The Optimal Growth Mode in the Relaxation to Statistical Equilibrium

TL;DR

This paper introduces the Optimal Growth Mode (OGM), a new method to analyze slow initial relaxation in systems approaching equilibrium, using Markov matrices to approximate phase space dynamics.

Contribution

The paper presents the OGM as a novel approach to capture slow relaxation, distinguishing it from traditional modes, with applications demonstrated on the Lorenz 63 model.

Findings

OGM captures slow initial relaxation in non-equilibrium systems.

Differences between OGM and traditional slowest decaying mode are clarified.

Implications for understanding system response to external forces are discussed.

Abstract

Systems far from equilibrium approach stability slowly due to "anti-mixing" characterized by regions of the phase-space that remain disconnected after prolonged action of the flow. We introduce the Optimal Growth Mode (OGM) to capture this slow initial relaxation. The OGM is calculated from Markov matrices approximating the action of the Fokker-Planck operator onto the phase space. It is obtained as the mode having the largest growth in energy before decay. Important nuances between the OGM and the more traditional slowest decaying mode are detailed in the case of the Lorenz 63 model. The implications for understanding how complex systems respond to external forces, are discussed.