Damage spreading in mode-coupling theory for glasses

We examine the problem of damage spreading in the off-equilibrium mode coupling equations. This study is conducted for the spherical p-spin model introduced by Crisanti, Horner and Sommers. For p > 2 we show the existence of a temperature transition T0 well above any relevant thermodynamic transition temperature. Above T0 the asymptotic damage decays to zero while below T0 it decays to a finite value independent of the initial damage. This transition is stable in the presence of asymmetry in the interactions. We discuss the physical origin of this peculiar phase transition which occurs as a consequence of the nonlinear coupling between the damage and the two-time correlation functions. The theoretical understanding of the dynamical behaviour of glasses is a long outstanding problem in statistical physics which has recently revealed new aspects related to the underlying mechanism responsible of the glass transition [1, 2]. The scenario for the dynamical behaviour of glasses can be summarized in two different temperatures which separate three different regimes. In the high-temperature regime T > Td the system behaves as a liquid and is described very well by the mode-coupling equations of Götze in the equilibrium regime [3]. A crossover takes place at Td where there is a dynamical singularity and the correlation functions do not decay to zero in the infinite-time limit (ergodicity breaking). This dynamical singularity is a genuine mean-field effect which turns out to be a crossover temperature when activated processes are taken into account. Below Td the relaxation time (or viscosity) starts to grow dramatically fast and seems to diverge at Ts where the configurational entropy apparently vanishes. The essentials of this scenario have been corroborated in the context of mean-field spin glasses, and in particular in those models with a one-step replica symmetry breaking transition [4]. The purpose of this paper is the study of the damage spreading in mode-coupling theory. Damage spreading is the study of the time propagation of a perturbation or damage in the initial condition of a system. This dynamical effect has deserved considerable attention in the past (especially in the context of dynamical systems, for instance, networks of Boolean automata [5]) because it allows us to explore the structure of the phase space of the system. To investigate damage spreading we consider two random initial configurations {σi, τi} with a given initial distance D0 for two identical systems which evolve under identical noise realizations and compute the distance D(t) as a function of time. Of particular interest is the asymptotic long-time behaviour of the distance D(t), i.e. D∞ = limt→∞D(t). In † E-mail address: [email protected] ‡ E-mail address: [email protected] 0305-4470/98/428423+07$19.50 c © 1998 IOP Publishing Ltd 8423 8424 M Heerema and F Ritort general, three different regimes can be distinguished. A high-temperature regime T > T0 whereD∞ = 0 independently of the initial distance D0. A intermediate regime T1 < T < T0 where D∞ = D∞(T ) is not zero but independent of the initial distance. And finally a lowtemperature regime T < T1 where D∞ = D∞(T ,D0) depends on both temperature and initial distance. Although it is widely believed that T1 corresponds to a thermodynamic phase transition it is not clear what the physical meaning of T0 is. Here we will show the existence of the temperature T0 in glasses well above Td and Ts in the high-temperature phase. We show that this new transition is a consequence of the nonlinear coupling between the damage and the corresponding two-time correlation function. This effect is an essential ingredient of the mode-coupling equations and should be generally valid even beyond the mean-field limit. We believe the appearance of this damage transition is a quite general result in glassy models (with and without disorder) where the scenario of Götze for modecoupling transitions is valid. The simplest solvable model described by the off-equilibrium mode-coupling equations is the spherical p-spin glass model [6]. In this case, the configurations are described by N continuous spin variables {σi; 1 6 i 6 N} which satisfy the spherical global constraint ∑N i=1 σ 2 i = N . The Langevin dynamics of the model is given by, ∂σi ∂t = Fi({σ })− μσi + ηi (1) where Fi is the force acting on the spin σi due to the interaction with the rest of the spins, Fi = −∂H ∂σi = 1 (p − 1)! ∑ (i2,i3,...,ip) J i2,i3,...,ip i σi2σi3 . . . σip (2) and H is a Hamiltonian. The J i2,i3,...,ip i are quenched random variables with zero mean and variance p!/(2Np−1) which we take to be symmetric under permutation of the different superindices. The calculations presented here can be easily generalized to asymmetric couplings [7]. Obviously, in this last case there is no energy H which drives the system to thermal equilibrium. The term μ in equation (2) is a Lagrange multiplier which ensures that the spherical constraint is satisfied at all times and the noise η satisfies the fluctuation– dissipation relation 〈ηi(t)ηj (s)〉 = 2T δ(t − s)δij where 〈· · ·〉 denotes the noise average. We define the overlap between two configurations of the spins σ, τ by the relation Q = 1 N ∑N i=1 σiτi so the Hamming distance between these two configurations is D = 1−Q 2 (3) in such a way that identical configurations have zero distance and opposite configurations have maximal distance. Then we consider two copies of the system {σi, τi} which evolve under the same noise (1) but with different initial conditions. Here we restrict ourselves to random initial configurations (i.e. equilibrium configurations at infinite temperature) with initial overlap Q(0). The different set of correlation functions which describe the dynamics of the system are given by