Mapping the dynamics of multi-dimensional systems onto a nearest-neighbor coupled discrete set of states conserving the mean first-passage times: a projective dynamics approach

Abstract

We consider classical and semi-classical dynamical systems that start from a given ensemble of configurations and evolve in time until the systems reach a certain fixed stopping criterion, with the mean first-passage time (MFPT) being the quantity of interest. We present a method, projective dynamics, which maps the dynamics of the system onto an arbitrary discrete set of states {ζk}, subject to the constraint that the states ζk are chosen in such a way that only transitions not further than to the neighboring states ζk ± 1 occur. We show that with this imposed condition there exists a master equation with nearest-neighbor coupling with the same MFPT and residence times as the original dynamical system. We show applications of the method for the diffusion process of particles in one- and two-dimensional potential energy landscapes and the folding process of a small biopolymer. We compare results for the MFPT and the mean folding time obtained with the projective dynamics method with those obtained by a direct measurement, and where possible with a semi-analytical solution.