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Space fractional Wigner equation and its semiclassical limit
1Institute of Physics, Karl-Franzens Universität Graz, A-8010 Graz, Austria.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2012
Summary
Space fractional quantum mechanics (SFQM) offers a quantum interpretation of Lévy flights. SFQM introduces anomalous kinetic terms and drift currents, reducing to classical forms when the Lévy index α equals 2.
Area of Science:
- Quantum Mechanics
- Statistical Physics
- Mathematical Physics
Background:
- Space fractional quantum mechanics (SFQM) provides a quantum mechanical framework for Lévy flight statistics.
- Lévy flights are crucial in modeling anomalous diffusion and transport phenomena.
Purpose of the Study:
- To investigate the physical manifestations of SFQM using the Wigner transform.
- To derive classical transport equations from the SFQM Schrödinger equation.
- To analyze the impact of anomalous kinetic terms on particle motion and drift currents.
Main Methods:
- Formulation of the SFQM Schrödinger equation with a Lévy index α.
- Application of the Wigner transform to the SFQM equation.
- Analysis of the limit h/Eτ → 0 for classical transport.
- Introduction of substitutions for |p|(α) and |p'|α in the von Neumann equation.
Main Results:
- SFQM introduces an anomalous kinetic energy term for free particle motion.
- Anomalous terms are observed in the drift current when an external potential is applied.
- The derived transport equations are consistent with Wigner transform criteria.
- All results correctly reduce to classical forms as α approaches 2.
Conclusions:
- SFQM offers a robust quantum mechanical interpretation of Lévy flight statistics.
- The study successfully links SFQM to anomalous transport phenomena through the Wigner formalism.
- The findings provide a foundation for further research into quantum transport in fractional systems.
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