Fractional kinetic model for chaotic transport in nonintegrable Hamiltonian systems

Abstract

We propose a kinetic model of transport in nonintegrable Hamiltonian systems, based on a fractional kinetic equation with spatially dependent diffusion coefficient. The diffusion coefficient is estimated from the remainder of the optimal normal form for the given region of the phase space. After partitioning the phase space into building blocks, a separate equation can be constructed for each block. Solving the kinetic equations approximately and estimating the diffusion time scales, we convolve the solutions to get the description of the macroscopic behavior. We show that, in the limit of infinitely many blocks, one can expect an approximate scaling relation between the Lyapunov time and the diffusion (or escape) time, which is either an exponential or a power law. We check our results numerically on over a dozen Hamiltonians and find a good agreement.