|The methods of atomistic
simulation at the level of Molecular Dynamics (MD) are becoming ever more
powerful, both due to the increasing amounts of computational power available,
and to the improvement of the algorithms that are used. Nevertheless there
are many contexts where the length and time scales arising in the physical,
chemical and biological questions of interest remain entirely beyond the
capabilities of MD simulations, and will remain out of reach for the foreseeable
future. Thus there is a need for reduced-order descriptions. Reduced-order
models can provide insights and numerical simulations for larger length
scales and longer time scales, but at the cost of discarding some level
of detail. The issue is to retain the appropriate level of information
so as to be able to address the fundamental scientific questions of interest
in a meaningful way.
One possible avenue to such reduced-order descriptions is provided by the methods of continuum mechanics. If a problem can be cast into the general framework of a continuum model there is a large, mature and extremely powerful body of work involving mathematical analysis and computational methods that can often be applied to great effect. However the traditional domains of application of continuum models have been macroscopic, at length scales of millimetres or more. In recent years there has been much activity in the mechanics and applied mathematics communities aimed toward applying classical continuum mechanics techniques at smaller and smaller scales, from micro-structure and models of grain structure in crystals, down to truly atomistic scales.
The main mathematical issues in relating continuum models to atomistic systems arise in identifying the appropriate averaging or homogenization techniques to obtain effective models at the dominant length and time scales. Discrete atom locations must be appropriately "smeared" in space to obtain continuous displacement fields, while very fast small-amplitude oscillations must be appropriately "smeared" in time to leave only larger amplitude, longer time-scale motions. Appropriate notions of stress (i.e. the appropriate continuum measure of the forces acting), and of strain (i.e. the appropriate continuum measure of the deformations arising) must be identified. Most importantly the constitutive relations that relate the stresses and strains within any continuum model should be derived from atomistic level information.
The talks and discussions at this conference will concern the derivation of continuum solid mechanics models from atomistic MD descriptions when the system has one of two special features: either the system is rather long and thin, for example a DNA molecule, or the system is rather flat and thin, for example a thin film of only a few atoms thickness. In mathematical terms such systems can be expected to have an asymptotic limit in which the ratio of the small and large length scales is sent to zero. The associated continuum models have (respectively) only one or two space dimensions plus time as their independent variables. In particular they are not space-filling which gives them distinctly different features from systems involving bulk gases, liquids or solids. Space-filling systems are often treated using the methods of statistical mechanics under assumptions such as isotropy and homogeneity, and assuming very simplified atomic/molecular interactions. In contrast, continuum models with fewer space variables are intrinsically anisotropic, account must be made for the interaction between the continuum and its surrounding environment, and comparatively large overall motions of the continuum may occur. Accordingly, rather detailed information must be extracted from the MD description in order to obtain an appropriate continuum model.
Our primary example of a long thin molecule will be DNA. DNA is such an important molecule that understanding its large scale or tertiary structures is perhaps in of itself sufficient justification for the conference. Certainly the case of developing mesoscopic continuum models for the deformations of DNA on the scale of a few hundred base pairs or more will be a central line of discussion. However the issues underlying the derivation of such continuum rod models are in principle similar for many other long, stiff polymers, for example alpha-helical proteins, or coiled-coiled molecules. In contrast the study of globular proteins and other flexible molecules with no mechanically distinguished direction, and no distinctive short-scale structural properties is beyond the scope of discussion in this meeting.
Similar problems of seeking continuum-type theories that inherit atomic level information arise in the case of thin films. One scheme that has recently been proposed is an asymptotic approach based on large width/thickness of the film. In this approach, weak convergence methods are used to identify all well-defined averages of atomic positions in the limit T/W -> 0. These are then selected as the variables of continuum theory. Atomic degrees of freedom are systematically eliminated via constrained energy minimization. This yields a kind of Cosserat membrane theory, whose energy density is evaluated by a certain atomic level computation (e.g. DFT) done under homogeneous conditions. The novelty of this approach is that the continuum variables are not postulated beforehand but emerge from the theory in a natural way. There is a need to compare these methods to those used in the context rod models of DNA, and to tackle the much more difficult problem of disparate time scales.
These theoretical developments in thin films are aimed at providing a mathematical and computational framework to complement the huge activity on the experimental side in microelectromechanical systems (MEMS). Theory, new concepts and computational methods are needed to guide the development of microscale (and emerging nanoscale) pumps, valves, vehicles for flight, and machines proposed for noninvasive biomedicine. One particularly topical area for both theory and experiment in this direction is the study of carbon sheets and nanotubes.
A discussion of constraints will be a third theme of the meeting. Imposing bond length constraints is one classic technique of obtaining a reduced order model in MD simulations. Similarly, entirely within the framework of continuum models, pointwise constraints such as inextensibility or incompressibility are sometimes adopted in order to eliminate some of the very fast modes of deformation. However both in the MD and continuum settings there remain several questions both about the precise nature of such an approximation, and the best way to handle the numerics arising for such differential algebraic equations.
The basic goal of the conference is to address the following question: if you believe an atomistic MD model, can a continuum version be derived that provides improved understanding, or larger, longer or faster numerical simulations? For reasons of limited time, and to keep a strong focus, various important related issues will not be discussed at any length. Specifically there is much current activity in actually deriving the potentials used in MD from quantum mechanical considerations, and debate as to the accuracy of the potentials that are currently being used, but we will not address such questions. Similarly the important issue of experimental verification of models will be discussed only in passing. The objective of this one week meeting is to address ways of bridging the gap between an MD description, which for the sake of discussion we will take to be our "truth", and a related reduced-order continuum mechanics based model.
There appears to be a growing consensus that an increased
interaction between mathematicians and computational chemists would be
extremely beneficial for all concerned, and could lead to significant
scientific advances (see for example the report : Mathematical
Challenges from Theoretical/Computational Chemistry, of the US
National Research Council, 1995). The idea of this conference is to
assemble, in a setting conducive to in-depth discussions, both senior
and younger researchers in mathematics and computational chemistry who
are interested in trying to bridge the gaps between the physical
length and time scales usually considered by the two disciplines. The
immediate goal is to foster increased understanding of different
viewpoints and areas of expertise, and hopefully to seed future
collaborations. The conference will seek to instigate progress through
the opportunity of having relaxed, detailed discussions on a rather
narrowly defined, but important choice of topics.
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