The Markovian approximation in the Theory of Coarse-Graining
Pep Espanol, UNED, Madrid, Spain

The theoretical basis of Non-Equilibrium Statistical Mechanics were lay down in the mid of last century by Onsager, Kirkwood, Green, Kubo, Mori, Zwanzig, and many others.  This theory deals with the dynamic behavior of large collections of microscopic constituents and it is, in fact, a theory of coarse-graining.  One of the major achievements of the theory was the formulation of a Fokker-Planck equation (FPE) for the dynamics of a set of variables used to describe a molecular system at a coarse-grained level. Zwanzig derived a formally exact integro-differential equation for the probability distribution of the coarse variables. Heuristically, the exact equation is usually approximated by "neglecting memory effects" and leads to the FPE. We present an argument that allows to formalize this Markovian approximation as a limit process and gives a recipe for the calculation of the drift and diffusion terms of the FPE. We will illustrate the method for a system of complex big molecules (star polymers) where the dynamics at the coarse level leads to the equations of Dissipative Particle Dynamics.