A theory for critically divergent fluctuations of dynamical events at non-ergodic transitions

TL;DR

This paper develops a theoretical framework to analyze divergent fluctuations at non-ergodic transitions, modeling them as saddle connection bifurcations and deriving critical exponents for fluctuations.

Contribution

It introduces a Ginzburg-Landau based phenomenological model for non-ergodic transitions, providing a systematic way to determine critical fluctuation properties.

Findings

Derived critical exponents for length, time, and amplitude divergences.

Established a connection between saddle connection bifurcations and non-ergodic transition fluctuations.

Compared theoretical results with previous studies on dynamic susceptibilities in glassy systems.

Abstract

We theoretically study divergent fluctuations of dynamical events at non-ergodic transitions. We first focus on the finding that a non-ergodic transition can be described as a saddle connection bifurcation of an order parameter for a time correlation function. Then, following the basic idea of Ginzburg-Landau theory for critical phenomena, we construct a phenomenological framework with which we can determine the critical statistical properties at saddle connection bifurcation points. Employing this framework, we analyze a model by considering the fluctuations of an instanton in space-time configurations of the order parameter. We then obtain the exponents characterizing the divergences of the length scale, the time scale and the amplitude of the fluctuations of the order parameter at the saddle connection bifurcation. The results are to be compared with those of previous studies of the four-point dynamic susceptibility at non-ergodic transitions in glassy systems.