Approximating invariant densities of metastable systems

We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs.Comment: 19 pages, 6 figure

Eigenvalue Bounds on Restrictions of Reversible Nearly Uncoupled Markov Chains

AbstractIn this paper we analyze decompositions of reversible nearly uncoupled Markov chains into rapidly mixing subchains. We state upper bounds on the 2nd eigenvalue for restriction and stochastic complementation chains of reversible Markov chains, as well as a relation between them. We illustrate the obtained bounds analytically for bunkbed graphs, and furthermore apply them to restricted Markov chains that arise when analyzing conformation dynamics of a small biomolecule

Sequential Change Point Detection in Molecular Dynamics Trajectories

Motivated from a molecular dynamics context we propose a sequential change point detection algorithm for vector-valued autoregressive models based on Bayesian model selection. The algorithm does not rely on any sampling procedure or assumptions underlying the dynamics of the transitions and is designed to cope with high-dimensional data. We show the applicability of the algorithm on a time series obtained from numerical simulation of a penta-peptide molecule

Rare events, escape rates and quasistationarity: some exact formulae

We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given

Conformation Dynamics

This article surveys the present state of the transfer operator approach to the effective dynamics of metastable dynamical systems and the variety of algorithms associated with it

Identification of Biomolecular Conformations from Incomplete Torsion Angle Observations by Hidden Markov Models

We present a novel method for the identification of the most important conformations of a biomolecular system from molecular dynamics or Metropolis Monte Carlo time series by means of Hidden Markov Models (HMMs). We show that identification is possible based on the observation sequences of some essential torsion or backbone angles. In particular, the method still provides good results even if the conformations do have a strong overlap in these angles. To apply HMMs to angular data, we use von Mises output distributions. The performance of the resulting method is illustrated by numerical tests and by application to a hybrid Monte Carlo time series of trialanine and to MD simulation results of a DNA-oligomer