Kramers equation effect

KRAMERS EQUATION—ARBITRARY FRICTION REGIME. In the presence Of inertial effects, the one-dimensional motion is determined by a bivariate Fokker-Planck equation. 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

Faced with these difficulties, we shall presently illustrate that if a generalization of the Klein-Kramers equation, first proposed by Barkai and Silbey [30], where the fractional derivatives do not act on the Liouville terms, is used, then the desired return to transparency at high frequencies is achieved. Moreover, the Gordon sum mle, Eq. (85), is satisfied. In conclusion of this subsection, we remark that the divergence of the integral absorption is not unusual in models that incorporate inertial effects. For example, in the well-known Van Vleck-Weisskopf model [88], the divergence results from the stosszahlansatz used by them, just as in the present problem. 

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