Fokker-Planck equation behavior

Planck equation. In this way the actual equations of motion need be solved only during At, which can be done by some perturbation theory. The Fokker-Planck equation then serves to find the long-time behavior. This separation between short-time behavior and long-time behavior is made possible by the Markov assumption. 

This equation is identified with the macroscopic equation of motion for the system, which is supposedly known. Thus the function A(y) is obtained from the knowledge of the macroscopic behavior. Subsequently one obtains B(y) by identifying (1.4) with the equilibrium distribution, which at least for closed physical systems is known from ordinary statistical mechanics. Thus the knowledge of the macroscopic law and of equilibrium statistical mechanics suffices to set up the Fokker-Planck equation and therefore to compute the fluctuations. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

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. 

Example 7.6 Fokker-Planck equation for Brownian motion in a temperature gradient short-term behavior of the Brownian particles The following is from Perez-Madrid et al. (1994). By applying the nonequilibrium thermodynamics of internal degrees of freedom for the Brownian motion in a temperature gradient, the Fokker-Planck equation may be obtained. The Brownian gas has an integral degree of freedom, which is the velocity v of a Brownian particle. The probability density for the Brownian particles in velocity-coordinate space is... 

It should be noted that if n is of the order 2, this is a cusp manifold (which is one which will show bifurcation behavior of the form discussed by Shore and Comins ). The analysis of this syst n presented by Ferrini et al. shows that this is indeed a system with multiple equilibrium states. Having this form for the potential, there exists a Fokker-Planck equation for the... 

In the investigation of non-thermal damage to dielectrics, the Fokker-Planck equation is applied to describe the transient behaviors of electron densities, and to predict the damage threshold fluences for various laser pulse widths ranging from 10 femtoseconds to 10 picoseconds [13]. This model includes the effects of electron avalanche and multiphoton ionization on the generation of electrons. 

The time behavior of the right-hand side of Eq. (90) indicates that the initial state Wo(x) decays slowly with a long-time tail unlike the exponential decay of normal diffusion, which is an indication of the fractal time character of the process. Equations (89) and (90) are fractional analogs of the conventional Fokker-Planck equation [Eq. (88)] giving rise to the Cole-Cole anomalous behavior. 

Another approach to fractionalize the Fokker-Planck equation incorporating Cole-Davidson behavior can now be given by extending a hypothesis of Nigmatullin and Ryabov [28]. They noted that the ordinary first-order differential equation describing an exponential decay... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

We remark that Eq. (262), unlike the form of the Rocard equation of the Levy sneaking model, Eq. (248), has an inertial term similar to the Rocard equation for normal diffusion, Eq. (249). This has an important bearing on the high-frequency behavior because return to transparency can now be achieved, as we shall demonstrate presently. The exact solution, Eq. (260), also has satisfactory high-frequency behavior. We further remark that, on neglecting inertial effects (y —> 0), Eq. (261) yields the Cole-Cole formula [Eq. (9)]—that is, the result predicted by the noninertial fractional Fokker-Planck equation. 

In Fig. 9, we show the dependence of this first approximation as a function of the Levy index a (dashed line), in comparison to the values determined from the numerical solution of the fractional Fokker-Planck equation (38) shown as the dotted line. The second-order iteration for the PDF, P2 k, t), can be obtained with maple6, whence the second approximation for the bifurcation time is found by analogy with the above procedure. The result is displayed as the full line in Fig. 9. The two approximate results are in fact in surprisingly good agreement with the numerical result for the exact PDF. Note that the second approximation appears somewhat worse than the first however, it contains the minimum in the a-dependence of the t 2 behavior. 

We have derived a master equation and two Fokker-Planck equations for channel cluster behavior in IP3 mediated Ca dynamics. Among the different approaches to approximate a master equation by a Fokker-Planck equation we have chosen van Kampen s II expansion and an ansatz based... 

Example 15.1 Fokker-Planck equation for Brownian motion in a temperature gradient short-term behavior of the Brownian particles... 

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