Markovian dissipative coarse grained molecular dynamics for a simple 2D graphene model

Abstract

Based upon a finite-element "coarse-grained molecular dynamics" (CGMD) procedure, as applied to a simple atomistic 2D model of graphene, we formulate a new coarse-grained model for graphene mechanics explicitly accounting for dissipative effects. It is shown that, within the Mori-projection operator formalism, the reversible part of the dynamics is equivalent to the finite temperature CGMD-equations of motion, and that dissipative contributions to CGMD can also be included within the Mori formalism. The CGMD nodal momenta in the present graphene model display clear non-Markovian behavior, a property that can be ascribed to the fact that the CGMD-weighting function suppresses high-frequency modes more effectively than, e.g., a simple center of mass (COM) based CG procedure. The present coarse-grained graphene model is also shown to reproduce the short time behavior of the momentum correlation functions more accurately than COM-variables and it is less dissipative than COM-CG. Finally, we find that, while the intermediate time scale represented directly by the CGMD variables shows a clear non-Markovian dynamics, the macroscopic dynamics of normal modes can be approximated by a Markovian dissipation, with friction coefficients scaling like the square of the wave vector. This opens the way to the development of a CGMD model capable of describing the correct long time behavior of such macroscopic normal modes.

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules

“…Now we turn to (28). For this purpose, we rewrite the exponent in the probability distribution as follows:…”

“…25 The main practical difficulty in implementing the GLE is the computation of the memory function. In some cases, Markovian approximations can be made [26][27][28] to reduce the GLE to a Langevin equation, or one may simply use exponential functions, 19 assuming a rapid decay. However, it is difficult to quantify and control the modeling error in such an ad hoc approximation.…”

Markovian equations of motion for non-Markovian coarse-graining and properties for graphene blobs

“…One atom i may also contribute to more than one CG variable µ. This is for example the case in the CGMD method [27,28]. The most general requirements on…”

“…The dynamics of the graphene model is described by Hamilton's equations of motion for the set of microscopic variables {r i , p i }, where m i is the mass, r i the position and p i the momentum of C-atom i. For the sake of simplicity, instead of more realistic potentials such as the Brenner bond order potential [37], we use a simple 2D potential of the same form as already used in [27]. The interaction potential between C-atoms is assumed to have the form U = U s + U a .…”

“…Early work was based on top-down CG CNT-models assuming a priori the DPD-form of the equations of motion for centre of mass (COM) variables of groups of carbon atoms [23,24]. In a systematic bottom-up approach for two-dimensional (2D) motion of graphene, a Markovian approximation was applied to GLEs obtained from Mori's or Zwanzig's projection operators [25][26][27]. COM-based relevant variables seemed to show not only less non-Markovian features than finite element based variables but also showed weaknesses in the reproduction of cross-correlation functions of the relevant variables or long-wavelength normal mode decay, while the reproduction of autocorrelation functions is nearly perfect.…”