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Fractional transport equations for Lévy stable processes.

E Lutz1

  • 1Max-Planck-Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany.

Physical Review Letters
|April 6, 2001
PubMed
Summary

This study derives quantum master equations and classical transport equations for systems with Lévy stable random forces. It introduces fractional extensions of Klein-Kramers and Smoluchowski equations, clarifying their interconnections.

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Non-equilibrium systems

Background:

  • The Feynman-Vernon influence functional method is a key tool for studying open quantum systems.
  • Lévy stable random forces introduce non-Markovian dynamics and long-range correlations.
  • Understanding transport phenomena in quantum systems under external noise is crucial.

Purpose of the Study:

  • To derive a quantum master equation for systems subjected to Lévy stable random forces.
  • To obtain classical transport equations for the Wigner function in weak and strong friction limits.
  • To clarify the relationships between different fractional transport equations.

Main Methods:

  • Application of the Feynman-Vernon influence functional method.
  • Derivation of quantum master equations.
  • Analysis of Wigner function transport in classical limits.
  • Identification of fractional extensions of known equations.

Main Results:

  • A quantum master equation for Lévy noise is obtained.
  • Fractional extensions of the Klein-Kramers and Smoluchowski equations are derived.
  • The fractional nature of position in Smoluchowski dynamics is linked to momentum in Klein-Kramers dynamics.
  • Connections among various fractional transport equations are elucidated.

Conclusions:

  • The study provides a theoretical framework for quantum systems with Lévy noise.
  • Fractional calculus is essential for describing transport under such noise.
  • The work clarifies the theoretical underpinnings of fractional transport equations.

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