitybreaking hypothesis, that has been proven to hold in a large class of systems, in particular the instances we are going todiscuss in the rest of the paper. Recent results showed that the general situation can be more intricate: in particularstarting the quench from finite (and not infinite) temperature it was shown that for mixed p -spin models [6] the It is possible to show that there is a unique solution respecting causality. Note that we always consider the case in which the thermodynamic limit is taken from the start, i.e. asymptotic values corresponds tothe specific limit order lim t,t (cid:48) →∞ lim N →∞ . long-time dynamics approach marginally stable free-energy minima that are not the threshold ones. In numericalsimulations in spin-glass models analysed on sparse graphs [7], the authors claimed that the weak ergodicity breakingdoes not hold. B. Aging and its two dynamical regimes
In order to analyse aging dynamics, in particular within mean-field, it is useful to make extensive use of long-timelimit analysis, which allows for a sharp timescale separation between a fast regime, in which rapid degrees of freedomequilibrate, and a slow regime displaying violation of fluctuation-dissipation relations and non-equilibrium phenomena[12, 39]. The existence of two-time sectors have been explicitly shown to hold for certain class of mean-field models[2, 40–42]. The exploitation of such time separation stands at the core of mean-field analysis of aging dynamics[2–4, 12, 39, 41, 43–46]. Cutting a long story short, we directly recall the form the correlation function takes in thelong time limit t, t (cid:48) → ∞ : C ( t, t (cid:48) ) = C TTI ( t − t (cid:48) ) + C A ( t, t (cid:48) ) . (8)In the time-translation invariant (TTI) sector, which corresponds to t , t (cid:48) (cid:29) with ( t − t (cid:48) ) of order one, only the firstterm on the RHS gives a non vanishing contribution and accordingly C TTI (0) = q d − q , C TTI ( ∞ ) = 0 . (9)The overlap q d is by definition equal to C ( t, t ) , whereas q corresponds to the plateau value of the correlation functionseparating TTI and aging regime, see Fig. 1.The aging sector corresponds to t , t (cid:48) (cid:29) , and ( t − t (cid:48) ) which diverges together with t, t (cid:48) . In this regime C TTI is zeroand the only contribution to C ( t, t (cid:48) ) is given by C A ( t, t (cid:48) ) , which satisfies the boundary condition: C A ( t, t ) = q . (10)The response function displays an analogous behavior which can be decomposed in a TTI and an aging contribution.In the TTI sector the response function verifies the fluctuation-dissipation theorem, i.e. R TTI ( τ ) = − T dC
TTI ( τ ) dτ . Thisis natural since degrees of freedom contributing to the TTI regime relax on a finite time-scale and, hence, equilibrateat long times.The behavior in the aging regime depends on the dynamical phase, in particular whether there is only one or multiplediverging time-scales. • One diverging time-scale: the RSB dynamical ansatz
In the simplest scenario, the aging regime isdescribed by a single diverging timescale. The corresponding dynamical Ansatz reads: C A ( t, t (cid:48) ) = C (cid:34) (cid:98) h ( t (cid:48) ) (cid:98) h ( t ) (cid:35) , R A ( t, t (cid:48) ) = ˙ (cid:98) h ( t (cid:48) ) R (cid:34) (cid:98) h ( t (cid:48) ) (cid:98) h ( t ) (cid:35) (11)where (cid:98) h ( t ) is a monotonously increasing function that corresponds to the relaxation timescale of the slow degreesof freedom (from now on we consider t > t (cid:48) ). It depends on t because the system is aging: the older is the system,the slower is the relaxation. An important and highly non-trivial relationship between correlation and responseholds in this regime: R A ( t, t (cid:48) ) = xT ∂ t (cid:48) C A ( t, t (cid:48) ) . This is a generalization of the fluctuation-dissipation relationfor the aging regime (with an effective temperature T eff defined by x = T /T eff ). This aging behavior has beenfound in models characterized by simple structural glass landscapes, and it is the dynamical counterpart of the RSB static Ansatz. • Infinitely many diverging time-scale: the Full RSB dynamical Ansatz
This case is characterised by infinitely many diverging timescales. A monotonously increasing function (cid:98) h i ( t ) isassociated to each timescale i . The aging contribution to the correlation function can be written as a combinationof rescaled functions C i associated to each timescale [39]: C A ( t, t (cid:48) ) = (cid:88) i C i (cid:34) (cid:98) h i ( t (cid:48) ) (cid:98) h i ( t ) (cid:35) (12) C ( t, t ) t − t q d aging q TTI regime
Figure 1. The correlation function C ( t, t (cid:48) ) , which depends generically on two times scales, displays a decreasing trend fromthe maximum value q d to the plateau value whose height coincides with q and signals the onset of non-ergodicity. The escapefrom the plateau is regulated by the function C A ( t, t (cid:48) ) . where C i (1) is equal to q i − q i − and C i (0) = 0 (remember that t > t (cid:48) ). Each C i describes the drop of thecorrelation from q i to q i − that takes place within the timescale i . A generalized fluctuation-dissipation relationis valid within each time-scale, i.e. R A ( t, t (cid:48) ) = x i T ∂ t (cid:48) C A ( t, t (cid:48) ) for t, t (cid:48) such that < (cid:98) h i ( t (cid:48) ) (cid:98) h i ( t ) < . We have describedthe aging Ansatz in terms of a discrete set of timescales. One can take the limit of an infinite number of timescaleassuming that that all the differences q i − q i − goes to zero and at the same time the number of timescales goesto infinity. In this case, it is useful to introduce the function x ( q ) which relates x i with q i . The aging behaviourjust described is the dynamical counterpart of the Full RSB static Ansatz. C. Fast and slow noises
Within DMFT the single spin stochastic equation is characterised by an effective noise that take into accountboth the thermal noise and the interaction with the rest of the system. Given that the system is characterised byseveral timescales, so does the noise. To make explicit these different contributions, we express ξ ( t ) as the sum of twoindependent Gaussian noise contributions, ξ TTI ( t ) and ξ A ( t ) , such that (cid:104) ξ TTI ( t ) ξ TTI ( t (cid:48) ) (cid:105) = 2 T δ ( t − t (cid:48) ) + p C TTI ( t − t (cid:48) ) + q ] p − − p q p − , (13) (cid:104) ξ A ( t ) ξ A ( t (cid:48) ) (cid:105) = p C A ( t, t (cid:48) ) p − . (14)By choosing the covariances in this way, the sum ξ ( t ) = ξ TTI ( t ) + ξ A ( t ) leads to a correct representation of the noisein the asymptotic limit t, t (cid:48) (cid:29) . The slow noise ξ A ( t ) can be further decomposed in multiple contributions if thereare many slow timescales. Using the notation introduced above for the Full RSB dynamical Ansatz, one can introduceindependent Gaussian noise contributions ξ A,i ( t ) for each slow timescale. In order to have a correct representation ofthe noise in the asymptotic limit, the covariance of the ξ A,i ( t ) s has to be chosen in the following way: (cid:104) ξ A,i ( t ) ξ A,i ( t (cid:48) ) (cid:105) = p (cid:32) C i (cid:34) (cid:98) h i ( t (cid:48) ) (cid:98) h i ( t ) (cid:35) + q i − (cid:33) p − − p q p − i − . (15) IV. FAST AND SLOW TIME-SCALES: ANALYSIS OF THE TTI AND AGING REGIMES
In the following we show how to disentangle the regimes of fast and slow time-scales in the analysis. As we shallshow, in the first regime the system is in quasi-equilibrium and one can study its corresponding quasi-equilibriumdynamics. In the second regime, instead, the system is evolving very slowly (the slower the older is the system). Thisleads to an almost adiabatic change of some of the parameters determining the fast dynamics. We will obtain theprobability distribution of such parameters along the aging dynamics. All that will allow us to find all the quantitiesof interest to characterise aging dynamics, except the effective temperature to which we come back in the next twosections.
A. TTI regime
In the following we will make extensive use of this aforementioned timescale separation, focusing first on ouranalytical derivation in the time-translational invariant regime. We consider the time evolution of the spin variable s ( t ) written in terms of an effective Langevin process: ˙ s ( t ) = − ∂V ( s ( t )) ∂s + p ( p − (cid:90) dt (cid:48)(cid:48) R ( t, t (cid:48)(cid:48) ) C p − ( t, t (cid:48)(cid:48) ) s ( t (cid:48)(cid:48) ) + ξ ( t ) (16)where V ( s ( t )) stands for a generic potential, whereas ξ ( t ) is a normally distributed coloured noise with zero meanand covariance defined by Eq. (6). We use timescale separation to decompose the second term on the RHS of Eq.(16), playing the role of a friction contribution: (cid:90) t dt (cid:48)(cid:48) R ( t, t (cid:48)(cid:48) ) C p − ( t, t (cid:48)(cid:48) ) s ( t (cid:48)(cid:48) ) (cid:39) (cid:90) tt − τ R TTI ( t − t (cid:48) ) [ C TTI ( t − t (cid:48) ) + q ] p − s ( t (cid:48) ) dt (cid:48) + (cid:90) t − τ R A ( t, t (cid:48) ) C p − A ( t, t (cid:48) ) s ( t (cid:48) ) dt (cid:48) (17)where τ /t ∼ o (1) , the response function R TTI ( t − t (cid:48) ) = T ddt (cid:48) C TTI ( t − t (cid:48) ) , R TTI ( t − t (cid:48) ) → as t − t (cid:48) → ∞ , and R A ( t, t (cid:48) ) is uniformly small but when integrated over time leads to a finite contribution. By integrating the contribution thataccounts only for the time-translation invariant regime by parts, we eventually obtain: (cid:90) tt − τ R TTI ( t − t (cid:48) ) [ C TTI ( t − t (cid:48) ) + q ] p − s ( t (cid:48) ) dt (cid:48) (cid:39) T ( p − (cid:2) ( C TTI (0) + q ) p − s ( t ) − ( C TTI ( τ ) + q ) p − s ( t − τ ) (cid:3) + (18) − T ( p − (cid:90) tt − τ [ C TTI ( t − t (cid:48) ) + q ] p − ˙ s ( t (cid:48) ) dt (cid:48) . (19)For very large τ , even if still much smaller than t , C T T I ( τ ) (cid:39) and the above equation becomes (cid:90) tt − τ R TTI ( t − t (cid:48) ) [ C TTI ( t − t (cid:48) ) + q ] p − s ( t (cid:48) ) dt (cid:48) (cid:39) T ( p − (cid:104) q p − d − q p − (cid:105) s ( t ) + − T ( p − (cid:90) tt − τ (cid:40) [ C TTI ( t − t (cid:48) ) + q ] p − − q p − (cid:41) ˙ s ( t (cid:48) ) dt (cid:48) (20) = 1 T ( p − (cid:104) q p − d − q p − (cid:105) s ( t ) − p ( p − (cid:90) tt − τ ν TTI ( t − t (cid:48) ) ˙ s ( t (cid:48) ) dt (cid:48) where ν TTI ( t − t (cid:48) ) ≡ p T [ C TTI ( t − t (cid:48) ) + q ] p − − p T q p − . (21)Recalling Eq. (13), we therefore also get: (cid:104) ξ TTI ( t ) ξ TTI ( t (cid:48) ) (cid:105) (cid:39) T δ ( t − t (cid:48) ) + T ν
TTI ( t − t (cid:48) ) . (22)The original Eq. (16) can thus be rewritten as ˙ s ( t ) (cid:39) − ∂V ( s ) ∂s + p T ( q p − d − q p − ) s ( t ) − (cid:90) tt − τ ν TTI ( t − t (cid:48) ) ˙ s ( t (cid:48) ) dt (cid:48) + ξ TTI ( t ) + h ( t ) (23)where the terms accounting for the slow (aging) dynamical behavior have been embedded into what can be considereda slowly evolving effective field h ( t ) ≡ p ( p − (cid:90) t − τ R A ( t, t (cid:48) ) C p − A ( t, t (cid:48) ) s ( t (cid:48) ) dt (cid:48) + ξ A ( t ) . (24)Eq. (23) shows that the original dynamical problem can be mapped into a stochastic process for the single variable s ( t ) in the presence of friction and subject to a quasi stationary effective potential V ( s, h ( t )) = V ( s ) − p T ( q p − d − q p − ) s − h ( t ) s . (25)Such a new stochastic process will therefore be associated to a quasi-stationary conditional probability distribution,which is nothing but the Boltzmann-Gibbs distribution at a given external temperature T , P ( s | h ( t )) = 1 Z ( h ) exp (cid:20) − V ( s, h ( t )) T (cid:21) (26)and to a quasi stationary free-energy obtained from the corresponding partition function Z ( h ) , F ( h ( t )) = − T ln( Z ( h )) .Here, we have followed [14] where a similar procedure was used in to study the motion of a particle moving in a randompotential and in contact with two thermal baths varying on very different timescales. As also shown in [14] the solutionof the full dynamics requires a detailed characterisation of the statistical properties of the quasi-static field h ( t ) . Wediscuss it for the present problem in the next section. B. Aging Regime
In the aging regime we assume that correlation and response obey generalised FDT relations with violation pa-rameter x and effective aging temperature T eff = T /x . Moreover, we can replace the dynamical variable s ( t (cid:48) ) , inthe integral of the first term of the slowly evolving field, with its average on the short times fluctuations, (cid:104) s ( t (cid:48) ) (cid:105) , orequivalently its average over the distribution in Eq. ((26)), which self-consistently depends on the field and can bedirectly expressed in terms of the free-energy F ( h ) h ( t ) (cid:39) p T eff (cid:90) t ∂∂t (cid:48) (cid:16) C p − A ( t, t (cid:48) ) (cid:17) (cid:104) s ( t (cid:48) ) (cid:105) h ( t (cid:48) ) + ξ A ( t ) = − p T eff (cid:90) t ∂∂t (cid:48) (cid:16) C p − A ( t, t (cid:48) ) (cid:17) ∂F ( h ) ∂h ( t (cid:48) ) + ξ A ( t ) . (27)Starting from this self-consistent equation on h ( t ) it is possible to show [14] that the corresponding slow non-Markoviandynamics coupled to a bath of temperature T eff is associated to the following stationary distribution for hP ( h ) = 1 Z exp − h (cid:16) p q p − (cid:17) − xT F ( h ) , (28)where Z is the the normalization factor.The stationary distribution of the slowly evolving field can now be used to explicitly characterise the full probabilitydistribution of the degrees of freedom: P ( s ) = (cid:90) dhP ( s | h ) P ( h ) (29)and of its moments. Except for the parameter x , which will be determined and discussed in the next two section, wecan now obtain all quantities of interest for aging dynamics, in particular the overlaps q and q d : (cid:104) s (cid:105) = 1 Z (cid:90) dh e (cid:32) − h ( p qp − ) − xT F ( h ) (cid:33) (cid:82) ∞−∞ ds e − V ( s ) T + p T ( q p − d − q p − ) s + hsT s (cid:82) ∞−∞ ds e − V ( s ) T + p T ( q p − d − q p − ) s + hsT ≡ q d , (30) (cid:104) s (cid:105) = Z (cid:90) dh e (cid:32) − h ( p qp − ) − xT F ( h ) (cid:33) (cid:82) ∞−∞ ds e − V ( s ) T + p T ( q p − d − q p − ) s + hsT s (cid:82) ∞−∞ ds e − V ( s ) T + p T ( q p − d − q p − ) s + hsT ≡ q (31)Remarkably, these equations do not involve the dynamics any longer and resemble the static ones obtained by thereplica method. This is not a coincidence, and it is at the basis of the correspondence between dynamic and staticapproaches, as discussed below in more details. C. Relationship with the statics
Replica aficionados will certainly realise that the two equations above coincides with the ones that one obtain bythe replica method for the overlaps within pure states (whether the phase is 1RSB or FRSB). In the following we0illustrate this relationship in the simple p = 2 case and assuming a 1RSB Ansatz. The generalization to larger valuesof p and to a FRSB Ansatz is straightforward.At equilibrium the usual way to obtain equations for the overlap parameters and the effective temperature is based onexploitation of the replica method [15], which allows one to compute the replicated free energy f = − lim N →∞ TN ln( Z ) by means of the following identity ln Z = lim n → Z n − n . (32)Instead of dealing with the disordered average of the logarithmic, one has just to compute and average the replicatedpartition function with n distinct copies of the original system.Generically, the replicated partition function can be eventually expressed in terms of an action S which is a functionof the overlap matrix Q ab : Z n = (cid:90) (cid:89) ( ab ) dQ ab √ π e S [ Q ab ] (33)given the usual definition of the overlap between two spin configurations labeled by the replica indices a , b : Q ab = 1 N (cid:88) i s ai s bi . (34)Different approximations can be introduced to correctly parametrise the overlap matrix Q ab , the simplest being thereplica symmetric (RS) one. This solution however is unable to describe the physics of disordered systems in the q d q q q q q n q Figure 2. Pictorial representation of a one-step replica symmetry breaking scheme for the overlap matrix Q ab . The n × n matrixis divided into n/x × n/x blocks, each of them of size x × x . low-temperature regime, whose correct solution is based on an iterative block structure of that matrix [15, 47–49].Figure 2 shows a representation of a RSB realisation of such structure in an n × n matrix parametrised by a diagonalvalue q d , and off-diagonal elements either equal to q if the replica indices belong to the same block B of size x × x (inlight green) or equal to q if the elements are outside the diagonal blocks (in dark green). This scheme can be iterated k times and used to construct a k RSB structure for the overlap matrix, which becomes Full RSB in the k → ∞ limit.Note that the correspondence in the notation between the parameters of the RSB structure and the moments of thedynamical variable within the DMFT approach is not accidental and hints at the identification of the correspondingdynamic and static quantities. For instance, the static replica computation of the overlap q in the RSB scheme isobtained by averaging single site variables from two replicas belonging to the same block B , with a weight given bythe replicated action S rewritten introducing an auxiliary variable z to decouple single replica integrals in the RSB ansatz .1For pairwise interactions ( p = 2 ), the final equation for q reads: q = (cid:104) s a s b (cid:105) = 1 Z n (cid:90) dz e − z q √ πq (cid:18)(cid:90) ds s e − T [ V ( s ) − zs − T ( q d − q ) s ] (cid:19) (cid:18)(cid:90) ds e − T [ V ( s ) − zs − T ( q d − q ) s ] (cid:19) x − ×× (cid:90) dz e − z q √ πq (cid:18)(cid:90) ds e − T [ V ( s ) − zs − T ( q d − q ) s ] (cid:19) x nx − (35)with Z n = (cid:90) dz e − z q √ πq (cid:18)(cid:90) ds e − T [ V ( s ) − zs − T ( q d − q ) s ] (cid:19) x nx . (36)Considering the analytical continuation n → , the equation becomes: q = (cid:82) dz (cid:18)(cid:82) ds s e − T [ V ( s ) − zs − T ( qd − q s ] (cid:19)(cid:18)(cid:82) ds e − T [ V ( s ) − zs − T ( qd − q s ] (cid:19) (cid:18) e − z q − xT F ( z ) (cid:19)(cid:82) dz e − z q − xT F ( z ) (37)with F ( z ) = − T ln (cid:18)(cid:90) ds e − T [ V ( s ) − zs − T ( q d − q ) s ] (cid:19) , (38)which is formally equivalent to the equation for q derived in the DMFT computation. The same correspondence holdsbetween the overlap on the diagonal of the RSB matrix and the equal time correlation of the dynamical variable,also called q d .Interestingly, comparing the DMFT and the replica equations, it also becomes evident a one to one correspondencebetween the dynamical aging field h and the auxiliary variable z . An intuitive understanding of such correspondencecan be acquired by re-deriving the static equations through the cavity approach [15, 50], where it clearly emerges thatsingle spin variables are effectively subject to random local fields z characterised by the same non trivial distributionas the aging fields in the DMFT approach.Finally this highlights the well known [39, 43, 51, 52] link between the FDT violation parameter and the RSBparameter x , as they play a formally identical role in the static and dynamic equations for q d and q . D. DMFT for Ising and spherical p -spin models In this Section, in order to provide simple examples and show connections with known results, we apply theformalism we developed to the spherical and the Ising cases introduced in Sec. II A.
1. Ising p -spin model For an Ising p -spin model V ( s ) is (formally) zero for s = ± and infinite otherwise, hence s = 1 and q d ≡ (cid:104) s (cid:105) = 1 .Moreover we have P ( s | h ( t )) = e h ( t ) sT h/T ) (39)with Z ( h ) = 2 cosh( h/T ) and F ( h ) = − T ln(2 cosh( h/T )) and for the only non trivial overlap q ≡ (cid:104) s (cid:105) = (cid:18) e h/T − e − h/T h/T ) (cid:19) = tanh ( h/T ) = (cid:82) dh e − h ( p qp − ) tanh ( h/T )[cosh( h/T )] x (cid:82) dh e − h ( p qp − ) [cosh( h/T )] x . (40)2
2. Spherical p -spin model To study the spherical p -spin model, we consider a soft spherical constraint implemented with the introduction of aquadratic potential V ( s ) = λs involving the spherical parameter λ . The conditional probability distribution for spindynamical variables thus becomes P ( s | h ( t )) = 1 Z ( h ) exp (cid:40) − (cid:104) λ − p T ( q p − d − q p − ) (cid:105) T s − h (cid:104) λ − p T ( q p − d − q p − ) (cid:105) + h T (cid:104) λ − p T ( q p − d − q p − ) (cid:105) (cid:41) , (41)with Z ( h ) = (cid:118)(cid:117)(cid:117)(cid:116) πT (cid:104) λ − p T ( q p − d − q p − ) (cid:105) exp (cid:40) h T (cid:104) λ − p T ( q p − d − q p − ) (cid:105) (cid:41) , (42)from which the long and short time limit of the correlation function turn out to be respectively q ≡ (cid:104) s (cid:105) = (cid:18)(cid:90) dsP ( s | h ) s (cid:19) = h (cid:104) λ − p T ( q p − d − q p − ) (cid:105) (43) q d ≡ (cid:104) s (cid:105) = (cid:90) P ( s | h ) s = Tλ − p T ( q p − d − q p − ) + h (cid:104) λ − p T ( q p − d − q p − ) (cid:105) . (44)The equation on their difference q d − q , evaluated at q d = 1 , becomes a condition on the spherical parameter λ to beimposed so that the spherical constraint is always satisfied during the dynamics λ − p T (1 − q p − ) = T − q . (45)Finally, by using the above equation on the spherical parameter λ and q d = 1 in Eq. (43) it is possible to rewrite theequation on q as follows q (cid:104) λ − p T (1 − q p − ) (cid:105) = p q p − − xT (cid:104) λ − p T (1 − q p − ) (cid:105) − (46)which after some passages becomes λ = T + p T (1 − q p (1 − x )) . (47)As we will show in detail in Appendix A, the equation for the Lagrange multiplier can be found in an alternativeway by introducing a virial equation, namely by multiplying every side of the equation of motion by s and averagingover the associated stochastic process. In the case of the Ising model, instead, the corresponding virial equation leadsto an automatically satisfied condition that trivially corresponds to the normalization of the distribution P ( h ) . For adetailed explanation, we refer the interested reader to the Appendix. V. EFFECTIVE TEMPERATURE IN THE RSB CASEA. A diagrammatic approach
As we have shown in the previous section, the dynamical aging Ansatz allows us to establish the equations satisfiedby the dynamical overlaps. The effective temperature, however, remains unknown. We will now present a generalapproach that allows one to derive the equation for the effective temperature in models for which the dynamical RSBAnsatz holds. Our procedure is based on the physical requirement that the dynamics in the TTI sector is marginal,3 i.e. the relaxation to the plateau q is power-law and not exponential.Our starting point is the stochastic Eq. (23), which describes relaxation dynamics in the TTI regime.We shall use standard diagrammatic perturbation theory following the procedure developed for equilibrium criticalspin-glass dynamics [53]. Eq. (23) can be rewritten as (cid:90) t −∞ R − ( t − t (cid:48) ) s ( t (cid:48) ) dt (cid:48) = − ∂V ( s ) ∂s + ξ TTI ( t ) + h ( t ) , (48)where R − ( t − t (cid:48) ) has a simple expression in the Fourier domain ( F ω denotes the Fourier transform): R − ( ω ) = − iω + F ω ( ∂ t ν TTI ( t )) . (49)We now present the method in the simplified case in which no field h ( t ) is present, and then later we explain how togeneralize it. The response function can be expressed in terms of the self-energy Σ as R TTI ( ω ) = 1 R − ( ω ) − Σ( ω ) . (50)This Schwinger-Dyson equation is generically represented in diagrammatic theory using a straight line for R and adashed circle for Σ : fcf = ff + fpf + fpfpf + · · · = ff1 − ( fp ) (51)The dashed circle corresponds to all self-energy diagrams generated when doing the perturbation theory in thecouplings corresponding to ∂V ( s ) ∂s .In cases where there are no conservation laws, as for the mean-field glassy systems we focus on, the responsefunction decreases exponentially to zero at large times in the high-temperature ergodic regime. This behavior changesfor marginal (or also critical [53]) dynamics where instead one expects a power-law relaxation. Accordingly, thebehavior at small ω of R − TTI ( ω ) is linear in the former case and power-law with an exponent less than one for marginaland critical dynamics. Hence, the condition encoding the existence of marginal dynamics is: lim ω → ∂R − TTI ( ω ) ∂ω = ∞ . (52)We now show that this requirement leads to a simple equation. In fact, using that at large times ∂ t ν TTI ( t ) (cid:39) p ( p − T q p − ∂ t C TTI ( t ) = − p ( p − q p − R TTI ( t ) , (53)one finds that for marginal dynamics and in the small ω limit: R − ( ω ) (cid:39) − iω − p ( p − q p − R TTI ( ω ) . (54)By taking the inverse of the Schwinger-Dyson equation and differentiating it, one gets in the small ω limit: ∂R − TTI ( ω ) ∂ω = ∂R − ( ω ) ∂ω − ∂ Σ( ω ) ∂ω (cid:39) ∂∂ω (cid:20) − iω − p ( p − q p − R TTI ( ω ) (cid:21) − ∂ Σ( ω ) ∂ω . (55)Using the identity: ∂R TTI ( ω ) ∂ω = − ∂R − TTI ( ω ) ∂ω R TTI ( ω ) (56)we finally obtain: ∂R − TTI ( ω ) ∂ω = − i − ∂ Σ( ω ) /∂ω − p ( p − q p − R TTI ( ω ) . (57)4It can be shown that the numerator is not singular ( e.g. to all orders in perturbation theory) [53]. In consequence,the divergence for ω → of the LHS — the condition for dynamical marginality — is given by the vanishing of thedenominator: p ( p − q p − R TTI ( ω ) (cid:12)(cid:12) ω =0 . (58)When a random field h is present in the stochastic equation, one has to redo the previous procedure introducinga h -dependent response function ˜ r ( ω, h ) , which when averaged over the static field h leads to the average responsefunction: ˜ r ( ω, h ) = R ( ω ) . By repeating the previous analysis, see [53] for the similar case of critical dynamics, onefinds: p ( p − q p − ˜ r (0 , h ) (59)By using FDT, which is valid in the TTI regime, one obtains ˜ r (0 , h ) = (cid:16) (cid:104) s (cid:105)−(cid:104) s (cid:105) T (cid:17) = (cid:16) ∂ (cid:104) s (cid:105) ∂h (cid:17) and hence a conditionfor marginal dynamics that depends only of the probability distribution of h and which therefore provides the extraequation allowing to fix the effective temperature: p ( p − q p − (cid:18) (cid:104) s (cid:105) − (cid:104) s (cid:105) T (cid:19) . (60)It can be shown that this is exactly the expression for the vanishing of the replicon in the RSB analysis of this model.This therefore completes the analysis of the aging dynamics in the RSB case, and shows how one can establish theconnection with the static formalism.As before, in order to show a simple application and the connection with known results, we apply the result above toi) Ising spins and p ≥ , ii) continuous variables and p > . B. Applications to Ising and spherical p -spin models with p >
1. Ising p -spin model Application of Eq. (60) to the Ising p -spin model requires the use of the previously derived Eq. (40) and q d = (cid:104) s (cid:105) = 1 to get ˜ r (0 , h ) = (cid:18) (cid:104) s (cid:105) − (cid:104) s (cid:105) T (cid:19) = 1 T (cid:2) − tanh ( h/T ) (cid:3) , (61)and therefore p ( p − T q p − (cid:104) − ( h/T ) + tanh ( h/T ) (cid:105) , (62)which coincides with the expression derived in [54] (see also [38]).The last condition, together with the Eq. (40) on q and q d = 1 , forms a closed system of equations derived in thiscase within the DMFT approach, which can be therefore used to determine the q and x that characterise agingdynamics for an Ising p -spin with p > . For p = 2 the situation will be different as a Full RSB phase is going tocontrol the aging behaviour of a relaxation dynamics after a quench. In this case a specific extension of DMFT mustbe considered as explained in Sec. VI. In this case, Eq. (62) evaluated at p = 2 , together with the condition x = 1 ,can be used to determine at what temperature aging dynamics would set in.
2. Spherical p -spin model To obtain a similar closed set of equations in the spherical case, we recall again the results of Sec. (IV D) and inparticular Eqs. (47) and (45). The addition of Eq. (60), which in this case gives p ( p − q p − (cid:18) (cid:104) s (cid:105) − (cid:104) s (cid:105) T (cid:19) = p ( p − T q p − (1 − q ) , (63)closes the set so that it is possible to determine λ , q and x within the DMFT approach. Note that this set of equationscorresponds to the one obtained in the works on the spherical p -spin model [2, 55].5 VI. AGING DYNAMICS IN THE FULL RSB REGIME
The aim of this section is to show how to tackle cases with an infinite number of slow time-scales. For the classof models we focus on, this happens for Ising spins and p = 2 , which corresponds to the Sherrington-Kirkpatrick(SK) model. The different nature of its transition (spin-glass like) largely affects the kind of aging behaviour takingplace in the long time dynamics. We have therefore to derive a new rule for the slow evolution of the external field.Conversely the description of the short time dynamics within a TTI framework will remain unchanged. In particularthe dynamical variable s ( t ) on short time scales evolves according to a stochastic process in the presence of frictionas described by Eq. (23) with q d = 1 , ν TTI ( t − t (cid:48) ) = 1 T C
TTI ( t − t (cid:48) ) , (64)(since p = 2 ), and associated to a quasi stationary conditional probability P ( s | h ( t )) of the form in Eq. (26), which inthis case becomes P ( s | h ( t )) = e h ( t ) sT h ( t ) /T ) , (65)and immediately implies (cid:104) s ( t ) (cid:105) = m ( t ) = tanh( h ( t ) /T ) . Recall that the slowly evolving external field h ( t ) was definedin Eq. (24) and in this case given by h ( t ) = ξ A ( t ) + (cid:90) t dt (cid:48) R A ( t, t (cid:48) ) m ( t (cid:48) ) (66)where (cid:104) ξ A ( t ) ξ A ( t (cid:48) ) (cid:105) = C A ( t, t (cid:48) ) . Finally, the slow evolution of such external field, controlled by aging dynamics, willset in as soon as the condition of marginal stability in Eq. (62) for p = 2 is satisfied: − T (cid:104) − ( h/T ) + tanh ( h/T ) (cid:105) . (67)In the following we derive an explicit equation that describes the evolution of the aging field h ( t ) along the dynamics.To this aim we notice that ∆ h ( t ) = h ( t + ∆ t ) − h ( t ) = ξ A ( t + ∆ t ) − ξ A ( t ) + (cid:90) t +∆ t dt (cid:48) R A ( t + ∆ t, t (cid:48) ) m ( t (cid:48) ) − (cid:90) t dt (cid:48) R A ( t, t (cid:48) ) m ( t (cid:48) )= ξ A ( t + ∆ t ) − ξ A ( t ) + (cid:20) R A ( t, t ) m ( t ) + (cid:90) t ∂ t R A ( t, t (cid:48) ) m ( t (cid:48) ) dt (cid:48) (cid:21) ∆ t . (68)Note that ∆ t represents a small change in unit of very large time-scales. There are three contributions to the changeof the slow field: the first is due to the evolution of the stochastic slow noise between t and t + ∆ t , the second dependson the state of the system at time t , and the last is obtained integrating over all the past behavior.By dropping the last term, which gives a sub-leading contribution, one can recognize that the equation on ∆ h ( t ) has the form of a stochastic equation with a drift term: D (1) ( t ) = R A ( t, t ) m ( t )∆ t = − xT ∂C A ( t, t (cid:48) ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t (cid:48) = t m ( t )∆ t (cid:39) xT [ C A ( t, t ) − C A ( t + ∆ t, t )] m ( t ) (69)and a noise term with variance that reads (keeping only terms up to the linear order in ∆ t ): D (2) ( t ) = ξ A ( t + ∆ t ) ξ A ( t + ∆ t ) + ξ A ( t ) ξ A ( t ) − ξ A ( t + ∆ t ) ξ A ( t ) − ξ A ( t ) ξ A ( t + ∆ t )= C A ( t + ∆ t, t + ∆ t ) + C A ( t, t ) − C A ( t + ∆ t, t ) . (70)In order to evaluate the differences of correlations in the drift and the variance of the noise we use the FRSB agingAnsatz discussed in Section III. The aging correlation function C A ( t, t (cid:48) ) with t ≥ t (cid:48) equals the intrastate overlap q ( x ) of the states reached at the largest timescale t and associated to FDT violation parameter x . Therefore the drift andthe variance can be rewritten in terms of the intrastate overlaps as follows D (1) ( t ) (cid:39) xT m ( h ) ( q ( x ) − q ( x − ∆ x )) (cid:39) xT m ( h ) ˙ q ( x )∆ x (71)6 t t ! Figure 3. Representation of k different time sectors (for simplicity with k = 6 ) showing the hierarchical organization oftimescales according to a k -RSB Ansatz. The Full RSB picture corresponds to sending the number of sectors k to infinity.In this limit, the piecewise function parametrising the overlap matrix is well defined and converges to the continuous function q ( x ) . D (2) ( t ) (cid:39) q ( x − ∆ x ) + q ( x ) − q ( x − ∆ x ) (cid:39) ˙ q ( x )∆ x . (72)Finally we note that the covariance of the noise at different timescales is zero since: ξ ( t + ∆ t ) ξ ( t (cid:48) + ∆ t ) + ξ ( t ) ξ ( t (cid:48) ) − ξ ( t + ∆ t ) ξ ( t (cid:48) ) − ξ ( t ) ξ ( t (cid:48) + ∆ t ) = q ( x − ∆ x ) + q ( x ) − q ( x − ∆ x ) − q ( x ) = 0 . (73)In conclusion we have obtained that the slow field h ( t ) satisfies a stochastic Langevin equation dh ( x ) dx = xT m ( h ) ˙ q ( x ) + (cid:112) ˙ q ( x ) z ( x ) . (74)where z ( x ) is a Gaussian white noise, z ( x ) z ( x (cid:48) ) = δ ( x − x (cid:48) ) , and the evolution is measured in terms of the change ofthe effective temperature x (each x corresponds to a time-scale t x as recalled in Fig. 3).Remarkably, such Langevin’s equation coincides with the one derived in the studies that focused on the thermody-namic FRSB phase [15, 56–60]. This shows that the distributions of the effective fields on the slow time-scales coincidewith the ones of the thermodynamic FRSB solution in the hierarchical clusters. Moreover, also m ( h ) and q ( x ) havethe same expression than in the static case: m ( h ) is the magnetization on the time-scale t x , hence it is averaged overall effective fields corresponding to smallest time-scales and it depends on h ( x ) , which is the effective field at time t x . Whereas q ( x ) is the overlap on the time-scale t x , i.e it is the square of the magnetization m ( h ( x )) average over h ( x ) . In order to solve equation 74, one has to know m ( h ) and q ( x ) which are determined self-consistently from thesolution of the stochastic equation itself. Given the one-to-one mapping with the static case, we refer to the classicoriginal paper on FRSB for more details [15, 56, 59, 60].In summary, our procedure shows (as it was expected from the solution of simplifed models [4, 61]) that agingdynamics in the FRSB case is strongly related to the static solution. Indeed, we have provided a purely dynamicalderivation of the stochastic equation (74) at the basis of FRSB. A. Relationship and analogies with Sompolinsky’s dynamical approach
The first who proposed a deep investigation of the spin glass phase using a dynamic approach was Sompolinsky inthe eighties [1, 13]. He proposed a way to obtain the properties of the spin-glass phase using an approach that takesinto account dynamics over diverging time-scales. Cutting a long story short, his main assumptions are: • there exists an infinite number of diverging timescales belowe T c in mean-field spin-glasses. • the spin-spin correlation function is affected by all those timescales, in particular for each time-scale t x , one gets q ( x ) = (cid:104) s i (0) s i ( t x ) (cid:105) . (75)7 • the equilibrium response function until the diverging timescale t x is given by: χ ( x ) = (cid:90) t x R ( t x , t (cid:48) ) dt (cid:48) = 1 T (∆( x ) + (1 − q (1))) (76)where the first term on the right hand side is by definition the contribution to the response from the divergingtime-scales, and the second term is the contribution due to the fast degrees of freedom ( q (1) is related to thespin-spin correlation, see Eq. (75)). • The anomalous part of the response and the correlation on diverging timescales are related by ˙∆( x ) = − x ˙ q ( x ) . (77)The physics behind Sompolinsky’s solution was never fully justified; nevertheless it was shown that these assumptionsallows to recover the Parisi solution of spin-glasses.In the following we show that the last assumption—the crucial one—is analogous to the generalized fluctuation-dissipation relation we used in the last section. In fact, within the dynamical aging ansatz the slow time-scales leadto an anomalous contribution to the aging response function for t, t (cid:48) (cid:29) : (cid:90) tt (cid:48) R ( t, t (cid:48)(cid:48) ) dt (cid:48)(cid:48) = 1 − q ( x max ) T + (cid:90) x max x xT ˙ q ( x (cid:48) ) dx (cid:48) (78)where we have used the same notation of the previous section, and t, t (cid:48) are taken in the time sector corresponding to x , i.e. < h x ( t (cid:48) ) h x ( t ) < ( x max is the value of x corresponding to the first plateau in the correlation function, and wehave traded the index i in h i ( t ) for the corresponding value of x ).Identifying the response in the LHS above with χ ( x ) , and taking the derivative with respect to x we discover thatthe Sompolinsky’s relation ˙∆( x ) = − x ˙ q ( x ) is mutatis mutandis the generalized fluctuation-dissipation relation.The interpretations of our and Sompolinsky approaches are clearly different: we study aging dynamics whereashe wasn’t discussing off-equilibrium. However, algebraically the two approaches are identical, and the assumption ˙∆( x ) = − x ˙ q ( x ) becomes under the lens of the off-equilibrium approach the generalized FDT discovered by Cugliandoloand Kurchan in their study of mean-field aging [2]. This result offers a new perspective on the Sompolinsky’s solution,and clarifies its algebraic equivalence with the Parisi’s solution of spin-glasses. VII. CONCLUSIONS AND PERSPECTIVES
Developing a mean-field procedure to study the dynamics of out-of-equilibrium systems has been a fundamentalstep in the theory of disordered and amorphous systems. It has allowed to address challenging questions aboutlow-temperature glassy behaviors, and to understand the role of complex landscapes in determining slow dynamics[11, 12, 23, 39].In this work, we have considered cases in which the slow dynamics is studied by Dynamical Mean-Field Theory(DMFT), and the resulting equations do not always simplify in integro-differential equations on response and cor-relation functions. Instead, one has to deal with the full-fledged self-consistent problem in which the thermal bathproperties are determined from the stochastic process induced by the bath itself. Our approach is based on the mean-field theory of aging dynamics [3, 11, 12, 39, 42, 46]. It relies on the hypothesis of well-defined timescale separation,between fast degrees of freedom, leading to a time translational invariant (TTI) regime, and slow degrees of freedom,leading to an aging regime. This separation of time scales feeds into the self-consistent stochastic process associatedto DMFT: it leads to generalized friction and noise that also have fast and slow contributions. Our main result isestablishing a procedure that allows to study this self-consistent dynamical problem and obtain the main quantitiesof interest, e.g. effective temperatures, correlations and responses on slow time-scales. The resulting equations makeexplicitly the link between aging dynamics and static replica computations, which was worked out in simplified models[2, 4, 15, 38, 43, 59, 62] and then assumed to valid more generally. They also display strong relationship with thequasi-equilibrium picture of glassy dynamics [63, 64].A natural extension of our work concerns the so-called Gardner phase [36], which appears in many mean-fieldmodels at low temperature, and it has been shown to have very important consequences in jamming physics [37]. Theassociated dynamical behavior is expected to be a combination of the RSB and Full RSB studied in this paper, soour results provide a good starting point to address it. Recent works have unveiled that the weak long-term memory • Dynamical theory of aging and shear of glasses in the limit of infinite dimensions . The DMFT equations totreat those cases have been established recently in [10, 16]. Our framework provides a way to analyze them andgeneralise the results obtained on these topics using simplified mean-field disordered systems [2, 65]. • Dynamical theory of slow dynamics in well mixed interacting ecosystems . Simple models of ecosystems with alarge number of species [8, 66, 67] have been shown to display aging and, in cases of symmetric interactions,Full RSB physics [68, 69]. Since DMFT naturally applies to them [8, 70], a generalization of our approach offersa promising way to develop a theory of such phenomena. • Inference and Machine Learning . Gradient descent algorithms are natural methods to deal with non-convexoptimization problems. Although they are widely used in practice, a theory of their algorithmic threshold islacking. Only very recently a first result has been obtained in a model of matrix-tensor PCA [17, 18]. The studyof the gradient descent dynamics in models such as the non-convex spherical perceptron [71], and generalizedlinear models [72] can be done by DMFT [9]. In consequence, our approach combined with the one of [17, 18]provides a guide to develop a theory of the algorithmic threshold of gradient descent in these contexts.
ACKNOWLEDGMENTS
We thanks L. Cugliandolo and J. Kurchan for helpful discussions. This work was supported by the Simons Foun-dation Grants No. (Giulio Biroli).
Appendix A: Alternative analysis of the spherical p-spin in terms of the virial equation1. 1st approach
We propose here a complementary approach to derive the expression of the Lagrange multiplier in the spherical p -spin model. Our starting point is the equation for the spin evolution, Eq. (16) of the main text, in which wemultiply both sides by s and average over the stochastic process. In this way, we can express the expectation valueof the constraining force term in an analytically treatable way. (cid:28) s dsdt (cid:29) = − (cid:28) s ∂V∂s (cid:29) + p ( p − (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) (cid:104) s ( t ) s ( t (cid:48) ) (cid:105) dt (cid:48) + (cid:104) ξ ( t ) s ( t ) (cid:105) (A1)Using a property of Gaussian integrals, we can simplify the term (cid:104) ξ ( t ) s ( t ) (cid:105) by means of Novikov theorem [73] (aka
Girsanov theorem in mathematical jargon). Keeping the discussion as general as possible, let us suppose to beinterested in computing the following value (cid:104) φ k F ( φ ) (cid:105) = 1 Z [0] (cid:90) dφ φ k F ( φ ) e − φAφ = − Z [0] (cid:88) n ∆ kn (cid:90) dφ F ( φ ) ∂∂φ n e − φAφ == 1 Z [0] (cid:88) n ∆ kn (cid:90) dφ ∂∂φ n F ( φ ) e − φAφ (A2)where ∆ = A − that in field theory corresponds also to the bare propagator, while the normalization factor is Z [0] = (cid:112) π/ det A . Using the aforementioned theorem, which remains valid as long as we consider Gaussian noises,the expectation value between the noise and the spin variable over the stochastic process can be rewritten in a morecompact way as (cid:104) ξ ( t ) s ( t (cid:48) ) (cid:105) = (cid:90) dt (cid:48)(cid:48) M ( t, t (cid:48)(cid:48) ) R ( t (cid:48) , t (cid:48)(cid:48) ) (A3)9where the kernel M ( t, t (cid:48) ) = 2 T δ ( t − t (cid:48) )+ p C p − ( t, t (cid:48) ) contains a non-interacting part satisfying the TTI hypothesis, andan interacting non-translational invariant contribution. Moreover, the first term on the LHS of Eq. (A1), accordingto Ito’s prescription, can be rewritten as ds dt = 2 s dsdt + 2 T (A4)which for a spherical model reduces to: s dsdt + T = 0 .Then, Eq. (A1) becomes: (cid:28) s dsdt (cid:29) = − (cid:28) s ∂V∂s (cid:29) + p ( p − (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) (cid:104) s ( t ) s ( t (cid:48) ) (cid:105) dt (cid:48) + (cid:104) ξ ( t ) s ( t ) (cid:105) ⇒− T = − λ + p ( p − (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) + (cid:90) t (cid:104) T δ ( t − t (cid:48)(cid:48) ) + p C p − ( t, t (cid:48)(cid:48) ) (cid:105) R ( t, t (cid:48)(cid:48) ) dt (cid:48)(cid:48) (A5)and, since R ( t, t ) is zero for causality, it implies − T = − λ + p (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) . (A6)This expression provides the well-known condition for the Lagrange multiplier in the case of a spherical p -spin model: λ = T + p (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) . (A7)If we integrate by part the argument in the integral we recover a compact expression for the spherical parameter: λ = T + p (cid:18) − q p T + q p T eff (cid:19) (A8)which is equivalent to Eq. (47) and can be related to the expression of the asymptotic energy E ∞ as well, as knownfrom Cugliandolo-Kurchan equations and corresponding to [2]: E ∞ = − T (cid:20) (1 − q p ) + pq p − (cid:90) dµ (cid:48)(cid:48) R ( µ (cid:48)(cid:48) ) C p − ( µ (cid:48)(cid:48) ) (cid:21) (A9)Note that the correlation and response depend now on the rescaled parameter µ ≡ t (cid:48) /t that implies C ( t, t (cid:48) ) = q C ( µ ) and tR ( t, t (cid:48) ) = R ( µ ) .
2. 2nd approach
We again consider the equation of motion and multiply both sides by s (cid:28) s dsdt (cid:29) = − (cid:28) s ∂V∂s (cid:29) + p ( p − (cid:90) t dt (cid:48) R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) (cid:104) s ( t ) s ( t (cid:48) ) (cid:105) + (cid:104) ξ ( t ) s ( t ) (cid:105) (A10)averaging then over the stochastic process. At this level, we do not need to specify the precise form of the potential.We proceed integrating by parts the first term in the RHS to keep the computation as general as possible. Thelast expectation value can be treated exactly as before using Novikov’s theorem . This term basically sums up to thefriction term on the RHS leading to − T = − Z (cid:40)(cid:90) ds (cid:20) − T s (cid:18) ∂∂s e − V ( s ) /T (cid:19) e p T (1 − q p − ) s + hsT (cid:21)(cid:41) + p (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) (A11) − T = − (cid:104) T + p T (1 − q p − ) (cid:104) s (cid:105) h + h (cid:104) s (cid:105) h (cid:105) + p T [1 − q p (1 − x )] ⇒ − p T (1 − q p − ) (cid:104) s (cid:105) h − h (cid:104) s (cid:105) h + p T [1 − q p (1 − x )] . (A12)0In the spherical model, the field distribution P ( h ) can be exactly computed and becomes P ( h ) = 1 Z exp − h (cid:16) p q p − (cid:17) − βxF ( h ) = 1 Z exp (cid:40) − h (cid:16) p q p − (cid:17) + βxh (cid:104) λ − p T (1 − q p − ) (cid:105) (cid:41) , (A13)where T eff = 1 / ( βx ) , while the normalization factor Z is Z = √ π (cid:113) q − p p + ( − q ) xβT . (A14)Using the additional condition on the spherical constraint, we can further simplify the denominator of the normaliza-tion factor and obtain a more compact expression for λ , as also reported in the main text: λ − p T (1 − q p − ) = T − q . (A15)Hence, the short and long-time limit of the correlation function, which correspond respectively to q d and q in a static RSB computation, are: (cid:104) s (cid:105) = 1 Z (cid:90) dh P ( h ) (cid:34) TT / (1 − q ) + h ( T / (1 − q )) (cid:35) = ( − q ) (cid:2) − q T − pq p ( − q )( − x ) (cid:3) T [2 q T + p ( − q ) q p xβ ] (A16)and (cid:104) s (cid:105) h = h (cid:2) λ − p T (1 − q p − ) (cid:3) = 1 Z (cid:90) dh P ( h ) h (1 − q ) T = − pq p ( − q )2 q T + pq p ( − q ) xβ . (A17)Then we use the spherical normalization condition, i.e. (cid:104) s (cid:105) = 1 , which in terms of Eq. (A16) leads to an additionalcondition for the breaking parameter x : x ∗ = q − − p (cid:104) pq p − pq p + pq p − q T (cid:105) p ( − q ) . (A18)Coming back to Eq. (A12) and inserting the obtained expression for x , we eventually get: − p T (1 − q p − ) (cid:104) s (cid:105) − h (cid:104) s (cid:105) + p T [1 − q p (1 − x )] ⇒ − p T (1 − q p − ) + pq p ( − q )2 q T + pq p xβ ( − q ) (cid:12)(cid:12)(cid:12)(cid:12) x ∗ + p (cid:18) − q p T + q p T eff (cid:19) (A19) p T (1 − q p − ) = − q T − q + p (cid:18) − q p T + q p T eff (cid:19) . (A20)To conclude this part of the computation we can resort to Eq. (A15) and re-express everything in terms of λ as: λ = T + p (cid:18) − q p T + q p T eff (cid:19) . (A21)
3. 3rd approach
We present now a third, alternative way based on integrating by parts the average spin values, which can be even-tually re-expressed in terms of single free-energy differentiation contributions. Taking advantage of the simplificationsperformed up to Eq. (A12), we can directly use the resulting equation − T = − (cid:104) T + p T (1 − q p − ) (cid:104) s (cid:105) + h (cid:104) s (cid:105) (cid:105) + p T [1 − q p (1 − x )] ⇒ − p T (1 − q p − ) (cid:104) s (cid:105) − h (cid:104) s (cid:105) + p T [1 − q p (1 − x )] . (A22)1Given a generic function F ( h ) , we can write the following expectation value as h (cid:104) s (cid:105) h = − (cid:90) dh P ( h ) ∂F∂h h = − (cid:90) dh e − h ( p qp − ) − βxF ( h ) ∂F∂h h == − (cid:16) p q p − (cid:17) (cid:90) dh e − h ( p qp − ) (cid:18) e − βxF ( h ) ∂F∂h (cid:19) (cid:48) == − (cid:16) p q p − (cid:17) (cid:90) dh e − h ( p qp − ) (cid:34) e − βxF ( h ) ( − βx ) (cid:18) ∂F∂h (cid:19) + e − βxF ( h ) ∂ F∂h (cid:35) . (A23)In this way, the he equation of motion (A22) becomes p T (1 − q p − ) = p q p − (cid:90) dh e − h ( p qp − ) e − βxF ( h ) ( − βx ) (cid:18) ∂F∂h (cid:19) + p q p − (cid:90) dh e − h ( p qp − ) e − βxF ( h ) (cid:18) ∂ F∂h (cid:19) ++ p T [1 − q p (1 − x )] (A24)where we have always used (cid:104) s (cid:105) = 1 . Specialising the analysis to the spherical model with the following free-energy F ( h ) = − T ln (cid:90) ds e − T [ ( λ − p T (1 − q p − ) ) s − hs ] = − T ln (cid:34) N e h T [ λ − p T (1 − qp −
11 ) ] (cid:35) (A25)whose normalization factor is N = √ π (cid:114) T (cid:104) λ − p T (1 − q p − ) (cid:105) , (A26)we have then proposed another way to derive the expression of the spherical parameter λ . Appendix B: Ising spin model: failure of the previously proposed approach
For a discrete model we can in principle apply the same procedure starting from the usual equation of motion (cid:28) s dsdt (cid:29) = − (cid:28) s ∂V∂s (cid:29) + p ( p − (cid:90) t dt (cid:48) R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) (cid:104) s ( t ) s ( t (cid:48) ) (cid:105) + (cid:104) ξ ( t ) s ( t ) (cid:105) , (B1)from which, by integrating by parts the first term on the RHS, we eventually obtain: − T = − Z (cid:40)(cid:90) ds (cid:20) s ( − T ) (cid:18) ∂∂s e − V ( s ) /T (cid:19) e p T (1 − q p − ) s + hsT (cid:21)(cid:41) + p (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) . (B2)At this level, we only need to determine the first two expectation values of the spins according to a RSB Ansatz: − p T (1 − q p − ) (cid:104) s (cid:105) − h (cid:104) s (cid:105) + p (cid:90) t xT ∂C∂t (cid:48) ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) . (B3)In the Ising spin case the conditional probability distribution P ( s | h ) reads P ( s | h ) = e hsT h/T ) , (B4)which is crucial to determine the only non-trivial expectation value (cid:104) s (cid:105) = q ≡ (cid:18) e h/T − e − h/T h/T ) (cid:19) = tanh ( h/T ) (B5)2as the other one is automatically known, (cid:104) s (cid:105) = 1 . The field distribution in the Ising case is also known, definedthroughout the parameter x ≤ : P ( h ) = 1 Z e − h ( p qp − ) (2 cosh( h/T )) x , (B6)which allows us to rewrite the equation of motion (B2) in the following form − p T (1 − q p − ) − (cid:90) dhZ e − h ( p qp − ) (2 cosh( h/T )) x tanh( h/T ) h + p (cid:90) xT ∂C∂t (cid:48) ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) . (B7)By integrating the second term on the RHS by parts and distinguishing the equilibrium and the off-equilibriumcontributions of the response function, we end up with − p T (1 − q p − ) − p q p − (cid:40)(cid:90) dhZ e − h ( p qp − ) (2 cosh( h/T )) x (cid:0) − tanh ( h/T ) (cid:1) T ++ (cid:90) dhZ e − h ( p qp − ) (2 cosh( h/T )) x (cid:16) xT (cid:17) tanh ( h/T ) (cid:41) + p T [1 − q p (1 − x )] (B8)in other words − p T (1 − q p − ) − p q p − (cid:16) − (1 − x )tanh ( h/T ) (cid:17) + p T [1 − q p (1 − x )] (B9)which however leads to a trivial condition. The case p = 2 . - . We can consider two simple limiting case, for x = 0 and x = 1 . By setting x = 0 and p = 2 inthe above equation, we simply recover an identity relationship for the overlap q , that is: T (cid:40) − (1 − q ) − q Z (cid:90) dh e − h / (2 q ) (cid:2) − tanh ( h/T ) (cid:3) + (1 − q ) (cid:41) (B10)where the normalization factor Z = √ πq . Therefore, we immediately find q = (cid:82) dhP ( h, x = 0) tanh ( h/T ) .Furthermore, if we consider the other straightforward limit for x = 1 – which essentially corresponds to an expansionaround the plateau, as performed in [74] – we find an automatically satisfied relation for the field distribution P ( h ) ,namely (cid:82) dh P ( h ) = 1 .
1. Double-well potential: perturbative expansion in the limit of a infinitely narrow double well
To better investigate the peculiarities of the different models, we have also considered a double-well potential V ( s ) = α ( s − (B11)where α is a tunable parameter that modulates the roughness of the given potential. In the limit of large α , we cansafely consider the saddle-point approximation and rewrite the potential as a function of the two different symmetriccontributions, i.e. P ( s, h ) = P (+1 , h ) + P ( − , h ) . We thus perform a harmonic expansion around each minimumobtaining to the leading order ˜ V ( s ) = 4 α ( s − + o (cid:0) ( s − (cid:1) (B12)and similarly for the other minimum, each being of O (1 /α ) . By neglecting the contribution of higher-order terms,the two normalization factors accounting respectively for the expansion around s = 1 and s = − turn out to berespectively: Z ∝ (cid:90) ∞−∞ du e [ − αT + p T ( q p − d − q p − ) ] u + [ p T ( q p − d − q p − )+ hT ] u + hT + p T ( q p − d − q p − ) (B13)3and Z − ∝ (cid:90) ∞−∞ du e [ − αT + p T ( q p − d − q p − ) ] u + [ − p T ( q p − d − q p − )+ hT ] u − hT + p T ( q p − d − q p − ) , (B14)where u has been introduced to denote the change of variable, i.e. u = s − in the first Z -contribution and u = s + 1 in the second one.The boundary term p T ( q p − d − q p − ) and the external field h contribute only to tilting the potential, hence favouringthe positive (or negative) minimum depending on the relative decrease of the free energy. Therefore, the resultingexpression of the free energy can be written as a sum of the two harmonic contributions: F ( h ) = − T ln (cid:40)(cid:90) ds e − T [ V ( s + ) − p T ( q p − d − q p − ) s − hs + ] + (cid:90) ds e − T [ V ( s − ) − p T ( q p − d − q p − ) s − − hs − ] (cid:41) = − T ln e h q qdT +4 αpqp qd − αpq qpd − hq qdTα − pqp qdT + pq qpdT +16 q qdT α (cid:113) − pq p q d + pq q pd +16 q q d T αq q d T + e h q qdT +4 αp ( qp qd − q qpd ) +16 hq qdTα − pqp qdT + pq qpdT +16 q qdT α (cid:113) − pq p q d + pq q pd +16 q q d T αq q d T (B15)from which, focusing on the h -dependent terms and neglecting irrelevant prefactors in the logarithm, we recover inthe limit α → ∞ the well-known relationship for the Ising model, i.e. F ( h ) = − T ln (2 cosh( h/T )) .To enter into the details of the computation, we consider as usual the effective equation of motion (cid:28) s dsdt (cid:29) = − (cid:28) s ∂V∂s (cid:29) + p ( p − (cid:90) t dt (cid:48) R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) (cid:104) s ( t ) s ( t (cid:48) ) (cid:105) + (cid:104) ξ ( t ) s ( t ) (cid:105) (B16)which, as long discussed before, can be simplified by integrating by parts. It eventually leads to − p T (cid:16) q p − d − q p − (cid:17) (cid:104) s (cid:105) − h (cid:104) s (cid:105) + p T [ q pd − q p (1 − x )] . (B17)which requires the computation of the following expectation value h (cid:104) s (cid:105) h = − (cid:90) dh P ( h ) ∂F∂h h = − (cid:90) dh e − h ( p qp − ) − βxF ( h ) ∂F∂h h == − (cid:16) p q p − (cid:17) (cid:90) dh e − h ( p qp − ) e − βxF ( h ) (cid:34) ( − βx ) (cid:18) ∂F∂h (cid:19) + ∂ F∂h (cid:35) . (B18)and therefore of the first two derivatives of F ( h ) w.r.t h to be conveniently expanded in powers of /α . According toEq. (B18), the two contributions respectively imply: (cid:18) ∂F∂h (cid:19) = (cid:40) − q q d T (cid:104) h + 8 α tanh (cid:16) hq q d α − pq p q d + pq q pd +16 q q d T α (cid:17)(cid:105) ( − pq p q d + pq q pd + 16 q q d T α ) (cid:41) , (B19) ∂ F∂h = − q q d T (cid:104) − pq p q d + pq q pd + 16 q q d T α + 128 q q d α sech (cid:16) hq q d α − pq p q d + pq q pd +16 q q d T α (cid:17)(cid:105) ( − pq p q d + pq q pd + 16 q q d T α ) , (B20)that, once they are expanded to the leading order in /α , yield: (cid:18) ∂F∂h (cid:19) (cid:39) tanh ( h/T )+ 18 q q d α (cid:20) tanh( h/T ) (cid:18) hq q d − p ( − q p q d + q q pd ) 1 T ( h/T sech ( h/T ) + tanh( h/T ) (cid:19)(cid:21) + O (cid:18) α (cid:19) (B21) ∂ F∂h (cid:39) − T sech ( h/T ) + 18 q q d α (cid:20) − p ( − q p q d + q q pd ) 1 T sech ( h/T ) ( − h/T tanh( h/T )) (cid:21) + O (cid:18) α (cid:19) (B22)4 ♠ In the simplest case, which corresponds to setting the diagonal value q d = 1 and the breaking parameter x = 1 ,the equation of motion reduces to p T − p T q p − = − p q p − (cid:90) dhP ( h ) xT (cid:18) ∂F∂h (cid:19) + p q p − (cid:90) dhP ( h ) ∂ F∂h + p T , − (cid:90) dhP ( h ) (cid:18) ∂F∂h (cid:19) + (cid:90) dhP ( h ) ∂ F∂h (B23)and focusing only on the leading order terms, we would get: − (cid:90) dhP ( h ) tanh ( h/T ) − (cid:90) dhP ( h ) sech ( h/T ) . (B24)Again, this equation results into an identity condition for the probability distribution P ( h ) . Going further in theexpansion of Eqs. (B19)-(B20) and including also higher-order terms in the computation − p T q p − (cid:90) dhP ( h ) (cid:20) α tanh( h/T )2 h + p ( q p − q )8 q T α h tanh( h/T ) sech ( h/T ) + p ( q p − q )8 q T α tanh ( h/T ) (cid:21) + − p T q p − (cid:90) dhP ( h ) (cid:20) p ( q p − q )8 q T α (cid:18) sech ( h/T ) − T h tanh( h/T ) sech ( h/T ) (cid:19)(cid:21) + O (cid:18) α (cid:19) (B25)we nevertheless notice that the last term – that might be possibly simplified by integration by parts – cancels outwith the same term of opposite sign in the first line.The problem is thus solved in the case of a spherical p -spin, but not in more general cases. Even for the correspondingsoft-spin version of an Ising model, based on the introduction of a tunable parameter α modulating the roughness ofthe double-well potential, the equation above appears to be needless and to provide only very basic information.From this analysis, we conclude that the virial equation (B16) is nothing more than an equation for the correlation C ( t, t (cid:48) ) at equal times that, in the case of an Ising model, is automatically satisfied whereas in the spherical modelleads to an additional condition useful to fix the spherical parameter. To determine the effective temperature andclose the system of equations, we need to define an additional condition to be mapped on the analog of an equationfor the response function.5 Appendix C: Connection between effective temperature and breaking parameter of the static solution
In the case of the spherical p -spin model, we have all the tools to show the underlying mapping between theeffective temperature T eff and the breaking point x of the static solution. The stationary field distribution has asimple quadratic dependence on the effective field and can be easily manipulated to get all other missing information. P ( h ) = exp (cid:40) − h T eff pq p − T eff − (cid:16) λ − p T (1 − q p − ) (cid:17) (cid:41) == exp (cid:40) − h T eff λ − p T (1 − q p − ) − p T eff q p − p T eff q p − (cid:16) λ − p T (1 − q p − ) (cid:17) (cid:41) (C1)In Secs. (IV B) and (IV D) we have shown that (cid:104) s (cid:105) = (cid:82) ∞−∞ P ( s | h ) s (cid:82) ∞−∞ P ( s | h ) = 1 λ − p T (1 − q p − ) (cid:20) p ( p − (cid:90) t R A ( t, t (cid:48) ) C A ( t, t (cid:48) ) p − s ( t (cid:48) ) dt (cid:48) + ξ A (cid:21) == hλ − p T (1 − q p − ) . (C2)to be eventually averaged over the effective field distribution P ( h ) . Using the information on the average spin value,we can rewrite the equation over the off-diagonal value of the overlap matrix, q , as q = p q p − (cid:104) λ − p T (1 − q p − ) (cid:105)(cid:104) λ − p T (1 − q p − ) (cid:105) (cid:104) λ − p T (1 − q p − ) − p T eff q p − (cid:105) (C3)and by simple algebraic manipulations get the following expression: (cid:104) λ − p T (1 − q p − ) (cid:105) (cid:20) λ − p T (1 − q p − ) − p T eff q p − (cid:21) = p q p − . (C4)The first parenthesis can be rewritten in a more straightforward way by using the condition for the Lagrange multiplier,as derived in Eq. (A15), i.e. λ − p T (1 − q p − ) = T / (1 − q ) . The above equation becomes then T (1 − q ) − T (1 − q ) p T eff q p − = p q p − → − (1 − q ) T p T eff q p − = p T q p − (1 − q ) . (C5)To extract a resulting equation for the effective temperature we can recall the condition obtained in the main text interms marginal stability, which has been imposed on the TTI dynamics. Then, using Eq. (63) and simply equatingthe RHS to / ( p − , we obtain: p − − (1 − q ) T p ( p − T eff q p − = 0 . (C6)We have thus recovered the relationship between the effective temperature and the breaking parameter x within the RSB approximation in the replica formalism for the spherical p -spin model: x ≡ TT eff = ( p − − q ) q . (C7)The resulting value of the breaking parameter x corresponds to those TAP states which are marginally stable, the so-called threshold states . The critical slowing down of the dynamics and related aging phenomena are then consequencesof the flatness of the free energy around these states.6 Appendix D: Connection with previous formalisms and identification of the anomaly
To prove the extreme generality of our approach we have also considered the problem of particle in a random manifoldthat has been extensively studied in the past [4, 5] and from which, under suitable assumptions, the usual equationsfor the spherical p -spin model can be recovered. The mean-field dynamical equations the two-time correlation andresponse functions can be expressed in the following form: ∂C ( t, t (cid:48) ) ∂t = − λC ( t, t (cid:48) ) + p (cid:90) t (cid:48) ds C p − ( t, s ) R ( t (cid:48) , s ) − p ( p − (cid:90) t ds C p − ( t, s ) R ( t, s ) [ C ( t, t (cid:48) ) − C ( s, t (cid:48) )] + 2 T R ( t (cid:48) , t ) , (D1) ∂R ( t, t (cid:48) ) ∂t = − λR ( t, t (cid:48) ) − p ( p − (cid:90) t ds C p − ( t, s ) R ( t, s ) [ R ( t, t (cid:48) ) − R ( s, t (cid:48) )] (D2)where the function must satisfy the following prescriptions according to the causality property and the Ito integrationscheme: R ( t, t ) = 0 , lim (cid:15) → R ( t, t − (cid:15) ) = 1 , R ( t (cid:48) , t ) = 0 if t > t (cid:48) (D3)Therefore, the last term in the equation for the correlation function vanishes and, as t (cid:48) → t , Eq. (D1) becomes d C ( t, t ) d t = − λC ( t, t ) + T + p (cid:90) t ds C p − ( t, s ) R ( t, s ) + p ( p − (cid:90) t ds C p − ( t, s ) R ( t, s ) [ C ( t, t ) − C ( s, s ) + B ( t, s )] (D4)or, equivalently, in terms of the mean-squared displacement B ( t, t (cid:48) ) , which is defined as B ( t, t (cid:48) ) ≡ C ( t, t ) + C ( t (cid:48) , t (cid:48) ) − C ( t, t (cid:48) ) = (cid:104) [ s ( t ) − s ( t (cid:48) )] (cid:105) . (D5)If the correlation is set to q , the above equation for the total correlation evaluated for t ≈ t (cid:48) reduces to λ ( t ) q = T + (cid:90) t ds (cid:20) p C p − ( t, s ) + p ( p − C p − ( t, s ) B ( t, s ) (cid:21) R ( t, s ) . (D6)The Lagrange multiplier has an explicit dependence on time and has to be properly fixed accordingly to the conditionon the spherical constraint. If we impose q = 1 and simplify the product of the different combinations of correlations,we recover exactly the same expression as in (A7), which has been obtained before in Eq. (A21) by using a virialexpansion, namely: λ = T + p (cid:90) t R ( t, t (cid:48) ) C p − ( t, t (cid:48) ) dt (cid:48) (D7)
1. Additional equation on the response function and derivation of the marginal stability condition
We want now to consider the second equation, for the response function, and thus derive the analogue of the anomaly , which accounts for those times that are not included in the asymptotic regime but for which aging effectsare nevertheless relevant. The anomaly essentially couples the asymptotic time regime, for which t − t (cid:48) is finite,with the non-asymptotic dynamical contribution. The anomaly is zero in the high-temperature phase and takes anon-vanishing contribution in the aging regime, associated with a finite value of the overlap parameter [4].The equation for the response function analysed in the aging regime implies (cid:104) − λ ∞ + p T (cid:16) − q p − (cid:17)(cid:105) R A ( t, t (cid:48) )+ p ( p − R A ( t, t (cid:48) ) C p − A ( t, t (cid:48) ) (1 − q ) T + p ( p − (cid:90) tt (cid:48) R A ( s, t (cid:48)(cid:48) ) C p − A ( s, t (cid:48)(cid:48) ) R A ( t (cid:48)(cid:48) , t (cid:48) ) ds (D8)which, for t (cid:48) ≈ t , becomes: R A ( t, t (cid:48) ) (cid:20) − λ ∞ + p T (1 − q p − ) + p ( p − C p − A ( t, t (cid:48) ) 1 − q T (cid:21) . 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