Computing the density of paths in complex systems

Abstract

Trajectories of duration tau joining two points q(0) and q(1) in the configuration space of a classical system satisfy Hamilton's principle: they are stationary points of the classical action. The second variation (fluctuations) of the action around the stationary point signals whether the latter is or not a minimum and delivers the density in trajectory space around the points q(0) and q(1). This concept of density of paths is of great importance in semiclassical quantum theory, where it weights the contribution to the propagator from the single classical trajectories. In this paper, two algorithms based on the concepts of molecular dynamics simulation are introduced for computing the density of paths, also called van Vleck [Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928)] determinant. Examples for realistic systems are presented, together with a suggestion about possible applications in the field of rare events in physics and chemistry.