Dynamical instantons and activated processes in mean-field glass models

Abstract

We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical p-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process.

Linear low energy excitations in fully-connected models of glasses

“…The study of the complexity suggests the existence of marginally stable minima in some intervals of energy above the level E mg that separates stable minima from marginal ones. These minima are described by replica symmetry breaking and could be the continuation of some high temperature states that undergo a Gardner transition [47,48,50,[56][57][58][59][60][61] at low temperature. Without much surprise, we can expect in these minima a divergent spin glass susceptibility, a square root spectral pseudogap and fully delocalized states.…”

Accelerating, to some extent, the $p$-spin dynamics

“…A related interesting question is how the instanton trajectories are modified by a nonzero γ in a finite size mean-field model, expanding on the analysis of the Kramers problem made in the present article. A relevant comparison could be provided by the study of Ros et al [48] where the instanton trajectory in the spherical p-spin model under equilibrium overdamped Langevin dynamics is characterized.…”

When you can't count, sample! Computable entropies beyond equilibrium from basin volumes

“…Such measurements could have deep fundamental implications, if one does not consider a granular (or zero-temperature) system, but a jammed liquid at a finite temperature. Indeed, in a finite-temperature glassy system, the shape of basins controls transition rates and relaxation times [54][55][56][57], which has prompted a large body of recent work on the enumeration and characterization of minima of rough landscapes [55,[58][59][60][61].…”