Coarse star polymers
Coarse graining theory and applications
Coarse graining preserving dynamical properties
Zwanzig theory of projection operators is a corner-stone of Non-Equilibrium Statistical Mechanics. However, the theory has been deemed a formal one in that it is usually very difficult to compute explicitly the different objects appearing in it. Also, the basic Markovian approximation that simplifies the theory is performed in an uncontrolled way, resulting in the well-known {\em plateau problem}. Here, we present an operational procedure that allows one to compute explicitly the different objects in the Fokker-Planck equation derived by Zwanzig. The method makes use of a constrained dynamics derived from a variational principle subject to constraints given by the macroscopic variables. The Green-Kubo formulas computed with the constrained dynamics do not suffer from the plateau problem. The methodology is illustrated for the coarse-graining of star polymer molecules in a melt through its center of mass, where the main message is that not only effective potentials, but also friction forces play a very important role in the dynamics. Finally, we also describe some intrinsic practical problems for the general use of Zwanzig theory, as the need to deal with functions of many variables, or the need to reconstruct microscopic information from macroscopic one in {\em on demand} simulations. Coworkers:
Pep Español (UNED)
Eric Vanden-Eijnden (Courant Institute, New York)