An operational procedure is presented to compute explicitly the different terms in the generalized Langevin equation (GLE) for a few relevant variables obtained within Mori-Zwanzig formalism. The procedure amounts to introducing an artificial controlled parameter which can be tuned in such a way that the so-called projected dynamics becomes explicit and the GLE reduces to a Markovian equation. The projected dynamics can be realised in practice by introducing constraints, and it is shown that the Green-Kubo formulae computed with these dynamics do not suffer from the plateau problem. The methodology is illustrated in the example of star polymer molecules in a melt using their center of mass as relevant variables. Through this example, we show that not only the effective potentials, but also the friction forces and the noise play a very important role in the dynamics.
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.
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