Inhomogeneous multiscale dynamics in harmonic lattices
D. Cubero and S. N. Yaliraki
Journal of Chemical Physics, 122, 034108 (2005).
ABSTRACT
We use projection operators to address the coarse-grained multiscale problem in
harmonic systems. Stochastic equations of motion for the coarse-grained
variables, with an inhomogeneous level of coarse graining in both time and
space, are presented. In contrast to previous approaches that typically start
with thermodynamic averages, the key element of our approach is the use of a
projection matrix chosen both for its physical appeal in analogy to mechanical
stability theory and for its algebraic properties. We show that thermodynamic
equilibrium can be recovered and obtain the fluctuation dissipation theorem a
posteriori. All system-specific information can be computed from a series of
feasible molecular dynamics simulations. We recover previous results in the
literature and show how this approach can be used to extend the quasicontinuum
approach and comment on implications for dissipative particle dynamics type of
methods. Contrary to what is assumed in the latter models, the stochastic
process of all coarse-grained variables is not necessarily Markovian, even
though the variables are slow. Our approach is applicable to any system in which
the coarse-grained regions are linear. As an example, we apply it to the
dynamics of a single mesoscopic particle in the infinite one-dimensional
harmonic chain.