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Mapping the dynamics of complex multi-dimensional systems onto a discrete set of states conserving mean first passage times: a Projective Dynamics approach

We consider any dynamical system that starts from a given ensemble of configurations and evolves in time until the system reaches a certain fixed stopping criterion, with the mean first-passage time the quantity of interest. We present a general method, Projective Dynamics, which maps the multi-dimensional dynamics of the system onto an arbitrary discrete set of states ${ζ_k}$, subject only to the constraint that the dynamics is restricted to transitions not further than the neighboring states $ζ_{k\pm 1}$. We prove that with this imposed condition there exists a master equation with nearest-neighbor coupling with the same mean first-passage time as the original dynamical system. We show applications of the method for Brownian motion of particles in one and two dimensional potential energy landscapes and the folding process of small bio-polymers. We compare results for the mean first passage time 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.