Time reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Time reparametrization invariance in arbitraryrange p-spin models: symmetric versusnon-symmetric dynamics
Gcina A. Mavimbela and Horacio E. Castillo
Department of Physics and Astronomy, Ohio University, Athens, Ohio, USA, 45701E-mail: [email protected], [email protected]
Abstract.
We explore the existence of time reparametrization symmetry in p-spinmodels. Using the Martin-Siggia-Rose generating functional, we analytically probethe long-time dynamics. We perform a renormalization group analysis where wesystematically integrate over short timescale fluctuations. We find three families ofstable fixed points and study the symmetry of those fixed points with respect totime reparametrizations. One of those families is composed entirely of symmetricfixed points, which are associated with the low temperature dynamics. The othertwo families are composed entirely of non-symmetric fixed points. One of these twonon-symmetric families corresponds to the high temperature dynamics.Time reparametrization symmetry is a continuous symmetry that is spontaneouslybroken in the glass state and we argue that this gives rise to the presence of Goldstonemodes. We expect the Goldstone modes to determine the properties of fluctuations inthe glass state, in particular predicting the presence of dynamical heterogeneity.PACS numbers: 64.70.Q-, 61.20.Lc, 61.43.Fs ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics
1. Introduction
Very slow dynamics is an essential feature of glasses [1]. In both structural glassesand spin glasses slow dynamics is manifested through a dramatic increase in relaxationtimes. This slowdown has been captured in mean field theories, such as the modecoupling theory for supercooled liquids [2] and the dynamical theory for mean field spinglass models [3, 4, 5, 6]. Even though mean field theories are useful in describing someaspects of glassy dynamics, they do not completely capture phenomena associated withfluctuations. Fluctuations have been shown, particularly with the discovery of dynamicalheterogeneities , to be central to an understanding of glassy dynamics [7].Dynamical heterogeneities - mesoscopic regions that evolve differently from eachother as well as from the bulk - have been found in experimental studies of materialsclose to the glass transition [8, 9, 10] and in simulations of both spin glasses andstructural glasses [11, 12, 13]. Their presence has been directly observed at themicroscopic level in experiments on colloidal glasses [10] and granular systems [14].Understanding the onset of heterogeneities without an apparent structural trigger isbelieved to be key to an understanding of the glass transition [7]. There have beenseveral theoretical attempts to explain the emergence of heterogeneous dynamics asthe glass transition is approached. One of them is a geometrical picture, accordingto which dynamical heterogeneities result from non-trivial structure in the space oftrajectories due to dynamical constraints [15]. Another proposed explanation is providedby the Random First Order transition (RFOT) approach, in which a liquid freezesinto a mosaic of aperiodic crystals [16]. Here we will explore a different theoreticalavenue to explain dynamical heterogeneities, which is based on time reparametrizationsymmetry [17, 18, 19].Time reparametrization symmetry (TRS), the invariance under transformationsof the time variable t → h ( t ), was discovered some years ago in the mean-field non-equilibrium dynamics of the Sherrington-Kirkpatrick model and the p-spin model [5, 6].The symmetry, which was shown to be present in the long-time limit of the mean fieldevolution, implies that the asymptotic equations do not have a unique solution [5, 6, 20].In more recent studies, TRS has been proved to be present in the long timedynamics of the glass state in a short range spin glass model, the Edwards-Andersonmodel [17, 18, 19]. In this last case, the proof of the symmetry is at the level of thegenerating functional, including all fluctuations. Using the renormalization group (RG),it was shown that the stable fixed point of the generating functional corresponding toglassy dynamics is invariant under reparametrizations of the time variable. Howevernot all models of interacting spins under Langevin dynamics show this behavior. Forexample, in a study of the O(N) model it was shown that the symmetry is not present,even for the long time limit of the low temperature dynamics [25]. The explanation fordynamical heterogeneities from TRS is derived from the fact that TRS is spontaneouslybroken by the correlations and responses in the glass state. A spontaneously brokencontinuous symmetry is expected to give rise to Goldstone modes, and these modes are ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics τ . Wesystematically increase the short time cutoff by integrating over the two-time fieldsassociated with the shortest time differences, thus following a procedure analogous toWilson’s approach to the RG. In our case, however, we integrate over fluctuationsthat are fast in time , not in space. We find three families of stable fixed points.The first family corresponds to fixed point actions containing the coupling to thethermal bath but not the spin-spin interactions. The fixed points in this family arenot time reparametrization invariant, and we believe that this family is associated withthe high temperature dynamics. A second family of stable fixed points that are nottime reparametrization invariant corresponds to fixed point actions containing both thecoupling to the thermal bath and the spin-spin interactions. For the third family, thespin-spin interaction term is marginal but the coupling to the thermal bath is irrelevant.The fixed points in this last family are time reparametrization invariant, and we believethat they represent the low temperature glassy dynamics of the model. After obtainingthese results, we discuss their connection with dynamical heterogeneity in the p-spinmodel, and we speculate on how a similar procedure may be applied to models ofstructural glasses, which have been shown to be connected to the p-spin model [21, 22].The rest of the paper is organized as follows: in Sec. 2 we start by givinga description of the model and an illustration of how we derive the Martin-Siggia-Rose generating functional; in Sec. 3 we show how we use Wilson’s approach to therenormalization group to get stable fixed points; in Sec. 4 we study the stable fixed pointgenerating functionals and determine which ones are invariant under reparametrizationsof the time variable; and in Sec. 5 we end with a discussion of our results and conclusions. ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics
2. Model and MSR generating functional
The p-spin Hamiltonian is given by H = − p ! X i ...i p J i ...i p φ i ...φ i p , (1)where the { φ } i =1 , ··· ,N are soft spins subject to the spherical constraint P Ni =1 [ φ i ( t )] = N ,and the couplings are assumed to be uncorrelated, Gaussian distributed, zero meanrandom variables, P { J } = Q i <...
3. Renormalization group analysis
We perform a renormalization group analysis on the time variables. For simplicity wetake t = 0 and t f = ∞ from now on. We focus on the two-time fields. First, weintroduce a cutoff in the integration of two-time fields, τ ≤ | t − t | . We then write theterms of the action affected by the cutoff: S [ Q, ˆ Q ] = i X i Z ≤ t ,t < ∞ τ ≤| t − t | dt dt X α i ,α ′ i ˆ Q α i ,α ′ i i ( t , t ) × (cid:18) Q α i ,α ′ i i ( t , t ) − φ α i i ( t ) φ α ′ i i ( t ) (cid:19) , (18) S J [ Q ] = − p p ! X i ...i p K i ...i p Z ≤ t ,t < ∞ τ ≤| t − t | dt dt C =1 ,C ′ =1 X α ir ,α ′ ir ∈{ , } p Y r =1 Q α ir ,α ′ ir i r ( t , t ) . (19)We define fast and slow fields respectively by Q α i ,α ′ i >i ( t , t ) = Q α i ,α ′ i i ( t , t ), for τ ≤| t − t | < bτ and Q α i ,α ′ i
1. Thisseparation of fast and slow parts of the fields results in a separation in the terms: S [ Q, ˆ Q, φ , φ ] = S [ Q > , ˆ Q > , φ , φ ] + S [ Q < , ˆ Q < , φ , φ ] , (20) S J [ Q ] = S J [ Q > ] + S J [ Q < ] . (21)Next we calculate the integral I > over fast fields. To do this we use the fact that thereare no cross-terms between fast and slow fields in the integral: I > = Z DQ > D ˆ Q > exp i X i Z τ ≤| t − t | i ( t , t ) × (cid:18) Q α i ,α ′ i >i ( t , t ) − φ α i i ( t ) φ α ′ i i ( t ) (cid:19) − p p ! X i ...i p K i ...i p Z τ ≤| t − t | i r ( t , t ) . (22) ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics Q > fields constitutes undoing the delta function integraltransformation we used to introduce the two-time fields for the fast modes. Hence, I > = exp − p p ! X i ...i p K i ...i p Z τ ≤| t − t |
0. The case of λ = 0 corresponds to freezing and the strict inequality corresponds to a decayingcorrelation. The terms in the action that are of interest for our analysis are the threeterms contained in S spin : the spin-spin interaction, the term containing a time derivativeand the term coupling the system to the thermal bath. As indicated in Eqs. (33), (38)and (39), those three terms have the scaling exponents λ J = 2(1 + λ + ( p − λ ), λ vel = λ + λ and λ T = 1 + 2 λ , respectively. By considering the cases in which onlyone of the terms is marginal we get the results summarized in Fig. 1.In the case in which the coupling to the thermal bath is marginal, we have a line λ = − / λ , λ ) plane. Considering the constraint λ ≤ λ J and λ vel we find that there is an interval on this line, λ < − p − , in which both the spin-spin interactions and the time derivative term are irrelevant. Since the coupling to the ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics λ λ Key
Scaling Exponents in the p−spin model for p=3 vel λ =0λ =0 T λ =0 J Figure 1.
The figure shows the different lines along which each one of the three termsin S spin is marginal for p = 3. The red line corresponds to a marginal coupling tothe thermal bath ( λ T = 0), the black line corresponds to a marginal time-derivativeterm ( λ vel = 0), and the blue line corresponds to a marginal spin-spin interaction term( λ J = 0). thermal bath is marginal then we have a family of stable high temperature fixed points.Second, we consider the case where the time derivative term is marginal, correspondingto the line λ = − λ in the ( λ , λ ) plane. Since we have λ ≤
0, the exponent λ T ofEq. (39) is always positive, i.e. the coupling to the thermal bath is always a relevantperturbation. Thus the fixed points that contain only the time derivative term arealways unstable. We then consider the case where the spin-spin term is marginal. Thishappens on the line described by λ = − − ( p − λ . In the interval − p − < λ ≤ λ = − p − , λ = − /
2, for which both the coupling to the thermal bath and the spin-spin interaction are marginal, but the time derivative term is irrelevant, thus allowingfor an additional family of stable fixed points.The above analysis shows that there is a subset of the ( λ , λ ) plane for whicha high temperature dynamical fixed point family is present. The effective generating ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics Z fp [ l, h ; T ] = h Z [ { l i } , { h i } ] i fp = Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N × exp i X i Z Z ∞ dt dt X α i ,α ′ i ˆ Q α i ,α ′ i i ( t , t ) (cid:18) Q α i ,α ′ i i ( t , t ) − φ α i i ( t ) φ α ′ i i ( t ) (cid:19) − T N X i =1 Z ∞ dt (cid:16) φ i ( t ) (cid:17) + Z ∞ dt [ l i ( t ) φ i ( t ) + ih i ( t ) φ i ( t )]+ i N X i =1 ˆ ϕ i [ φ i ( t ) − ϕ i ] + i Z ∞ dt ˆ N ( t ) " N X i =1 ( φ i ( t )) − N . (42)There is another subset of the ( λ , λ ) plane for which a low temperature interaction-dominated fixed point family is present. The effective generating functional for thisfamily of fixed points is Z fp [ l, h ; J ] = h Z [ { l i } , { h i } ] i fp = Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N × exp i X i Z Z ∞ dt dt X α i ,α ′ i ˆ Q α i ,α ′ i i ( t , t ) (cid:18) Q α i ,α ′ i i ( t , t ) − φ α i i ( t ) φ α ′ i i ( t ) (cid:19) − p p ! X i ...i p K i ...i p Z Z ∞ dt dt C =1 ,C ′ =1 X α ir ,α ′ ir ∈{ , } p Y r =1 Q α ir ,α ′ ir i r ( t , t )+ Z ∞ dt [ l i ( t ) φ i ( t ) + ih i ( t ) φ i ( t )]+ i N X i =1 ˆ ϕ i [ φ i ( t ) − ϕ i ] + i Z ∞ dt ˆ N ( t ) " N X i =1 ( φ i ( t )) − N . (43)We note that the segment representing stable low temperature fixed points in the ( λ , λ ) plane includes the point λ = 0 and λ = −
1. This is the only point in the segmentthat represents freezing of the correlation, a property of glasses.
4. Time reparametrization symmetry
We now evaluate the effect of a reparametrization t → s ( t ) of the time variable on thestable fixed point generating functionals. For this purpose we consider a monotonouslyincreasing function with the boundary conditions s (0) = 0 and s ( ∞ ) = ∞ , whichinduces the following transformations on the sources,˜ l i ( t ) = ∂s∂t l i ( s ( t )) , (44)˜ h i ( t ) = h i ( s ( t )) . (45) ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics Z fp [˜ l, ˜ h ; T ] = Z D ˜ QD ˜ˆ QDψ Dψ D ˜ˆ ϕD ˜ N × exp i X i Z Z ∞ dt dt X α i ,α ′ i ˜ˆ Q α i ,α ′ i i ( t , t ) (cid:18) ˜ Q α i ,α ′ i i ( t , t ) − ψ α i i ( t ) ψ α ′ i i ( t ) (cid:19) − T N X i =1 Z ∞ dt (cid:16) ψ i ( t ) (cid:17) + Z ∞ dt [˜ l i ( t ) ψ i ( t ) + i ˜ h i ( t ) ψ i ( t )]+ i N X i =1 ˜ˆ ϕ i [ ψ i ( t ) − ˜ ϕ i ] + i Z ∞ dt ˜ N ( t ) " N X i =1 ( ψ i ( t )) − N . (46)Here we have used new dummy variables ψ α , ˜ˆ ϕ , ˜ˆ Q , ˜ Q and ˜ N , instead of φ α , ˆ ϕ , ˆ Q , Q and ˆ N , respectively, in the functional integral. We now perform the following change ofvariables ψ αi ( t ) = ∂s∂t ! α φ αi ( s ( t )) , (47)˜ Q α,α ′ i ( t, t ′ ) = ∂s∂t ! α ∂s∂t ′ ! α ′ Q α,α ′ i ( s ( t ) , s ( t ′ )) , (48)˜ˆ Q α,α ′ i ( t, t ′ ) = ∂s∂t ! α ∂s∂t ′ ! α ′ ˆ Q α,α ′ i ( s ( t ) , s ( t ′ )) , (49)˜ N ( t ) = ∂s∂t ˆ N ( s ( t )) , (50)˜ˆ ϕ = ˆ ϕ. (51)The change of variables results in Jacobians in the differentials, D ˜ QD ˆ˜ Q = DQD ˆ Q J D ˜ QDQ D ˆ˜ QD ˆ Q , (52) Dψ Dψ D ˜ N = Dφ Dφ D ˆ N J " Dψ Dφ Dψ Dφ D ˜ ND ˆ N , (53) D ˜ˆ ϕ = D ˆ ϕ. (54)Since the field transformations are linear, the Jacobians depend only on thereparametrization s ( t ). Therefore, they are independent of the fields and sources, andcan be taken outside the integral as common factors.By inserting the values of the transformed sources and dummy variables back into ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics Z fp [˜ l, ˜ h ; T ] = J J Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N exp i X i Z Z ∞ dtdt ′ X α i ,α ′ i ∂s∂t ! α i + α i ∂s∂t ′ ! α ′ i + α ′ i ˆ Q α i ,α ′ i i ( s ( t ) , s ( t ′ )) − T Z ∞ dt ∂s∂t ! (cid:16) φ i ( s ( t )) (cid:17) + Z ∞ dt " ∂s∂t l i ( s ( t )) φ i ( s ( t )) + ih i ( s ( t )) ∂s∂t φ i ( s ( t )) + i N X i =1 ˆ ϕ i [ φ i ( s (0)) − ϕ i ] + i Z ∞ dt ∂s∂t ˆ N ( s ( t )) " N X i =1 ( φ i ( s ( t ))) − N . (55)So then the transformed fixed point generating functional is Z fp [˜ l, ˜ h ; T ] = J J Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N exp i X i Z Z ∞ dsds ′ X α i ,α ′ i ˆ Q α i ,α ′ i i ( s, s ′ ) (cid:18) Q α i ,α ′ i i ( s, s ′ ) − φ α i i ( s ) φ α ′ i i ( s ′ ) (cid:19) − T Z ∞ ds (cid:16) φ i ( s ) (cid:17) ∂t∂s ! − + Z ∞ ds [ l i ( s ) φ i ( s ) + ih i ( s ) φ i ( s )]+ i N X i =1 ˆ ϕ i [ φ i (0) − ϕ i ] + i Z ∞ ds ˆ N ( s ) " N X i =1 ( φ i ( s )) − N . (56)Here we have used the fact that α + α = 1. We notice that the term describing thecoupling to the bath is not invariant with respect to the transformation t → s ( t ), exceptin the trivial case s ( t ) = t . So the high temperature fixed points are not invariant underreparametrizations of the time variable. For the same reason, the fixed point actionscontaining both the coupling to the thermal bath and the spin-spin interaction are notinvariant under time reparametrizations.Finally, we consider the fixed point generating functional for the low temperaturefixed point family. We evaluate the fixed point generating functional for the new sources Z fp [˜ l, ˜ h ; J ] = Z D ˜ QD ˜ˆ QDψ Dψ D ˜ˆ ϕD ˜ N × exp i X i Z Z ∞ dt dt X α i ,α ′ i ˜ˆ Q α i ,α ′ i i ( t , t ) (cid:18) ˜ Q α i ,α ′ i i ( t , t ) − ψ α i i ( t ) ψ α ′ i i ( t ) (cid:19) − p p ! X i ...i p K i ...i p Z Z ∞ dt dt C =1 ,C ′ =1 X α ir ,α ′ ir ∈{ , } p Y r =1 ˜ Q α ir ,α ′ ir i r ( t , t )+ Z ∞ dt [˜ l i ( t ) ψ i ( t ) + i ˜ h i ( t ) ψ i ( t )]+ i N X i =1 ˜ˆ ϕ i [ ψ i ( t ) − ˜ ϕ i ] + i Z ∞ dt ˜ N ( t ) " N X i =1 ( ψ i ( t )) − N . (57) ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics J and J .By inserting the values of the transformed sources and dummy variables back intothe fixed point generating functional we obtain, Z fp [˜ l, ˜ h ; J ] = J J Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N exp i X i Z Z ∞ dtdt ′ X α i ,α ′ i ∂s∂t ! α i + α i ∂s∂t ′ ! α ′ i + α ′ i ˆ Q α i ,α ′ i i ( s ( t ) , s ( t ′ )) × (cid:18) Q α i ,α ′ i i ( s ( t ) , s ( t ′ )) − φ α i i ( s ( t )) φ α ′ i i ( s ( t ′ )) (cid:19) − p p ! X i ...i p K i ...i p Z Z ∞ dtdt ′ C =1 ,C ′ =1 X α ir ,α ′ ir ∈{ , } p Y r =1 ∂s∂t ! α ir ∂s∂t ! α ′ ir Q α ir ,α ′ ir i r ( s ( t ) , s ( t ′ ))+ Z ∞ dt " ∂s∂t l i ( s ( t )) φ i ( s ( t )) + ih i ( s ( t )) ∂s∂t φ i ( s ( t )) + i N X i =1 ˆ ϕ i [ φ i ( s (0)) − ϕ i ] + i Z ∞ dt ∂s∂t ˆ N ( s ( t )) " N X i =1 ( φ i ( s ( t ))) − N . (58)We now use the fact that α + α = 1 and that the constraints C = C ′ = 1 ensure that p Q r =1 (cid:16) ∂s∂t (cid:17) α ir (cid:16) ∂s∂t ′ (cid:17) α ′ ir = ∂s∂t ∂s∂t ′ , to write the transformed fixed point generating functional Z fp [˜ l, ˜ h ; J ] = J J Z DQD ˆ QDφ Dφ D ˆ ϕD ˆ N exp i X i Z Z ∞ dsds ′ X α i ,α ′ i ˆ Q α i ,α ′ i i ( s, s ′ ) (cid:18) Q α i ,α ′ i i ( s, s ′ ) − φ α i i ( s ) φ α ′ i i ( s ′ ) (cid:19) − p p ! X i ...i p K i ...i p Z Z ∞ dsds ′ C =1 ,C ′ =1 X α ir ,α ′ ir ∈{ , } p Y r =1 Q α ir ,α ′ ir i r ( s, s ′ )+ Z ∞ ds [ l i ( s ) φ i ( s ) + ih i ( s ) φ i ( s )]+ i N X i =1 ˆ ϕ i [ φ i (0) − ϕ i ] + i Z ∞ ds ˆ N ( s ) " N X i =1 ( φ i ( s )) − N . (59)In other words, we have shown that Z fp [˜ l, ˜ h ; J ] = J J Z fp [ l, h ; J ] . (60)We know that in the absence of sources, the transformation leaves the generatingfunctional unchanged. This implies that J J = 1, but since J and J are independentof the values of the sources, then for any value of the sources the fixed point generatingfunctional is unchanged by the transformation, i.e., Z fp [˜ l, ˜ h ] = Z fp [ l, h ] . (61) ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics
5. Discussion and conclusion
In our long time renormalization group analysis we have shown that there are threefamilies of stable fixed point dynamic generating functionals for the Langevin dynamicsof the p-spin model: (i) a family of high temperature fixed points, which are not invariant under global reparametrizations of the time variable, characterized by thepresence in the action of the coupling to the thermal bath and the absence of the spininteraction term; (ii) a family of low temperature fixed points with actions containing thespin interaction term but not the coupling to the bath, which are invariant under globaltime reparametrizations in the long time limit; and (iii) a third family of stable fixedpoints, for which both terms are present in the action, and thus the action is not invariantunder time reparametrizations. Since not all of the stable fixed points in the model areinvariant, it is clear that time reparametrization symmetry is a nontrivial property ofthe low temperature, interaction dominated dynamics. It should also be pointed outthat in another interacting spin model, the O(N) ferromagnet, the symmetry is notpresent in the asymptotic long time Langevin dynamics, even in the low temperaturecase [25].The proof of invariance for the low temperature, long time dynamics only assumesthat the couplings J i ...i p are uncorrelated Gaussian random variables with zero mean,but no condition is imposed on the variance K i ...i p of the couplings, thus allowing themto have an arbitrary space dependence. In particular, the proof applies to both short-range and long-range models. Since some versions of the p-spin model share many ofthe main features of structural glass phenomenology [21, 22], we expect that analyticaltools similar to the ones used here can uncover the presence of time reparametrizationsymmetry in models of structural glass systems.As discussed in Refs. [17, 18, 19], time reparametrization symmetry is aspontaneously broken symmetry in a glass. The symmetry is broken by correlationsand responses. To illustrate the spontaneous breaking of the symmetry, we consider thecorrelation function C ( t , t ). If correlations were invariant under the transformationwe would have C ( t, t ′ ) = C ( h ( t ) , h ( t ′ )) for all t and t ′ and all reparametrizationsand the only way this is possible is when the correlation function is independent oftime. This is not the case in glasses because the correlation decays with time. Thepresence of a broken continuous symmetry in the absence of long range interactions orgauge potentials is expected to give rise to Goldstone modes [27]. In the case of theglass problem, the Goldstone modes should be associated with smoothly varying localfluctuations t → h r ( t ) in the time reparametrization [17, 18, 19]. These fluctuationscan be interpreted as representing local fluctuations of the age of the sample [17, 18].Support for this point of view comes from simulation results both in the Edwards- ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics Acknowledgments
We thank L. Cugliandolo and M. Kennett for useful discussions, and particularlyC. Chamon for his help in starting to formulate some of the ideas presented in Sec. 3.2.This work was supported in part by DOE under grant DE-FG02-06ER46300, by NSFunder grants PHY99-07949 and PHY05-51164, and by Ohio University. H. E. C.acknowledges the hospitality of the Aspen Center for Physics and the Kavli Institutefor Theoretical Physics, where parts of this work were performed. [1] P. G. Debenedetti, F. H. Stillinger, 2001 Nature Spin Glasses and Random Fields edited by A. P. Young (Singapore: World Scientific)[5] L. F. Cugliandolo, J. Kurchan, 1994 J. Phys. A (1) 173[7] M. D. Ediger, 2000 Annu. Rev. Phys. Chem. R15112[9] E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, 2000 Science S359[11] G. Parisi, 1999 J. Phys. Chem. B (24) 5526[17] C. Chamon, M. P. Kennett, H. E. Castillo, L. F. Cugliandolo, 2002 Phys. Rev. Lett. ime reparametrization invariance in arbitrary range p-spin models: symmetric versus non-symmetric dynamics [20] L. F. Cugliandolo, 2002 arxiv:cond-mat/0210312v2[21] T. R. Kirkpatrick, P. G. Wolynes, 1987 Phys. Rev. A