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

摘要

The calculation of the friction matrix in the coarse-grained (CG) description of an atomistic system is a crucial issue, in order to properly account for the dissipative effects inherent to any reduced representation of the atomistic dynamics. Within the Mori-Zwanzig projection operator approach to CG, there are several possibilities for the definition of the friction matrix, depending on the projector that is being used. In this paper, the connection of two of these projectors (Mori's and Zwanzig's) is discussed and the corresponding merits and limitations are analysed. Moreover, three different ways of computing the friction matrix in the Mori's framework are presented, along with a discussion of their mutual connections. By the example of CG centre of mass blob variables in the graphene lattice, it is shown that, even though the three approaches are equivalent from a theoretical viewpoint, they may differ considerably in terms of practical implementation for computer simulation purposes. In the given example, which is representative for atomic lattices, it turns out that a linear regression of the velocity–velocity correlation function, inspired by the Einstein–Helfand approach, is the least error-prone against disturbances from optical modes.