Optimal response to non-equilibrium disturbances under truncated Burgers-Hopf dynamics

TL;DR

This paper develops a dynamical optimization approach to predict the average response of truncated Burgers-Hopf systems to finite disturbances, showing strong agreement with simulations in near-equilibrium conditions.

Contribution

It introduces a novel optimal control framework for modeling non-equilibrium responses in Burgers-Hopf dynamics, extending fluctuation-dissipation concepts.

Findings

Excellent match between optimal predictions and simulations for moderate perturbations.

Optimal response theory acts as a predictive tool near equilibrium.

Method effectively recovers non-equilibrium averages using geodesic paths in probability space.

Abstract

We model and compute the average response of truncated Burgers-Hopf dynamics to finite perturbations away from the Gibbs equipartition energy spectrum using a dynamical optimization framework recently conceptualized in a series of papers. Non-equilibrium averages are there approximated in terms of geodesic paths in probability space that best-fit the Liouvillean dynamics over a family of quasi-equilibrium trial densities. By recasting the geodesic principle as an optimal control problem, we solve numerically for the non-equilibrium responses using an augmented Lagrangian, non-linear conjugate gradient descent method. For moderate perturbations, we find an excellent agreement between the optimal predictions and the direct numerical simulations of the truncated Burgers-Hopf dynamics. In this near-equilibrium regime, we argue that the optimal response theory provides an approximate yet predictive counterpart to fluctuation-dissipation identities.