Computer Simulations

Traditionally, research has been divided into two categories: "Experiment" and "Theory". Experiments provide, through observation of physical reality, insight into processes of unknown mechanism, allowing deductions regarding the physical nature of the observed phenomenon. Theory, on the other hand, works mathematically from a set of completely specified assumptions and connects them with properties of interest; theory alone does not provide information on physical systems, but does retain a mechanistic connection between assumption and result.

A new category is "Computer Experiments", which assumes a basic description of microscopic physical nature and then simulates the macroscopic phenomena consistent with this "nature". Typically, everything is known about the microscopic processes, but nothing is known a priori about the way in which macroscopic observables depend on these processes. The result can be as unexpected as in a true experiment. The advantage of simulations lies inthe possibility to inquire about the effect of microscopic features on macroscopic observables in a systematic way, especially where no theoretical connection is known.

Thus one should be careful to distinguish between reality, models of reality, and the behavior of these models in relation to real behavior. Experiments yield measurements of real behavior. Theory invokes a model of reality and a number of approximations to render the behavior of the model mathematically tractable. Simulations invoke a model and then attempt to follow the "pseudo-real" behavior of that model. Thus simulations have, historically, served two purposes:

  1. to test theories - the assumptions invoked within the theory are tested by comparison of theoretical behavior to simulated behavior, for the same identical model.
  2. to test models - the "goodness" of a model as a representation of reality is tested by comparison of simulated behavior to real observed behavior.

The computer simulations used by engineers to estimate material properties involve many atoms and molecules and are usually performed assuming the Laws of Classical Mechanics. The sytems simulated are conservative, meaning that state functions like energy are conserved. In this situation, the internal energy of a system of particles or molecules consists of a kinetic energy contribution, which depends only on the velocities or momenta of the particles, and a potential energy contribution, which depends only on the position (coordinates) of the particles. To compute the potential energy, one uses an empirical force field which provides analytical equations to estimate the interactions energies associated with a particular configuration of particles (or molecules). Computer experiments fall into four classes:

  1. Molecular Mechanics (aka molecular statics) potential energy is minimized; there is no ensemble averaging.
  2. Lattice Dynamics: free energy is minimized, with configurational fluctuations limited to strictly harmonic motion.
  3. Molecular Dynamics: a deterministic set of differential equations is solved numerically. Usually, one obtains a time evolution at constant energy, consistent with the integration of equations of a conservative system.
  4. Monte Carlo: a set of configurations are sampled from phase space in a stochastic, non-deterministic manner, corresponding to a numerical integration very similar to molecular dynamics. Usually, this sampling is weighted according to the Boltzmann factor in order to produce canonical averages.

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Last modified 6/20/97 - GCR