Fractional transport equations for Lévy stable processes

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.