| Given an autonomous system of ordinary differential equations (ODE), we consider developing practical models for the deterministic, slow/coarse behavior of the ODE system. Two types of coarse variables are considered.;The first type consists of running finite-time averages of phase functions. We explore two 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 and a 'Forced' Lorenz system. This method is suitable for which there is (near) absence of time-scale separation. In the other, the choice of our coarse variables automatically guarantees them to be 'slow' in a precise sense. This allows their evolution to be phrased in terms of averaging utilizing limit measures (probability distributions) of the fast flow under the condition that there is a clear separation of timescales between the slow and fast dynamics. Two 'toy' examples are tested: a 'Forced' Lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium. Finally, the developed multi-scale techniques are implemented on a molecular dynamical (MD) system undergoing phase transition. Three different scenarios are considered, i.e., Newtonian dynamics with no energy dissipation, dynamics with small viscosity and dynamics with thermostat. Coarse variables that describe some macroscopic features of the system are defined (e.g. temperature, number of phase interfaces and averaged strain). The coarse (macroscopic) dynamics is obtained and tested against coarse response of the 'microscopic' models. The gain of computational time from the coarse model 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 functions that are required by design to evolve autonomously, to the extent possible, with the goal of being candidate state functions for unambiguously initializable coarse dynamics. The question motivates a mathematical restatement in terms of a first-order PDE. A computational approximation is developed and tested on the Lorenz system and the Hald Hamiltonian system. |
