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Article

Spectral Properties of Effective Dynamics from Conditional Expectations

1
Center for Theoretical Biological Physics and Department of Chemistry, Rice University, Houston, TX 77005, USA
2
Institute of Mathematics, Universität Paderborn, 33098 Paderborn, Germany
3
Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany
4
Institute for Quantitative and Computational Biosciences and Department of Microbiology, Immunology and Molecular Genetics, University of California Los Angeles, Los Angeles, CA 90095, USA
5
Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
*
Author to whom correspondence should be addressed.
Entropy 2021, 23(2), 134; https://doi.org/10.3390/e23020134
Received: 15 December 2020 / Accepted: 18 January 2021 / Published: 21 January 2021
The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting. View Full-Text
Keywords: stochastic differential equations; coarse graining; infinitesimal generator; spectral analysis; extended dynamic mode decomposition; Kramers–Moyal formulae; Langevin dynamics stochastic differential equations; coarse graining; infinitesimal generator; spectral analysis; extended dynamic mode decomposition; Kramers–Moyal formulae; Langevin dynamics
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MDPI and ACS Style

Nüske, F.; Koltai, P.; Boninsegna, L.; Clementi, C. Spectral Properties of Effective Dynamics from Conditional Expectations. Entropy 2021, 23, 134. https://doi.org/10.3390/e23020134

AMA Style

Nüske F, Koltai P, Boninsegna L, Clementi C. Spectral Properties of Effective Dynamics from Conditional Expectations. Entropy. 2021; 23(2):134. https://doi.org/10.3390/e23020134

Chicago/Turabian Style

Nüske, Feliks; Koltai, Péter; Boninsegna, Lorenzo; Clementi, Cecilia. 2021. "Spectral Properties of Effective Dynamics from Conditional Expectations" Entropy 23, no. 2: 134. https://doi.org/10.3390/e23020134

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