On the variational principle for fractional kinetic theory

TL;DR

This paper discusses the variational principle for fractional kinetic equations, clarifying the implicit normalization constraint through a transformation property of the action functional.

Contribution

It reveals that the normalization constraint in the variational principle is inherently incorporated via a transformation property of the action functional.

Findings

The action functional has a specific transformation property.

The normalization constraint is implicitly imposed.

The approach clarifies the structure of the variational principle.

Abstract

In a recent paper (Abe S 2013 Phys. Rev. E 88 022142), a variational principle has been formulated for spatiotemporally-fractional Fokker-Planck equations and applied to derivations of their approximate analytic solutions based on the L\'evy Ansatz. Here, the problem of the constraint associated with normalization condition on a probability distribution behind the principle is discussed. It is shown that the action functional possesses a specific transformation property in terms of an auxiliary field and the constraint turns out to have already been imposed implicitly in terms of such a structure.