Temporal Coarse-Graining of High-frequency Dynamics Using Parameterized Locally Invariant Manifolds and Practical Time Averaging

Given an autonomous system of ordinary differential equations (ODE), we considerdeveloping practical models for the deterministic, slow/coarse behavior of the ODEsystem. Two types of coarse variables are considered.The first type consists of running finite-time averages of phase functions. We exploretwo strategies to construct the coarse dynamics for such variables. In one, we compute(locally) invariant manifolds of the fast dynamics, parameterized by the slow variables.The tested problems include an atomic chain with nonlinear interatomic potentials anda ‘Forced’ Lorenz system. This method is suitable for which there is (near) absenceof time-scale separation. In the other, the choice of our coarse variables automaticallyguarantees them to be ‘slow’ in a precise sense. This allows their evolution to be phrasedin terms of averaging utilizing limit measures (probability distributions) of the fast flowunder the condition that there is a clear separation of timescales between the slow andfast dynamics. Two ‘toy’ examples are tested: a ‘Forced’ Lorenz system and a singu-larly perturbed system whose fast flow does not necessarily converge to an equilibrium.Finally, the developed multi-scale techniques are implemented on a molecular dynami-cal(MD) system undergoing phase transition. Three different scenarios are considered,i.e., Newtonian dynamics with no energy dissipation, dynamics with small viscosity anddynamics with thermostat. Coarse variables that describe some macroscopic featuresof the system are defined (e.g. temperature, number of phase interfaces and averagedstrain). The coarse (macroscopic) dynamics is obtained and tested against coarse re-sponse of the ‘microscopic’ models. The gain of computational time from the coarsemodel becomes significant in the regime where the separation of timescales is large.The second type of coarse variables are defined as (non-trivial) scalar state functionsthat are required by design to evolve autonomously, to the extent possible, with thegoal of being candidate state functions for unambiguously initializable coarse dynamics.The question motivates a mathematical restatement in terms of a first-order PDE. Acomputational approximation is developed and tested on the Lorenz system and theHald Hamiltonian system.iii