In this Master thesis, we study the different scales of modeling, and we specifically
focus on methods that aim to bridge the microscopic and macroscopic scale. The
mesoscopic length scale applies to structures that occur on between the atomic and
the macroscopic length scales. Such a method is the Dissipative Particle Dynamics
(DPD), a technique invented to carry out particle based simulations of hydrodynamic
behavior. The DPD method is based on the Langevin dynamics and the lattice gas
algorithms, with respect to fulfilled the Galilean invariance and the conservation of
the momentum.
This thesis is organized as follows: we begin from scratch, reviewing the basic properties
of dynamical systems, and the basic concepts of Statistical Mechanics, in
Section 1. Then in Section 2, we give an introduction on the multi-scale modeling.
In Section 3, we provide a review of the the basic modeling techniques we mention in
this thesis, the Molecular Dynamics, the Langevin equation and the DPD model. An
alternative simulation technique is presented in Section 4, based on the Generalized
Langevin Equation (GLE). We present the derivation of GLE from the microscopic
equations of motions using the Mori-Zwanzig projection operator formalism. Moreover,
we discuss two common approximations made in practical applications of the
GLE, the Markovian approximation and the Modified Projected Dynamics method,
which give rise to the DPD type of equations. In Section 5, we carry out and present
the results of the computer simulations of two case studies, of coarse-grained systems,
most specifically systems of star polymers, part of an on-going work. Finally,
in Section 6, we summarize the work that have be done so far, and discuss the
relative on going projects. In the Appendix, in which we provide some theoretical
foundations that are used in this work.
(EN)
University of Crete
