Hamiltonian formulation of fractional kinetics

Abstract

Fractional kinetic theory plays a vital role in describing anomalous diffusion in terms of complex dynamics generating semi-Markovian processes. Recently, the variational principle and associated Lévy Ansatz have been proposed in order to obtain an analytic solution of the fractional Fokker–Planck equation. Here, based on the action integral introduced in the variational principle, the Hamiltonian formulation is developed for the fractional Fokker–Planck equation. It is shown by the use of Dirac’s generalized canonical formalism how the equation can be recast in the Liouville-like form. A specific problem arising from temporal nonlocality of fractional kinetics is nonuniqueness of the Hamiltonian: it has two different forms. The non-equal-time Dirac-bracket relations are set up, and then it is proven that both of the Hamiltonians generate the identical time evolution.