Glassy dynamics on networks: local spectra and return probabilities

Abstract

The slow relaxation and aging of glassy systems can be modelled as a Markov process on a simplified rough energy landscape: energy minima where the system tends to get trapped are taken as nodes of a random network, and the dynamics are governed by the transition rates among these. In this work we consider the case of purely activated dynamics, where the transition rates only depend on the depth of the departing trap. The random connectivity and the disorder in the trap depths make it impossible to solve the model analytically, so we base our analysis on the spectrum of eigenvalues λ of the master operator. We compute the local density of states ρ(λ|τ ) for traps with a fixed lifetime τ by means of the cavity method. This exhibits a power law behaviour ρ(λ|τ ) ∼ τ |λ| T in the regime of small relaxation rates |λ|, which we rationalize using a simple analytical approximation. In the time domain, we find that the probabilities of return to a starting node have a power law-tail that is determined by the distribution of excursion times F (t) ∼ t −(T +1) . We show that these results arise only by the combination of finite configuration space connectivity and glassy disorder, and interpret them in a simple physical picture dominated by jumps to deep neighbouring traps. arXiv:1906.07434v1 [cond-mat.dis-nn]

Linear stability analysis of large dynamical systems on random directed graphs

“…Detectability thresholds of spectral algorithms often depend on the location of the leading and subleading eigenvalue [90,[95][96][97][98]. A fourth example of an application is the analysis of stochastic processes with spectra of Laplacian or Markov matrices [99][100][101][102][103]: the stationary state of a Markov process is the right (or left) eigenvector associated with the leading eigenvalue of a Markov matrix [103], the relaxation time is provided by the spectral gap [104][105][106], and the cumulant generating function of a timeadditive observable can be expressed in terms of the leading eigenvalues of a sequence of Markov matrices [107][108][109][110][111]. A fifth application is the study of nonHermitian quantum mechanics on random graphs [85,112,113].…”

From entropic to energetic barriers in glassy dynamics: The Barrat-Mézard trap model on sparse networks

“…For time domain properties we obtain stochastic simulation results directly for the thermodynamic limit N → ∞, by generating an effectively infinite tree on the fly, during the course of a stochastic simulation using the Gillespie algorithm. The network construction method is identical to that used for the Bouchaud trap model in [22]; its key advantage is that it allows us to study time-dependent properties without any finite size effects. The Gillespie simulation algorithm consists of repeated evaluation the following steps:…”

“…Our main finding is that below the glass transition temperature the slowest relaxation modes, which determine the long time behavior, have a density that follows a divergent power law (Eq. ( 30)), which is characteristic of activated processes [19,22]. This contrasts with the mean field case of infinite connectivity (c → ∞), where the spectral density is constant for the slowest modes.…”

“…Arguably the simplest trap model is the Bouchaud model, which has Arrhenius transition rates that presume activation to some common threshold energy level and depend only on the departing trap. For this model recent studies [19,22] have considered in detail the effect of the structure of the network of traps, with a focus on the spectral and time domain properties of the master operator that defines the dynamics. In this paper we use a similar set of tools to analyze the Barrat-Mézard (BM) trap model on Random Regular Graphs (RRG).…”

Effective traplike activated dynamics in a continuous landscape

“…Much progress along these lines was made in a series of works culminating with the proof that the random energy model (REM), a simple model with glassy behavior, exhibits trap-like dynamics [6][7][8]23,24]. Another consists of studying the influence of phase space connectivity on the dynamics [25,26]. In our approach, we show that the TM paradigm also applies to a very simple model of a continuous N-dimensional landscape, where each dimension represents an independent coordinate in a fictitious space with well-defined metrics.…”