IMA Development and Analysis of Multiscale Methods
November 3-7, 2008
University of Minnessota, Minneapolis, MN
http://www.ima.umn.edu/2008-2009/W11.3-7.08/

Organizers:
Anne M. Chaka Physical and Chemical Properties, National Institute of
Standards and Technology
Gero Friesecke  Mathematics, University of Warwick
Kurt Kremer  Max-Planck Institut fr Polymerforschung
Yousef Saad  Computer Science and Engineering, University of Minnesota
Gregory A. Voth   Theoretical and Physical Chemistry, University of Utah

Description:

Theoretical, computational and experimental approaches to problems in
natural sciences typically focus on particular aspects of the studied
phenomena or systems. This is linked to the need to structure the questions
with respect to the most relevant length and time scales. This need comes
from the limited range of applicability of specific experimental as well as
theoretical tools. In the past this has created huge progress and is the
basis of our current understanding of physical, chemical and biological
systems. For example, in the area of phase transitions and critical
phenomena renormalization group theory has shown that many properties such
as critical exponents or ratio of critical amplitudes does not depend on
microscopic details of the studied system. This means that within each
universality class, for many properties it is sufficient to study highly
idealized model systems. However details of the models determine the
transition temperature or the absolute amplitudes. Similar examples could be
given in many other areas, for example the mechanical response of bulk
solids, thin films or biological membranes are to a large extent governed by
a small number of universal models but constitutive parameters depend
crucially on the details of the underlying microstructure. In a computer
simulation, in principle it would be possible to study systems on huge
length scales and for long times (i.e. fracture mechanics based on an all
atom simulation, function of a membrane protein in a fully fluctauting
membrane etc.) if all the interactions would be fully treated and infinite
CPU time would be available. While neither of the two is the case, such an
ansatz probably also would produce too much information, obscuring a more
general understanding.

Out of this, for several years now scale bridging or multiscale simulations
methods are developed at many places. They are still in their infancy and
the many different ideas did not converge into one or several generally
accepted and validated schemes. Because of that, this fairly young and
critical area of computational science can benefit greatly from advances in
mathematics. Conversely, emerging computational experience on truly
multiscale systems can serve as a great stimulus to mathematical
understanding, which at present remains at its most thorough for two-scale
systems (as treated, e.g., in classical and stochastic homogenization theory
or Gamma-convergence).

Examples of truly multiscale systems include biological ion channels,
proteins, emulsions, functional materials and quantum dots. They require new
methods to address challenges such as hierarchies of structual organisation,
fluctuating (electrostatic) fields, simultaneous treatment and
interdependencies of very short and very long range interactions, and
approximate Hamiltonians to model dynamics and reactivity of tens of
thousands of atoms. In order to achieve this coupling schemes between
different scales have to be developed which includes systematic coarse
graining strategies, appropriate interaction potentials and force fields,
and methods to link studies on different scales and tune the resolution of
the computer model to coarser and finer resolution as needed. Ultimately
such schemes have to include classical as well as quantum methods. All this
requires new approaches beyond conventional computer modeling.

The workshop aims to address a number of exemplary questions. How does one
parameterize coarse grained interaction potentials for bonded and nonbonded
interactions? The latter is especially delicate for soft matter, because of
the huge size of the molecules. What is the best point or regime in
parameter and phase space to hand over from one to another level of
description? How do errors propagate from one level to the next and what are
the consequences when one wants to finegrain again? How specific or
transferable are models and methods or are there general strategies to
follow? Do we have strategies and general criteria for validation beyond
trivial tests? Coarse graining means mapping of scales, but how does this
work for nonequilibrium systems and time scales, i.e., for studying
dynamics? All these questions will be addressed and discussed in terms of
basic concepts as well as specific applications.

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