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Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

Please always quote using this URN: urn:nbn:de:0297-zib-63822
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  • 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.

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Author:Andreas Bittracher, Péter Koltai, Stefan Klus, Ralf Banisch, Michael Dellnitz, Christof Schütte
Document Type:ZIB-Report
Tag:effective dynamics; embedding; metastability; reaction coordinate; slow dynamics; transfer operator; transition manifold
MSC-Classification:47-XX OPERATOR THEORY / 47Bxx Special classes of linear operators / 47B38 Operators on function spaces (general)
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H35 Computational methods for stochastic equations [See also 65C30]
82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER / 82Cxx Time-dependent statistical mechanics (dynamic and nonequilibrium) / 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10]
Date of first Publication:2017/05/03
Series (Serial Number):ZIB-Report (17-22)
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