Connection between Nonlinear Energy Optimization and Instantons

Abstract

How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently-developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift-Hohenberg equation which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances -- minimal seeds -- for transition to specific attractors from the ground state. Then, after generalising the technique to consider multiple, equally-spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton.