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.

28 pages, annotated codes in the ancillary files page

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• Probing chaos in the spherical p-spin glass model
• Competition between Barrier- and Entropy-Driven Activation in Glasses
• On-Site Potential Creates Complexity in Systems with Disordered Coupling
• Clever algorithms for glasses work by time reparametrization
• Comment on "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$"
• Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons
• Instantons in ${\varphi }^4$ Theories: Transseries, Virial Theorems and Numerical Aspects
• Overlaps between eigenvectors of spiked, correlated random matrices: from matrix PCA to random Gaussian landscapes