Physics and Astronomy Calendar

We describe a mathematical framework for the sampling of path space. This framework, in which the Onsager-Machlup functional plays a central role in defining a probability density on the space of paths, allows a unified viewpoint for the formulation of a number of important problems including signal processing, data assimilation, data interpolation, and the sampling of rare events.

In many molecular models, we would like to understand phenomena that happen on a variety of time scales. The time scales of the motion are a reflection of the free-energy landscape. Typically, the phase space that the system "visits" will be in a basin of one of the many free-energy minima. On short time scales, the system explores the phase space in one of the basins, and this describes the evolution of the "fast degrees of freedom." On a longer time scale, the system will progress over a barrier and into another basin; this progression governs the "slow degrees of freedom." A natural question is whether we can understand, and quantify, the motion of the particles on the longer time scales. In this modeling, we would like to incorporate the effects of the fast degrees of freedom, as we describe the evolution of the slower moving variables.

The approach taken here is to employ a Langevin equation in path space, calculated from the Onsager-Machlup functional, to sample these rare events. In particular, we look at paths in phase space, conditioned to cross a relevant free-energy barrier. Such paths allow us to investigate the crossing of such barriers in the presence of representative thermal motions.

The particular system we are studying is a collection of 89 particles in a two dimensional container. The 10 X 9 lattice holds one vacancy. We are investigating the infrequent movement of a particle into the vacancy as it is buffeted by the thermal motions of the other particles. We use the motions of the particle, as described by the space of sampled paths, to extract an effective single-particle potential particle from the many-body problem.