One realm where computer simulations are beginning to find great use is in the field of understanding the behavior and interaction between atoms and/or molecules in various circumstances and conditions. Molecular dynamics is one such method for understanding these interactions.

The concept of a molecular dynamics simulation begins with a simple combination of Newton’s laws of motion and a Taylor series expansion of a particle’s position r(t+dt):

From this form, the formula can manipulated to provide various algorithms for computing r(t) at the next timestep. The simplest form, called the Verlet algorithm, is given by summing r(t+dt) and r(t-dt):

which depends on r(t) (the current timestep), r(t-dt) (the previous timestep), and force (mass is neglected for units purposes).

Velocity can also be determined by taking the difference of r(t+dt) and r(t-dt) (not shown).

Additionally, higher-order algorithms can be used so that the time increment can be increased with a lesser significant loss in accuracy.

The final element that is necessary to compute r for the next timestep is the force component. Most programs use simple (or complicated) potential equations to determine this force. Seeing as

any potential can easily be evaluated for each x, y and z direction and used in the Verlet algorithm to find the next position of each particle.

Since the force calculation generally requires at least a pairwise interaction, the force calculation is generally the longest step in an MD simulation seeing as it requires looping over pairs of atoms/molecules. The total number of iterations for such loops is equal to n(n-1)/2 for n molecules, which scales as the square of n. For this reason, potentials for atoms with a greater separation than some arbitrary cutoff value are usually neglected. Certain optimization techniques such as coarse graining and nearest-neighbor lists can also be used to avoid superfluous iterations. Coarse graining can be applied to reduce the algorithm to order n.

Also, to be useful, the system running a simulation must be observable. To this end, simple values such as temperature (ie. total kinetic energy), can be computed easily by summing all velocities, potential energy can be summed from the force calculation step and total energy can be taken as the sum of these values. Note that a good program should conserve total energy. More complicated parameters such as the diffusion constant can also be evaluated.

The advantages of MD simulations (and other computer modeling methods) is that the behavior of atoms or molecules in some structure can be modeled without going through the pains of setting up a physical experiment. This allows for enhanced control of the situation. For example, the experimenter can implement his own defects into a lattice (say, remove an atom/molecule or insert an impurity). Additionally, the experimenter can simulate bombarding a structure with particles or scale temperature in the middle of a simulation. Computer simulations are also useful for observing behavior in transition. For example, the transition between various solid phases in metal lattices can be observed by taking finite timesteps and observing atoms in transition.

As methods become more precise and efficient and computers become more powerful, computer simulations in this field will become much more effective. Already, massively parallel supercomputers are in use for such techniques, and as technology improves, computers will contribute more to this field.

sources:
http://www.cse.scitech.ac.uk/ccg/software/Democritus/Basic/Contents.html

images taken from Wikipedia.