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Related Experiment Videos

Time-ordered fluctuation-dissipation relation for incompressible isotropic turbulence.

K Kiyani1, W D McComb

  • 1School of Physics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 9, 2005
PubMed
Summary

This study introduces a renormalized response function and fluctuation-dissipation relation (FDR) to model time-dependent systems. The new method reconciles covariance and causality, offering exponential forms for system dynamics.

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Area of Science:

  • Statistical mechanics
  • Turbulence modeling
  • Theoretical physics

Background:

  • The Kraichnan-Wyld perturbation expansion is a key tool for analyzing complex systems.
  • Representing time-dependent covariance and response functions has been a long-standing challenge.
  • Existing theories like Edwards' self-consistent field theory and renormalization group approaches offer partial solutions.

Purpose of the Study:

  • To develop a unified framework for describing time-dependent covariances and response functions.
  • To reconcile the time symmetry of covariance with the causality of response.
  • To provide a new Langevin equation model for turbulence.

Main Methods:

  • Utilizing the Kraichnan-Wyld perturbation expansion.
  • Introducing a renormalized response function connecting covariances at different times.

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  • Employing time ordering and a counterterm to ensure causality.
  • Specializing the relationship to a fluctuation-dissipation relation (FDR).

    • Main Results:

    • The derivative of the covariance with respect to difference time vanishes at the origin.
    • The renormalized response function exhibits transitivity with respect to intermediate times.
    • A novel Langevin equation model for turbulence is presented.
    • The formulation allows for studying relationships between spectral closures and time-independent theories.

    Conclusions:

    • The developed formulation offers a solution for representing time-dependent covariance and response using exponential forms.
    • The time-ordering procedure has broad applicability in fluctuation-dissipation relation (FDR) applications.
    • This work bridges the gap between time-dependent and time-independent theoretical approaches in statistical physics and turbulence.