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Generalized Langevin equation with fractional derivative and long-time correlation function.

Kwok Sau Fa1

  • 1Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 16, 2006
PubMed
Summary

This study explores particle motion using a generalized Langevin equation, detailing how fractional derivatives and nonlocal forces affect dynamics. We derived variance expressions and analyzed asymptotic behaviors for correlated systems.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • The generalized Langevin equation (GLE) describes complex systems with memory effects.
  • Fractional derivatives and nonlocal forces are crucial for modeling systems with long-range interactions and memory.
  • Understanding particle dynamics in such systems is vital for various scientific fields.

Purpose of the Study:

  • To investigate particle dynamics governed by a GLE with fractional derivatives and nonlocal dissipation.
  • To derive general expressions for variances under a linear external force.
  • To analyze the asymptotic behaviors of these variances for power-law correlated systems.

Main Methods:

  • Utilized the generalized Langevin equation framework.
  • Incorporated fractional derivatives to model memory effects.

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  • Derived analytical expressions for particle variances.
  • Performed asymptotic analysis for specific correlation functions.

    • Main Results:

    • Obtained general formulas for variances in the presence of a linear external force.
    • Characterized the long-time behavior of particle motion.
    • Demonstrated the influence of fractional derivatives and nonlocal dissipation on system dynamics.

    Conclusions:

    • The derived expressions provide a comprehensive description of particle motion under generalized Langevin dynamics.
    • The study elucidates the impact of non-Markovian effects on system variances.
    • Findings are applicable to diverse physical systems exhibiting memory and long-range correlations.