Path integral approach unveils role of complex energy landscape for activated dynamics of glassy systems

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.

Automated summary

The article discusses computating the exponentially small probability of a system jumping from one metastable state to another, focusing on the evaluation of path integrals using mean-field models and saddle-point methods, and solving the resulting dynamical equations with numerical algorithms.