60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Refine
Document Type
- ZIB-Report (3)
Language
- English (3)
Has Fulltext
- yes (3)
Is part of the Bibliography
- no (3)
Keywords
- Frobenius Perron operator (1)
- Koopman operator (1)
- Markov o (1)
- Markov operator (1)
- almost invariant aggregates (1)
- aperiodic (1)
- asymptotically periodic (1)
- asymptotically stable (1)
- conformation (1)
- conformational dynamics (1)
Institute
- ZIB Allgemein (3)
The statistical behavior of deterministic and stochastic dynamical systems may be described using transfer operators, which generalize the notion of Frobenius Perron and Koopman operators. Since numerical techniques to analyze dynamical systems based on eigenvalues problems for the corresponding transfer operator have emerged, bounds on its essential spectral radius became of interest. This article shows that they are also of great theoretical interest. We give an analytical representation of the essential spectral radius in $L^{1}\!(\mu)$, which then is exploited to analyze the asymptotical properties of transfer operators by combining results from functional analysis, Markov operators and Markov chain theory. In particular, it is shown, that an essential spectral radius less than $1$, constrictiveness and some weak form'' of the so--called Doeblin condition are equivalent. Finally, we apply the results to study three main problem classes: deterministic systems, stochastically perturbed deterministic systems and stochastic systems.
Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of $T$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.
The topic of the present paper bas been motivated by a recent computational approach to identify chemical conformations and conformational changes within molecular systems. After proper discretization, the conformations show up as almost invariant aggregates in reversible nearly uncoupled Markov chains. Most of the former work on this subject treated the direct problem: given the aggregates, analyze the loose coupling in connection with the computation of the stationary distribution (aggregation/disaggregation techniques). In contrast to that the present paper focuses on the inverse problem: given the system as a whole, identify the almost invariant aggregates together with the associated transition probabilites. A rather simple and robust algorithm is suggested and illustrated by its application to the n-pentane molecule.