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.

Comparison of D-Wave Quantum Annealing and Classical Simulated Annealing for Local Minima Determination

“…At reasonably low temperatures, the model is still useful, and one can then consider the paths from one LV to another and the merging of BoAs as the temperature gets larger. Novotny and collaborators have performed work on the algorithms associated with such transitions [27]- [29]. Computational methods for classical MC systems to understand the highdimensional transitions between LVs include Transition Path Theory (TST) [30], the string or elastic band method for calculating the most probable paths at low temperatures between LVs [31][32] for continuum systems and MC with Absorbing Markov Chains (MCAMC) [27] for discrete models.…”

An iterative aggregation and disaggregation approach to the calculation of steady state distributions of continuous processes

“…To reach a convergence better than = 10 −5 the procedure has to be repeated eight more times, leading to x (9) = 0.1959 0.2041 0.2123 0.1102 0.2775 .…”

“…(2) is an expression of p(j, t + dt|i, t)p(i, t) = p(j, t + dt; i, t), where p(j, t + dt; i, t) is the joined probability of finding the system at time t in state i and at time t + dt in state j. More details concerning the notation of the mapping procedure can be found in[9,10].…”

Improved coarse-graining of Markov state models via explicit consideration of statistical uncertainty

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“…A major challenge in coarse-graining MSMs is dealing with uncertainty. The most common methods for coarse-graining MSMs are Perron Cluster Cluster Analysis (PCCA) [10,11] and PCCA+ [12], though a number of new methods have been published recently [7,[13][14][15][16]. Most all of these methods operate on the maximum-likelihood estimate of the transition probability matrix and do not account for statistical uncertainty in these parameters due to finite sampling.…”