TY - JOUR

T1 - Glassy dynamics on networks

T2 - local spectra and return probabilities

AU - Margiotta, Riccardo Giuseppe

AU - Kühn, Reimer

AU - Sollich, Peter

PY - 2019/9/27

Y1 - 2019/9/27

N2 - 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.

AB - 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.

KW - Aging

KW - Cavity and replica method

KW - Energy landscapes

KW - Glassy dynamics

KW - Networks

KW - Random graphs

KW - Slow relaxation

UR - http://www.scopus.com/inward/record.url?scp=85086997721&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/ab3aeb

DO - 10.1088/1742-5468/ab3aeb

M3 - Article

AN - SCOPUS:85086997721

SN - 1742-5468

VL - 2019

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

IS - 9

M1 - 093304

ER -