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

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.