A large part of modern computational chemistry is based on the solution of the equations of classical mechanics for many- body systems. For these problems, the stand ard numerical integration techniques found in classic textbooks only provide a simple framework for application and analysis. This is because most of the theory of numerical analysis provides criteria for long-time stability for these smaller systems wit h regular dynamics. Nevertheless, it is clear that for most many-body chemical systems, the differential equations may have chaotic solutions primarily. Thus, it is of little use to talk about the stability of an individual trajectory for long times, particularly when methods with some stochastic elements (e.g., Langevin dynamics) are involved.
A necessary mathematical advance is an understanding of error estimation and long-time stability for chaotic systems. In fact, one seeks a way of characterizing the accuracy in some statistical sense from such a simulation, or collection of simulations , since an individual trajectory's details are certainly predicted incorrectly. The classical theory of error estimates in numerical analysis is clearly inadequate for most purposes and the theory of long-time stability for complex systems is rat her at its infancy (Stuart and Humphries, 1994).
A most striking example is in the old simulations of hard-sphere molecular dynamics showing trajectories for times encompassing many, many collisions of a dilute gas (e.g., Alder and Wainwright, 1970). One can easily show that the trajectories were num erically inaccurate beyond the limit of machine precision, even after only 10 collisions. Nevertheless, when the velocity correlation function was computed, it exhibited a long-time tail that persisted out to 30 collision times, and this long-time tail's form as well as amplitude agreed precisely with kinetic theory calculations. Thus, statistical properties can be accurate even when individual trajectories are entirely incorrect.
A similar problem enters when one considers the use of "stochastic dynamics" methods in simulation (e.g., pseudorandom forces are added to the systematic force to mimic a thermal reservoir). When only a subsystem of a larger system (e.g., biomolecule p lus solvent) is being studied, the appropriate equations of motion are stochastic or Langevin equations. The most familiar example is the diffusion of biomolecules in water. The theory of quadratures for such stochastic equations has long been of intere st, but compre- hensive analyses of associated error estimates have been developed mainly from a relatively simple point of view, such as via analysis of harmonic oscillators (Pastor et al., 1988; Tuckerman and Berne, 1991).
However, there are many important mathematical issues that arise in this connection. What is the meaning of "error analysis" in this stochastic framework? Is it appropriate to compare results to parallel approaches, such as molecular dynamics, where t he stochastic forces are zero? Are asymptotic results as the time step approaches zero relevant to practical problems? As discussed elsewhere in this report, the complexity of biomolecular systems involves multiple conformations and numerous biologicall y relevant pathways. And, given the severity of the time step problem in molecular dynamics (see pages 54-58), how can qualitative and quantitative theories for evaluating simulation results be merged? Clearly, there is a strong need for both aspects, bu t one can imagine that different numerical models in combination with different integration or propagation methods could be designed to address various aspects of the dynamical problems. Therefore, an organized theory for simulation evaluation, including local error analysis, long-time behavior, and some kind of broader or "global" framework for evaluation, would be extremely valuable. Such a theory could clearly aid in evaluating new models and methods as they arise, identifying the appropriate tests f or the various simulation protocols, and putting in perspective the biological results that may emerge from any computer simulation.
References
Alder, B.J., and T.E. Wainright, 1970, Phys. Rev. A 1:18-21.
Pastor, R.W., B.R. Brooks, and A. Szabo, 1988, An analysis of the accuracy of Langevin and molecular
dynamics algorithms, Mol. Phys. 65:1409-1419.
Stuart, A.M., and A.R. Humphries, 1994, Model problems in numerical stability theory for initial value
problems, SIAM Review 36:226-2571.
Tuckerman, M., and B.J. Berne, 1991, J. Chem. Phys. 95:4389.
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