Hamiltonian formulation of fractional kinetics

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 Levy 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.

Sumiyoshi Abe · 2017