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One Dimensional Turing-Like Handshake Test for Motor Intelligence

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Mori-Zwanzig formalism as a practical computational tool.

Carmen Hijón¹, Pep Español, Eric Vanden-Eijnden

¹Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, Aptdo. 60141, E-28080 Madrid, Spain.

Faraday Discussions

|February 18, 2010

Summary

This study presents a new method to explicitly calculate terms in the generalized Langevin equation (GLE) by reducing it to a Markovian equation. This approach resolves the plateau problem in Green-Kubo formulas, crucial for polymer dynamics.

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Area of Science:

Statistical Mechanics
Polymer Physics
Computational Chemistry

Background:

The generalized Langevin equation (GLE) is a powerful tool for describing complex dynamics, but explicitly computing its terms is challenging.
The Mori-Zwanzig formalism provides a theoretical framework for deriving the GLE, yet practical computation remains difficult.
Existing methods for calculating Green-Kubo formulas can suffer from numerical instabilities like the plateau problem.

Purpose of the Study:

To develop an operational procedure for explicitly computing terms in the generalized Langevin equation (GLE).
To demonstrate how to simplify the GLE to a Markovian equation through projected dynamics.
To address and resolve the plateau problem encountered in Green-Kubo formula calculations.

Main Methods:

Introducing an artificial controlled parameter to make projected dynamics explicit.
Tuning the parameter to reduce the generalized Langevin equation (GLE) to a Markovian equation.
Implementing constraints to realize projected dynamics in practical simulations.
Applying the methodology to star polymer molecules in a melt, using their center of mass as relevant variables.

Main Results:

The presented procedure successfully computes GLE terms explicitly.
The generalized Langevin equation (GLE) is effectively reduced to a Markovian equation.
Green-Kubo formulas calculated using the projected dynamics do not exhibit the plateau problem.
For star polymers, effective potentials, friction forces, and noise are shown to be critical dynamic components.

Conclusions:

The developed operational procedure offers a robust method for analyzing complex molecular dynamics via the GLE.
This approach provides a solution to the plateau problem, enhancing the reliability of Green-Kubo formula calculations.
The study highlights the significant influence of potentials, friction, and noise on the dynamics of star polymer systems.