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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 50; no. 17; pp. 175502 - 175523
• Main Authors: Thalabard, Simon, Turkington, Bruce
• Format: Journal Article
• Language: English
• Published: IOP Publishing 28.04.2017
• Subjects: fluctuation-dissipation; Galerkin dynamics; non-equilibrium response; optimal control; self-thermalization
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aa651b

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.