Stochastic approach to diffusion inside the chaotic layer of a resonance

Area of Science:

• Physics
• Applied Mathematics
• Dynamical Systems

Background:

• Chaotic diffusion is a key phenomenon in Hamiltonian systems.
• Understanding diffusion in high-dimensional systems is crucial for fields like celestial mechanics and accelerator physics.

Purpose of the Study:

• To model and analyze chaotic diffusion in a 4D symplectic map.
• To develop a seminumerical method for calculating diffusion coefficients.
• To investigate the applicability of Fokker-Planck equations in describing diffusion processes.

Main Methods:

• Utilizing a theorem for stochastically perturbed integrable Hamiltonian systems.
• Explicitly defining a map coupling a free rotator (FR) and a standard map (SM).
• Calculating the diffusion coefficient for the action I of the FR.

Main Results:

• A seminumerical method was developed to compute the diffusion coefficient.
• The probability density function for action I was well-interpolated by a Fokker-Planck (FP) equation in the case of a thick chaotic layer.
• A nonconstant time shift was observed with respect to the FP solution in the case of a thin chaotic layer, suggesting anomalous diffusion.

Conclusions:

• The study provides a method for calculating diffusion coefficients in 4D symplectic maps.
• Results indicate that Fokker-Planck equations can describe chaotic diffusion, but anomalous time scales may arise.
• The findings contribute to understanding slow diffusion in celestial mechanics and accelerator physics.