Abstract

Fractional calculus is an old branch of mathematics which deals with fractionalorder derivatives, e.g., d1=2=dt1=2. Davidson’s group (Weizmann) has recorded the spatialdiffusion of cold atoms in optical lattices, fitting the results to the solution of a fractionaldiffusion equation@βP(x; t)@tβ = KµrµP(x; t):Within the semi classical theory of Sisyphus cooling we derive this fractional equationand discuss its meaning and its limitations [1,2]. An asymptotically weak friction force,induced by the laser field, is responsible for the large deviations from normal transporttheory (and from Boltzmann-Gibbs equilibrium concepts [3]) at least below a critical valueof the depth of the optical lattice.1. D. A. Kessler, and E. Barkai Theory of fractional-L´evy kinetics for cold atoms diffusing in optical lattices Phys. Rev. Lett. 108, 230602 (2012).2. E. Barkai, E. Aghion, and D. Kessler From the area under the Bessel excursion toanomalous diffusion of cold atoms Physical Review X 4 , 021036 (2014)3. A. Dechant, D. A. Kessler and E. Barkai Deviations from Boltzmann-Gibbs equilibrium in confined optical lattices Phys. Rev. Lett. 115, 173006 (2015).