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This article is concerned with the averaging principle and its extensions
for stochastic dynamical systems with fast and slow degrees of
freedom. It is demonstrated how the \conventional" averaging principle
results from asymptotic multiscale analysis, how one can construct
an indicator for its (in-)appropriateness, and how, if inappropriate, it
may be extended into an improved approximation. The conventional
scheme contains averages over the entire accessible state space of the
fast degrees of freedom and may thus fail if these fast degrees of freedom
exhibit long-term (auto-)correlations. In contrast, the improved
scheme combines several conditional averages with a Markov jump process
that is designed to represent the
ipping process between the
conditional averages and thus incorporates the important long-term
correlations. All important steps of the derivation are illustrated by
numerical experiments. Application to problems from molecular dynamics
is discussed.
We consider a particle constrained to a submanifold ? of the
configuration space Rm. Using that the notion of holonomic constraints coincides
with integrability of the corresponding vector field, we show how this property
naturally determines local coordinates on ?. We give a rigorous justification for
the calculation of the mean force along a constrained coordinate, and we provide
a concise geometrical interpretation of the different contributions to the mean
force in terms of the unconstrained vector field and extrinsic curvature properties
of ? in Rm. Our approach gives rise to a Hybrid Monte-Carlo based algorithm
that can be used to compute the mean force acting on selected coordinates in the
context of thermodynamic free energy statistics.
In 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.
We present a formal procedure for structure-preserving model reduction of linear second-order control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems.
It turns out that the Hamiltonian structure, stability and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.
We propose a nonequilibrium sampling method for computing free energy profiles along a given reaction coordinate. The method consists of two parts: a controlled Langevin sampler that generates nonequilibrium bridge paths conditioned by the reaction coordinate, and Jarzynski’s formula for reweighting the paths. Our derivation of the equa- tions of motion of the sampler is based on stochastic perturbation of a controlled dissipative Hamiltonian system, for which we prove Jarzynski’s identity as a special case of the Feynman-Kac formula. We illustrate our method by means of a suitable numerical example and briefly discuss issues of optimally choosing the control protocol for the reaction coordinate.
We revisit the problem of the linear response of a constrained mechanical system. In doing so we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville’s theorem and the Green-Kubo relations naturally follow.
We propose a robust and efficient numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant measure is precisely the cell approximation of the Boltzmann distribution of the original process. Moreover the resulting jump process preserves the detailed balance property of the original stochastic process.
he generalized Langevin equation is useful for modeling a wide
range of physical processes. Unfortunately its parameters,
especially the memory function, are difficult to determine for
nontrivial processes. In this paper, relations between a
time-discrete generalized Langevin model and discrete multivariate
autoregressive (AR) or autoregressive moving average models (ARMA)
are established. This allows a wide range of discrete linear
methods known from time series analysis to be applied. In
particular, the determination of the memory function {\it via} the
order of the respective AR or ARMA model is addressed. The method
is illustrated on a one-dimensional test system and subsequently
applied to the molecular dynamics of a biomolecule which exhibits
an interesting relationship between the solvent method used, the
molecular conformation and the depth of the memory.
Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics
of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have
seen widespread use in recent years. This approach has many appealing characteristics compared
to straightforward molecular dynamics simulation and analysis, including the potential to mitigate
the sampling problem by extracting long-time kinetic information from short trajectories and the
ability to straightforwardly calculate expectation values and statistical uncertainties of various
stationary and dynamical molecular observables. In this article, we summarize the current state of
the art in generation and validation of MSMs and give some important new results. We describe
an upper bound for the approximation error made by modeling molecular dynamics with an MSM
and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast
to previous practice, it becomes clear that the best MSM is not obtained by the most metastable
discretization, but the MSM can be much improved if non-metastable states are introduced near
the transition states. Moreover, we show that it is not necessary to resolve all slow processes
by the state space partitioning, but individual dynamical processes of interest can be resolved
separately. We also present an efficient estimator for reversible transition matrices and a robust
test to validate that an MSM reproduces the kinetics of the molecular dynamics data.
Markov State Models (MSMs) have become the tool of choice to analyze large amounts of molec-
ular dynamics data by approximating them as a Markov jump process between suitably predefined
states. Here we investigate ”Core Set MSMs”, a new type of MSMs that builds on metastable core
sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core
Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transi-
tion Path Theory (TPT). As a result, Core Set MSMs can now be used to extract phenomenological
rate constants between the metastable sets of the system and to approximate the evolution of certain
key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed
and illustrated on a model potential and the torsion angle dynamics of Alanine dipeptide.