Kramer equation

Zhu S-B, Lee J, Robinson G W and Lin S H 1988 A microscopic form of the extended Kramers equation. A simple friction model for cis-trans isomerization reactions Chem. Phys. Lett. 148 164-8... 

Sun Y-P and Saltiel J 1989 Application of the Kramers equation to stiibene photoisomerization in / -alkanes using translational diffusion coefficients to define microviscosity J. Phys. Chem. 93 8310-16... 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

We consider11 the slowing down of a Brownian particle as described by the Kramers equation (p, momentum m, mass)... 

The last one is taken equal to the coefficient in (4.6) because the external force does not affect the collisions with the gas molecules. Thus we have found the Kramers equation )... 

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

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. 

Exercise. Construct the Langevin equation that corresponds to the Kramers equation (VIII.7.4). 

Exercise. A pendulum obeying the equation Mx = — sin x is suspended in air, which causes damping and fluctuations. Show that it obeys the bivariate nonlinear Fokker-Planck equation, or Kramers equation,... 

Exercise. In the Kramers equation (VIII.7.4) allow y and T to depend on the position x. The expansion for large y is again possible and produces the diffusion equation for inhomogeneous temperature510... 

The Kramers equation (VIII.7.4) has the form (4.1) with two variables, nonlinear 4,(x), and constant Bijt which is semi-definite, while... 

So far we studied the first passage of Markov processes such as described by the Smoluchowski equation (1.9). On a finer time scale, diffusion is described by the Kramers equation (VIII.7.4) for the joint probability of the position X and the velocity V. One may still ask for the time at which X reaches for the first time a given value R, but X by itself is not Markovian. That causes two complications, which make it necessary to specify the first-passage problem in more detail than for diffusion. 

Thus we have derived the Kramers equation (VIII.7.4) as an approximation for short tc. It becomes exact in the white noise limit (3.12). The coefficient of the fluctuation term is the integrated autocorrelation function of the fluctuating force, in agreement with (IX.3.5) and (IX.3.6).110... 

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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