November 2, 2008
Polymers are prototypical examples of soft-matter systems. The talk
will first focus on equilibrium statistical mechanics, and introduce
basic concepts like the random walk and the self-avoiding walk. This
is complemented by a discussion of the notion of coarse-graining and
scale invariance, which is at the basis of modeling polymers in terms
of simple bead-spring models. The second part will then discuss the
basics of polymer dynamics, in terms of the fundamental Rouse, Zimm,
and reptation models. The third part is devoted to a brief overview
over Monte Carlo and Molecular Dynamics models and simulation
algorithms, which are directly based upon the insight into the
essential physics. If time permits, a brief outlook on the physics
of membranes will be added.
Joint work with Eric Vanden-Eijnden.
In many problems of multiscale modeling,
we are interested in capturing the macroscale behavior of the system
with the help of some accurate microscale models, bypassing the need
of using empirical macroscale models.
This paper gives an overview of the recent efforts on establishing
general strategies for designing such algorithms.
After reviewing some important classical examples, the Car-Parrinello
molecular dynamics, the quasicontinuum method for modeling the deformation
of solids and the kinetic schemes for gas dynamics, we discuss three
attempts that have been made for designing general strategies:
Brandt's renormalization multi-grid method (RMG),
the heterogeneous multiscale method (HMM)
and the "equation-free" approach.
We will discuss the relative merits and difficulties
with each strategy and we will make
an attempt to clarify their similarities and differences.
We will then discuss a general strategy for developing seamless
multiscale methods for this kind of problems.
We will end with a discussion of the applications to free energy
calculations
and a summary of the challenges that remain in this area
Beginning with some observations about the periodic and
nonperiodic
structures commonly adopted by elements in the periodic table, I will
introduce a definition ("objective structures") of a mathematically
small but physically well represented class of molecular structures.
This definition will be seen to have an intimate relation to the
invariance of the equations of quantum mechanics. The resulting
framework can be used to design various multiscale methods, and gives a
new perspective on some of the fundamental solutions in continuum
mechanics for solids and fluids. Open mathematical problems will be
highlighted.