Connection between Nonlinear Energy Optimization and Instantons

TL;DR

This paper links energy optimization techniques with instanton theory to identify minimal disturbances causing state transitions in nonlinear systems, demonstrating that discrete perturbations can effectively approximate instantons.

Contribution

It introduces a method connecting energy optimization with instanton trajectories, enabling minimal disturbance identification with fewer perturbations in complex systems.

Findings

Energy optimization can identify minimal seeds for state transitions.

Instanton trajectories are recovered as limits of the energy optimization with many perturbations.

Few discrete perturbations can capture key instanton features, aiding practical diagnostics.

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.