Hamiltonian formulation of fractional kinetics

TL;DR

This paper develops a Hamiltonian formulation for fractional kinetic equations, specifically the fractional Fokker-Planck equation, using Dirac's formalism to handle nonlocal temporal effects and demonstrating the equivalence of different Hamiltonian forms.

Contribution

It introduces a Hamiltonian framework for fractional kinetics based on the variational principle, addressing nonlocality and nonuniqueness issues in the fractional Fokker-Planck equation.

Findings

Hamiltonian formulation for fractional Fokker-Planck equation established.

Nonuniqueness of Hamiltonian due to temporal nonlocality analyzed.

Both Hamiltonian forms produce identical dynamics.

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 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.