The workshop this week brought together specialists from four different groups:

• Those who wish to study the long-term simulation of stochastic differential equations from a numerical-analytic point of view.
• Those who develop and use symplectic integrators, in particular in the context of molecular dynamics.
• Those who wish to use stochastic methods and/or symplectic integrators to do molecular dynamics simulations
• Those who develop and use kinetic Monte-Carlo situations. In particular, they would like to study a network of chemical reactions and determine rate constants.

A process is called stochastic (originating from a Greek root meaning "aim" or "guess") if it is completely random: any given state gives no information about the next state. Many processes in nature are stochastic. Stochastic integrals, and stochastic differential equations, are mathematical tools that have been developed by Norbert Wiener and others to study these phenomena.

A summary statement of the general point of view at this workshop is:

If you need to simulate a stochastic system over a long period of time, e.g. to do sampling in the context of molecular dynamics or to understand the long-time properties of a chemical kinetic network, what are the numerical methods that you can use? Which properties should one method have to win over another?

Some particular questions that interested all participants are

• Can stochastic differential equations represent a viable alternative to thermostats such as Nosé-Hoover or Andersen for the simulation of systems in ensembles other than the microcanonical one?
• How can one guarantee the ergodicity of integrators for such stochastic differential equations and assess their rates of convergence, accuracy, and efficiency?
• How can one incorporate holonomic constraints into contexts such as those just described? [Here a system is holonomic if it is highly over-determined.]

The method of stochastic simulation is used to sample the states for the given system. One of the key tools in this subject is computer simulation of the system. One takes a chemical reaction that might have two-second duration and breaks it up into tiny steps (perhaps one femto-second--or one millionth of one billionth of a second). One uses techniques of numerical analysis to iterate the reaction two million billion times (to simulate a two-second reaction) in order to numerically predict a possible outcome state.

The complication arises that there are so many iterations in this numerical procedure that we may as well be dealing with infinitely many steps. Most classical numerical analysis techniques are designed for finitely many steps; in particular, the known error-control methods depend on knowing in advance the number of steps in the process. So new methods must be devised to get meaningful results from the computer work being described here.

It is useful to devise shortcuts to make these calculations more feasible. As matters stand, a typical calculation can take two or more months--even on a large cluster of powerful computers. One also would like to find simple examples that provide good tests for the methods.

Typical computer output consists of numerics and high-dimensional pictures of the data. The data typically illustrates the state that the system is in after the chemical reaction has run. Then there is a great deal of work to interpret the computer output and convert it to meaningful mathematics.

This workshop was unusual in several respects. Participants were invited to contribute open problems and questions before the workshop began, and these were posted on the workshop Web site. These included specific problems on which there is hope of making some progress during the workshop, as well as some more ambitious projects that may influence future activity for some time to come.

Participants in this workshop came from backgrounds in probability theory, differential equations, computing, and several other fields. One of the points of the workshop was to get people with different sets of tools to interact and to work together productively. The small working groups were useful vehicles for promoting new collaboration, and many new ideas were generated.