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Effective Dynamics Along Given Reaction Coordinates, and Reaction Rate Theory

Please always quote using this URN: urn:nbn:de:0297-zib-59706
  • In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: We first show that if we start with an ergodic diffusion processes whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Zwanzig-Mori, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the ”equation-free” approach and the ”heterogeneous multiscale method” can be seen as special cases of our approach.

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Author:Wei Zhang, Carsten HartmannORCiD, Christof Schütte
Document Type:ZIB-Report
Tag:Ergodic diffusion; effective dynamics; equation-free; heterogeneous multiscale method; model reduction; reaction coordinate
MSC-Classification:58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J65 Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
62-XX STATISTICS / 62Pxx Applications [See also 90-XX, 91-XX, 92-XX] / 62P35 Applications to physics
Date of first Publication:2016/06/22
Series (Serial Number):ZIB-Report (16-35)
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