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We consider any dynamical system that starts from a given ensemble ofconfigurations and evolves in time until the system reaches a certain fixedstopping criterion, with the mean first-passage time the quantity of interest.We present a general method, Projective Dynamics, which maps themulti-dimensional dynamics of the system onto an arbitrary discrete set ofstates ${\zeta_k}$, subject only to the constraint that the dynamics isrestricted to transitions not further than the neighboring states $\zeta_{k\pm1}$. We prove that with this imposed condition there exists a master equationwith nearest-neighbor coupling with the same mean first-passage time as theoriginal dynamical system. We show applications of the method for Brownianmotion of particles in one and two dimensional potential energy landscapes andthe folding process of small bio-polymers. We compare results for the meanfirst passage time and the mean folding time obtained with the ProjectiveDynamics method with those obtained by a direct measurement, and where possiblewith a semi-analytical solution.

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