8 research outputs found
A weak characterization of slow variables in stochastic dynamical systems
We present a novel characterization of slow variables for continuous Markov
processes that provably preserve the slow timescales. These slow variables are
known as reaction coordinates in molecular dynamical applications, where they
play a key role in system analysis and coarse graining. The defining
characteristics of these slow variables is that they parametrize a so-called
transition manifold, a low-dimensional manifold in a certain density function
space that emerges with progressive equilibration of the system's fast
variables. The existence of said manifold was previously predicted for certain
classes of metastable and slow-fast systems. However, in the original work, the
existence of the manifold hinges on the pointwise convergence of the system's
transition density functions towards it. We show in this work that a
convergence in average with respect to the system's stationary measure is
sufficient to yield reaction coordinates with the same key qualities. This
allows one to accurately predict the timescale preservation in systems where
the old theory is not applicable or would give overly pessimistic results.
Moreover, the new characterization is still constructive, in that it allows for
the algorithmic identification of a good slow variable. The improved
characterization, the error prediction and the variable construction are
demonstrated by a small metastable system
Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics
We consider complex dynamical systems showing metastable behavior but no
local separation of fast and slow time scales. The article raises the question
of whether such systems exhibit a low-dimensional manifold supporting its
effective dynamics. For answering this question, we aim at finding nonlinear
coordinates, called reaction coordinates, such that the projection of the
dynamics onto these coordinates preserves the dominant time scales of the
dynamics. We show that, based on a specific reducibility property, the
existence of good low-dimensional reaction coordinates preserving the dominant
time scales is guaranteed. Based on this theoretical framework, we develop and
test a novel numerical approach for computing good reaction coordinates. The
proposed algorithmic approach is fully local and thus not prone to the curse of
dimension with respect to the state space of the dynamics. Hence, it is a
promising method for data-based model reduction of complex dynamical systems
such as molecular dynamics
Spectral Properties of Effective Dynamics from Conditional Expectations
The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting
Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms
In this paper, we study Jarzynski's equality and fluctuation theorems for
diffusion processes. While some of the results considered in the current work
are known in the (mainly physics) literature, we review and generalize these
nonequilibrium theorems using mathematical arguments, therefore enabling
further investigations in the mathematical community. On the numerical side,
variance reduction approaches such as importance sampling method are studied in
order to compute free energy differences based on Jarzynski's equality.Comment: journal versio
Consistency and Convergence of Non-parametric Estimation of Drift and Diffusion Coefficients in SDEs from Long Stationary Time-series
We study the efficiency of non-parametric estimation of stochastic differential equations driven by Brownian motion (i.e. diffusions) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients for SDEs. These estimators involve time- and space-discretization parameters for computing discrete analogs of expected values from discretely-sampled stationary data. Number of observational points is the third important computational parameter. Next, we derive bounds for the asymptotic behavior of L2 errors for the drift and diffusion estimators. The asymptotic behavior is characterized when the number of observational points becomes infinite and observational time-step and bin size for spatial discretization of drift and diffusion coefficients tend to zero. Using our asymptotic analysis we are able to obtain practical guidelines for selecting computational parameters. Finally, we perform a series of numerical simulations which support our analytical investigation and illustrate practical guidelines for selecting near-optimal and computationally efficient values for computational parameters
Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics
Many interesting rare events in molecular systems, like ligand association, protein folding or conformational changes, occur on timescales that often are not accessible by direct numerical simulation. Therefore, rare event approximation approaches like interface sampling, Markov state model building, or advanced reaction coordinate-based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches. How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so we also allow for understanding deep connections between the different approaches
Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics
Many interesting rare events in molecular systems, like ligand association, protein folding or conformational changes, occur on timescales that often are not accessible by direct numerical simulation. Therefore, rare event approximation approaches like interface sampling, Markov state model building, or advanced reaction coordinate-based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches. How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so we also allow for understanding deep connections between the different approaches.
Keywords:
molecular dynamics; eigenproblem; effective dynamics; Galerkin method; variational approach; Markov state model; reaction coordinat