Fractional dynamics using an ensemble of classical trajectories

Area of Science:

• Theoretical Physics
• Stochastic Processes
• Fractional Calculus

Background:

• Fractional dynamics and Lévy flights are crucial for modeling complex systems.
• Existing methods for simulating fractional dynamics can be computationally intensive or lack precision.
• Understanding barrier crossing phenomena in systems driven by anomalous noise is essential.

Purpose of the Study:

• To develop a novel trajectory-based formulation for fractional dynamics.
• To introduce a new class of estimators for the Riesz fractional derivative using confluent hypergeometric functions.
• To investigate barrier crossing in bistable potentials driven by Lévy noise.

Main Methods:

• Deterministic generation of trajectories for fractional dynamics.
• Derivation of estimators for the Riesz fractional derivative using the confluent hypergeometric function (_{1}F_{1}).
• Simulation of free and confined Lévy flights.
• Analysis of barrier crossing in a bistable potential driven by Lévy noise.

Main Results:

• The trajectory-based formulation shows excellent agreement with exact numerical and analytical results for Lévy flight simulations.
• The derived estimators accurately represent the Riesz fractional derivative.
• The study provides insights into barrier crossing dynamics influenced by Lévy noise.

Conclusions:

• The proposed trajectory-based formulation offers an effective and accurate approach to fractional dynamics.
• The use of confluent hypergeometric functions provides a powerful tool for representing fractional derivatives.
• This work enhances the understanding of Lévy flight behavior and barrier crossing in complex systems.