Aging processes in systems with anomalous slow dynamics

Abstract

Recently, different numerical studies of coarsening in disordered systems have shown the existence of a crossover from an initial, transient, power-law domain growth to a slower, presumably logarithmic, growth. However, due to the very slow dynamics and the long-lasting transient regime, one is usually not able to fully enter the asymptotic regime when investigating the relaxation of these systems toward equilibrium. We here study two simple driven systems-the one-dimensional ABC model and a related domain model with simplified dynamicsthat are known to exhibit anomalous slow relaxation where the asymptotic logarithmic growth regime is readily accessible. Studying two-times correlation and response functions, we focus on aging processes and dynamical scaling during logarithmic growth. Using the time-dependent growth length as the scaling variable, a simple aging picture emerges that is expected to also prevail in the asymptotic regime of disordered ferromagnets and spin glasses.

Relaxation processes in a system with logarithmic growth

“…For the 600 × 1 and 600 × 10 systems we are able to access in the microscopic model the asymptotic growth regime for all values of q between 0.9 and 0.1. In agreement with what has been observed for the interface model [27,45], we find also for the microscopic model that the plot of L x (t) versus ln t yields a slop that depends on q in the following way:…”

“…5 for the system containing 600 × 10 sites. The value of κ is much smaller than the value κ ≈ 2.0 found for the interface model in one dimension [45], in agreement with the expectation that the simplified dynamics in that model leads to a change in the value of the unit of time.…”

“…Anomalous coarsening with logarithmic domain growth has been identified in a variety of situations with dynamical constraints, as for example in the ABC model introduced by Evans et al [26,27], where three different particle types swap places asymmetrically, in the model discussed by Lahiri and Ramaswamy [28,29], where two sublattices are considered with two types of particles on each sublattice, or in the driven two-lane particle system studied by Lipowski and Lipowska [30]. Among these models, the one-dimensional ABC model has enjoyed much attention in recent years [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], due to its unique properties combined with its amenability to exact calculations in some special cases. Thus the ABC model provides an opportunity to study exactly the long-range correlations in a system undergoing a non-equilibrium phase transition as well as to elucidate ensemble inequivalence far from equilibrium.…”

“…In [49] slow coarsening was shown to also exist in a two-dimensional version of the ABC model. Finally, in [45] the focus was on the interface model in one dimension whose study provided some insights into aging and dynamical scaling in presence of a logarithmically growing length.…”

Physical Aging of Classical Oscillators

“…Another set of systems, in which aging is observed, shares a different kind of slow processes, which are related to growth [9,10], coarsening [11], diffusion, and subdiffusion [12,13], leading to a critical slowing of the dynamics, often with a similar type of dynamical scaling as in the first set of models. Aging in a Hamiltonian system of coupled rotators was also observed for conservative dynamics, remarkably without a thermal bath and without disorder or frustration [14], but for a particular family of initial conditions in the specific limits of infinite-range couplings, and the thermodynamic limit N → ∞ taken before the large time limit t → ∞.…”