Path Integral Approach Unveils the Role of Complex Energy Landscape for Activated Dynamics of Glassy Systems

TL;DR

This paper introduces a path integral approach to analyze the activated dynamics in glassy systems, revealing how complex energy landscapes influence metastable state transitions and ergodicity restoration.

Contribution

It develops a saddle-point path integral method to compute transition probabilities between metastable states in mean-field models, providing new insights into glassy dynamics.

Findings

Numerical solutions of the boundary value problem illustrate transition pathways.

The approach elucidates the role of energy landscape ruggedness in ergodicity.

Results apply to models like the spherical p-spin glass, relevant in various fields.

Abstract

The complex dynamics of an increasing number of systems is attributed to the emergence of a rugged energy landscape with an exponential number of metastable states. To develop this picture into a predictive dynamical theory I discuss how to compute the exponentially small probability of a jump from one metastable state to another. This is expressed as a path integral that can be evaluated by saddle-point methods in mean-field models, leading to a boundary value problem. The resulting dynamical equations are solved numerically by means of a Newton-Krylov algorithm in the paradigmatic spherical $p$-spin glass model that is invoked in diverse contexts from supercooled liquids to machine-learning algorithms. I discuss the solutions in the asymptotic regime of large times and the physical implications on the nature of the ergodicity-restoring processes.