Gaussian distribution, Fokker-Planck

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

These methods are appealing since the fundamental equation of motion is for the phase-space distribution itself rather than for individual trajectories. The structure of the Fokker-Planck equation in effect carries out a number of averages that must otherwise be performed by generating suitable trajectory ensembles. A preliminary application of the Fokker-Planck method to gas-surface scattering has been made [3.37]. In this application it was assumed that the full phase-space distribution was Gaussian in character with time-dependent first and second moments. Consequently the Fokker-Planck equation produced a set of first-order differential equations for these moments [3.48]. Integration of these equations was essentially... 

Starting from the Langevin equation As t)IAt = v(t) where, by assumption, the translocation velocity v(f) follows Gaussian statistics, we derived a Fokker-Planck equation for the distribution W(5, t) ... 

In [3] the master equation is approximated by a Fokker-Planck equation, which is linearized close to the deterministic limit cycle trajectory the probability distribution in the degree of freedom perpendicular the hmit cycle trajectory becomes a Gaussian distribution. A comparison of the numerical (Monte Carlo) results with those of the Fokker-Planck equation is given in Fig. 19.2. 

The stationary solution of the Fokker-Planck equation, which includes the friction force F=— /3v, and the momentum diffusion coefficient (eqn 5.22), is a 3D Gaussian distribution... 

The Fokker-Planck equation accurately captures the time evolution of stochastic processes whose probahihty distribution can be completely determined by its average and variance. For example, stochastic processes with Gaussian probahihty distributions, such as the random walk, can be completely described with a Fokker-Planck equation. 

One can assume a probability distribution type, e.g., Gaussian, and eliminate high-order moments. In such a case the master equation reduces to a Fokker-Planck equation, as discussed in Chapter 13, for which a numerical solution is often available. Fokker-Planck equations, however, cannot capture the behavior of systems that escape the confines of normal distributions. 

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