Laplace transform Kramers equation

If the potential of mean force is parabolic (w(q) = - imco q ) then the GLE (Eq. 1) may be solved using Laplace transforms. Denoting the Laplace transform of a function f(t) as f(s) = dte - f(t), taking the Laplace transform of the GLE and averaging over realizations of the random force (whose mean is 0) one finds that the time dependence of the mean position and velocity is determined by the roots of the Kramers-Grote-Hynes equation ... 

Kramers and Fokker-Planck equations can be expressed in terms of its Brownian analogue, Wi, according to Eq. (49). Application of relation (49) to the Laplace transform p(u) = (rK + u) l of the exponential survival probability, Eq. (62a), produces... 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

The parabolic barrier plays a special role in rate theory. The GLE (with space-independent friction) may be solved analytically using Laplace transforms. The two-dimensional Fok-ker-Planck equation derived from the Langevin equation may be solved analytically, as was done by Kramers in his famous paper of 1940. In this section we present some of the analytic results for the parabolic barrier dynamics. These results are important from both a conceptual and a practical point of view. Later we shall see how one returns to the parabolic barrier case as a source of comprehension, approximation, etc. 

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