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

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.