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Time scale separation leads to position-dependent diffusion along a slow coordinate.

Alexander Berezhkovskii1, Attila Szabo

  • 1Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health, Bethesda, Maryland 20892, USA. david.tew@bristol.ac.uk

The Journal of Chemical Physics
|August 25, 2011
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Summary
This summary is machine-generated.

Adiabatic elimination of fast variables in anisotropic Langevin dynamics can be improved. Introducing position-dependent friction provides a more accurate Markovian description of slow dynamics, enhancing predictions for complex systems.

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Area of Science:

  • Statistical Mechanics
  • Physical Chemistry
  • Computational Physics

Background:

  • Adiabatic elimination is a standard technique for simplifying complex dynamical systems by removing fast variables.
  • Conventional methods in anisotropic Langevin dynamics result in a position-independent friction constant for the slow coordinate.
  • This simplification can lead to inaccuracies in describing the system's long-term behavior.

Purpose of the Study:

  • To develop a more accurate, yet still Markovian, description of slow dynamics in systems with separated time scales.
  • To investigate the role of position-dependent friction in enhancing the description of slow variables.
  • To generalize the findings to multidimensional systems and arbitrary slow coordinates.

Main Methods:

  • Applying adiabatic elimination to anisotropic Langevin dynamics in multiple dimensions.
  • Deriving a position-dependent friction coefficient using a Kirkwood-like formula based on force autocorrelation functions.
  • Generalizing the approach for slow coordinates that are arbitrary functions of Cartesian coordinates.
  • Analyzing the case where fast variables are effectively one-dimensional to obtain a closed-form expression for additional friction.

Main Results:

  • A more accurate Markovian description of slow dynamics is achieved using position-dependent friction.
  • The derived friction coefficient is related to the time integral of the autocorrelation function of the force difference.
  • The method is generalized to multidimensional systems, yielding a position-dependent diffusion coefficient.
  • For specific channel geometries, the results align with the multidimensional Zwanzig-Bradley formula.

Conclusions:

  • Position-dependent friction offers a significant improvement over constant friction in describing slow dynamics after adiabatic elimination.
  • The generalized framework provides a powerful tool for analyzing complex systems in various scientific disciplines.
  • This approach enhances the predictive power of models involving anisotropic Langevin dynamics and separated time scales.