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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Langevin approach to fractional diffusion equations including inertial effects
1Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany.
The Journal of Physical Chemistry. B
|September 14, 2007
Summary
Fractional Kramers-Fokker-Planck equations are linked to Langevin equations through subordination. This method, based on Fogedby
Area of Science:
- Statistical Physics
- Nonlinear Dynamics
- Stochastic Processes
Background:
- The Kramers-Fokker-Planck equation describes the evolution of probability density functions for stochastic systems.
- Recent advancements include fractional generalizations to model anomalous diffusion and complex dynamics.
- Understanding the relationship between fractional equations and their stochastic differential equation counterparts is crucial.
Purpose of the Study:
- To establish a connection between fractional Kramers-Fokker-Planck equations and Langevin equations.
- To demonstrate the utility of the subordination procedure in unifying these descriptions.
- To provide a framework for analyzing systems exhibiting anomalous diffusion.
Main Methods:
- Utilized Fogedby's approach for relating fractional equations to stochastic processes.
- Applied the subordination procedure to connect fractional Kramers-Fokker-Planck equations with Langevin equations.
- Mathematical derivation and analysis of the transformation.
Main Results:
- Demonstrated that fractional Kramers-Fokker-Planck equations can be derived from specific Langevin equations via subordination.
- Established a clear link between fractional dynamics and the underlying stochastic processes.
- The subordination method provides a unified perspective on these fractional generalizations.
Conclusions:
- The subordination procedure offers a powerful tool for understanding fractional dynamics.
- Fractional Kramers-Fokker-Planck equations and Langevin equations are intrinsically related through this method.
- This work facilitates the study of complex systems with anomalous diffusion properties.
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