Dilatation symmetry of the Fokker-Planck equation and anomalous diffusion

TL;DR

This paper explores the dilatation symmetry of the Fokker-Planck equation for atomic motion in optical lattices, linking it to anomalous diffusion and Tsallis statistics, and generalizes the conditions for power-law decay in nonstationary cases.

Contribution

It introduces a symmetry-based framework for understanding anomalous diffusion in optical lattices and extends previous results to nonstationary distributions.

Findings

Identifies dilatation symmetry in the Fokker-Planck equation for optical lattices.

Connects anomalous transport to Tsallis statistics in high-energy regimes.

Derives conditions for power-law decay in nonstationary solutions.

Abstract

Based on the canonical formalism, the dilatation symmetry is implemented to the Fokker-Planck equation for the Wigner distribution function that describes atomic motion in an optical lattice. This reveals the symmetry principle underlying the recent result obtained by Lutz [Phys. Rev. A 67, 051402(R) (2003)] on the connection between anomalous transport in the optical lattice and Tsallis statistics in the high-energy regime.Lutz's discussion is generalized to the nonstationary case, and the condition, under which the solution distribution decays as a power law, is derived.