Markovian approximation in a coarse-grained description of atomic systems

Abstract

The Markovian assumption stating that memory effects can be neglected is a crucial assumption in the theory of coarse-graining. We investigate the coarse-graining of a one-dimensional chain of oscillators where the atoms are grouped into clusters or blobs. When the interaction between oscillators is through Hookean springs, the cluster dynamics is non-Markovian, as has been recently noted by Cubero and Yaliraki [J. Chem. Phys. 122, 03418 (2005)]. When the oscillators interact through a nonlinear potential of the Lennard-Jones type, the dynamics turns out to be Markovian. The different behavior in both types of interactions is attributed to the persistence of sound waves in the harmonic case, which are strongly suppressed in the nonlinear case.

Three Routes to the Friction Matrix and Their Application to the Coarse‐Graining of Atomic Lattices

“…The time t F is the time of the initial decay of the microscopic blob force-force correlation hF m F n ðtÞi (FF). It was shown [32][33][34] that, if the dynamics of the relevant variables is approximately Markovian and the FF has a dominant rapidly decaying part with small t F and just small terms evolving on macroscopic time scales then it is possible to calculate the friction matrix through the Kirkwood expression [18] G…”

Response to “Comment on ‘Markovian approximation in a coarse-grained description of atomic systems’ ” [J. Chem. Phys. 128, 147101 (2008)]

“…These two terms have been neglected in Ref. 5 and this has been questioned by Cubero in his Comment. We will assume that the very short range decaying part of C͑t͒ may be modeled with a delta function.…”

“…Cubero has detected a mathematical error in Ref. 5 that led him to conclude that our claim that a Markovian description is valid for a high temperature Lennard-Jones fluid is not correct. We have to admit that non-Markovian effects cannot be discarded in this model, and many of the observations made by Cubero are correct.…”

“…In a series of papers [1][2][3][4][5] a debate about the validity of the Markovian assumption in a coarse-grained blob description of one-dimensional interacting particles has arisen. Cubero has detected a mathematical error in Ref.…”

Parametric reduced models for the nonlinear Schrödinger equation

“…A simple approach proposed by [7,17,18] is to approximate the memory kernel in the GLE model with a delta function (but with a carefully chosen damping parameter). This certainly introduces additional modeling error that is difficult to quantify.…”