Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems

TL;DR

This paper develops a field theory framework using functional renormalization group to analyze the glassy dynamics of pinned elastic systems, revealing complex barrier distributions and relaxation behaviors.

Contribution

It introduces a hierarchy of FRG equations and a thermal boundary layer ansatz to describe non-trivial relaxation time distributions in glassy elastic systems.

Findings

Relaxation times follow a size-dependent distribution, not a simple exponential.

A random friction model yields a log-normal distribution of relaxation times.

The dynamical effective action reveals broad, non-log-normal barrier distributions.

Abstract

We study thermally activated, low temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. A naive single relaxation time approximation for each wavevector is found to be unsatisfactory. A second approximation based on a random friction model, yields a size (L) dependent log-normal distribution of relaxation times (mean barriers ~L^\theta and variance ~ L^{\theta/2}) and a procedure to estimate dynamical scaling functions. Finally, we study the full structure of the running dynamical effective action within the field theory. We find that relaxation time distributions are non-trivial (broad but not log-normal) and encoded in a closed hierarchy of FRG equations. A thermal boundary layer ansatz (TBLA) appears as a consistent solution. It extends the one discovered in the statics which was shown to embody droplet thermal fluctuations. Although perturbative control remains a challenge, the structure of the dynamical TBLA which encodes barrier distributions opens the way for deeper understanding of the field theory approach to glasses.