Automated Model Reduction for Complex Systems Exhibiting Metastability

Abstract

We present a novel method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The novel approach represents the effective dynamics of the full system by a Markov jump process between metastable states and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models with output behavior given by SDEs. The numerical effort of the method is linear in the length of the given time series and quadratic in terms of the number of metastable states. The performance of the resulting method is illustrated by numerical tests and by application to molecular dynamics time series of a trialanine molecule.

Metastable Walking Machines

“…For readers searching for additional references on metastability, we suggest the following small, representative set of the literature on metastable system dynamics (Hanggi et al, 1990;Huisinga, Meyn, & Schutte, 2004;Muller et al, 1997;Gaveau & Schulman, 1998), tools for estimating stochastic stability in discrete cases (Markov chains) (Talkner et al, 1987;Bovier, Eckhoff, Gayrard, & Klein, 2000;Bovier, 2004;Boyd, Diaconis, & Xiao, 2004;Jain & Jain, 1994;Larralde & Leyvraz, 2005;Weber, Kube, Walter, & Deuflhard, 2006), and issues of model order reduction (Horenko, Dittmer, Fischer, & Schutte, 2006;Vijayakumar, D'Souza, & Schaal, 2005;Au, 2004). Additionally, two recommended texts on stochastic processes are (Gardiner, 2004;Kampen, 2007).…”

Optimal Langevin modeling of out-of-equilibrium molecular dynamics simulations

“…The extraction of the optimal Langevin parametrization is carried out within a maximum likelihood approach, a framework profitably used in other previous approaches [7,10,13]. We start by assuming that for a suitable choice of the discretization time interval, dt, the CV's evolution is describable as a Markovian process.…”

“…Several approaches have been developed over the years to perform dimensional reduction in specific contexts [6,7,8,9,10,11,12,13,14,15]. The validity of the overdamped Langevin dynamics is very commonly assumed a priori for describing the dynamical evolution of the reduced system.…”

“…[12] where Kopelevich et al estimate the Langevin drift and diffusion coefficients from short trajectories with different initial conditions. In other commonly-employed approaches the Langevin equation parameters are derived from a maximum likelihood principle [7,10,13]. Specifically, the free energy and diffusion coefficient profiles are chosen in such a way that the time evolution of the collective variables actually observed in the molecular dynamics trajectory has the highest realization probability.…”

“…As shown by Hummer [10], this scheme can also be generalized by allowing for a position-dependent diffusion coefficient. Another powerful related approach is the one of Horenko et al [13] where the evolution of a system is assimilated to a diffusive process in a series of harmonic free energy wells. Transitions between the wells are described as discontinuous "jump" processes, with a suitable transition probability per unit time.…”