Fractional transport equations for Levy stable processes

Abstract

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a Brownian system subjected to a Levy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limit of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the fractional character acquired by the position in the Smoluchowski equation follows from the fractional character of the momentum in the Klein-Kramers equation. Connections among fractional transport equations recently proposed are clarified.

4 pages

References in corpus (4)

• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Quantum Lévy Processes and Fractional Kinetics
• Levy Anomalous Diffusion and Fractional Fokker--Planck Equation
• Universality of quantum Brownian motion

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• Stable Equilibrium Based on Lévy Statistics: A Linear Boltzmann Equation Approach
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• Stationary states for underdamped anharmonic oscillators driven by Cauchy noise
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