Klein-Kramers equation

Also called Klein-Kramers equation see O. Klein, Arkiv Mat. Astr. Fys. 16, No. 5 (1922). It was used by H.A. Kramers for describing chemical reactions, see XIII.6. 

V. The Fractional Klein-Kramers Equation Fractional Dynamics in Phase Space... 

V. THE FRACTIONAL KLEIN-KRAMERS EQUATION FRACTIONAL DYNAMICS IN PHASE SPACE... 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

Fractional dynamics emerges as the macroscopic limit of the combination of the Langevin and the trapping processes. After straightforward calculations based on the continuous-time version of the Chapman-Kolmogorov equation [75, 114] which are valid in the long-time limit t max r, t, one obtains the fractional Klein-Kramers equation... 

The integration of the fractional Klein-Kramers equation (69) over the position coordinate leads in, the force-free limit, to the fractional Rayleigh equation... 

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