Refine
Language
- English (23) (remove)
Keywords
- metastability (4)
- rare events (3)
- cluster analysis (2)
- dynamical systems (2)
- molecular dynamics (2)
- Monte Carlo simulation (1)
- conformation dynamics (1)
- conformational space (1)
- hypercubes (1)
- rate matrix (1)
Project
- A4 (23) (remove)
Application Area
- A (23)
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.
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 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.
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.
e propose an algorithm for the fast and efficient simulation of polymers represented by
chains of hard spheres. The particles are linked by holonomic bond constraints.
While the motion of the polymers is free (i.e., no collisions occur) the equations
of motion can be easily integrated using a collocation-based partitioned Gauss-Runge-Kutta method.
The method is reversible, symplectic and preserves energy. Moreover the numerical scheme allows the integration using much longer time steps than any explicit integrator such as the popular Verlet method. If polymers collide the point of impact can be determined to arbitrary precision by simple nested intervals. Once the collision point is known the impulsive contribution can be computed analytically. We illustrate our approach by means of a suitable numerical example.
We study Balanced Truncation for stochastic differential equations. In doing so, we adopt ideas from large deviations theory and discuss notions of controllability and obervability for dissipative Hamiltonian systems with degenerate noise term, also known as Langevin equations. For partially-observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.
or stable linear input-output systems, the method of balanced truncation (B.C.~Moore, {\em IEEE Trans. Auto. Contr.} {\bf AC-26}, 17--32, 1981) consists in finding a coordinate transformation such that modes which are least sensitive to the external input (controllability) also give the least output (observability) and therefore can be neglected. A drawback is that projecting the original equations of motion onto the subspace of interest typically fails to preserve the problem's physical structure, e.g., if the original equations are of second-order form or Hamiltonian. For Hamiltonian systems, a natural way of restricting a system to a subspace is by means of constraints, and we show, employing singular perturbation arguments, that balanced truncation can be done in a structure-preserving fashion. The thus obtained reduced Hamiltonian system preserves stability and passivity and satisfies the usual balanced truncation error bound.
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.
For an analysis of a molecular system from a computational statistical thermodynamics point of view, extensive
molecular dynamics simulations are very inefficient. During this procedure, at lot of redundant data is generated. Whereas
the algorithms spend most of the computing time for a sampling of configurations within the basins of the potential energy
landscape of the molecular system, the important information about the long-time behaviour of the molecules is given by
transition regions and barriers between the basins, which are sampled rarely only. Thinking of molecular dynamics trajectories,
researchers try to figure out which kind of dynamical model is suitable for an efficient simulation. This article suggests to
change the point of view from extensive simulation of molecular dynamics trajectories to more efficient sampling strategies
of the conformation dynamics approach.
For the treatment of equilibrated molecular systems in a heat bath we
propose a transition state theory that is based on conformation dynamics.
In general, a set-based discretization of a Markov operator P does not
preserve the Markov property. In this article, we propose a discretization
method which is based on a Galerkin approach. This discretization
method preserves the Markov property of the operator and can be interpreted
as a decomposition of the state space into (fuzzy) sets. The
conformation-based transition state theory presented here can be seen as
a first step in conformation dynamics towards the computation of essential
dynamical properties of molecular systems without time-consuming
molecular dynamics simulations.