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Spectral Properties of Effective Dynamics from Conditional Expectations.

Feliks Nüske1,2, Péter Koltai3, Lorenzo Boninsegna1,4

  • 1Center for Theoretical Biological Physics and Department of Chemistry, Rice University, Houston, TX 77005, USA.

Entropy (Basel, Switzerland)
|January 26, 2021
PubMed
Summary

This study introduces a method to simplify complex scientific models by focusing on essential variables. It provides a new error bound for reduced models and shows that spectral computations are robust, enabling effective dynamics for underdamped Langevin systems.

Keywords:
Kramers–Moyal formulaeLangevin dynamicscoarse grainingextended dynamic mode decompositioninfinitesimal generatorspectral analysisstochastic differential equations

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Area of Science:

  • Applies to physical sciences, computational modeling, and data analysis.
  • Focuses on stochastic dynamics and diffusion processes.

Background:

  • Reducing high-dimensional systems to effective models is crucial in science.
  • The conditioning approach is a standard method for finding effective equations for stochastic dynamics.
  • Understanding the spectrum of the reduced generator is key to analyzing effective dynamics.

Purpose of the Study:

  • To analyze the spectrum of the generator for effective dynamics derived from a conditioning approach.
  • To compare the spectrum of the reduced generator with that of the full generator.
  • To investigate the accuracy and robustness of spectral computations for reduced models.

Main Methods:

  • Proving a new relative error bound using eigenfunction approximation error for reversible systems.
  • Utilizing Kramers-Moyal (KM) approximations to compute the spectrum of the reduced generator.
  • Analyzing systems driven by underdamped Langevin dynamics.

Main Results:

  • A novel relative error bound is established for the spectrum of effective dynamics.
  • Numerical examples show Kramers-Moyal estimators are insensitive to the time window.
  • Effective dynamics can be meaningfully defined for underdamped Langevin systems.

Conclusions:

  • The developed error bound enhances the reliability of reduced models.
  • The robustness of KM estimators simplifies spectral analysis of reduced generators.
  • This work provides a framework for defining effective dynamics in complex systems like those governed by Langevin equations.