Linear Fokker-Planck equation

When the linear model is inserted into (6.46), a linear Fokker-Planck equation (Gardiner 1990) results ... 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

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. 

We have introduced the Fokker-Planck equation as a special kind of M-equation. Its main use, however, is as an approximate description for any Markov process Y(t) whose individual jumps are small. In this sense the linear Fokker-Planck equation was used by Rayleigh 0, Einstein, Smoluchowskin), and Fokker, for special cases. Subsequently Planck formulated the general nonlinear Fokker-Planck equation from an arbitrary M-equation assuming only that the jumps are small. Finally Kolmogorov8 provided a mathematical derivation by going to the limit of infinitely small jumps. 

This phenomenological identification of A and B has been utilized by Einstein and others with great success (section 3), but only for linear Fokker-Planck equations. If the macroscopic law is nonlinear a difficulty arises, first pointed out by D.K.C. MacDonald. The flaw in the argument lies in the identification of the coefficient A y) with the macroscopic law. The two may well differ by a term of the same order as the fluctuations once one neglects the fluctuations such a term is invisible anyway. The consequence was that different authors obtained different, but equally plausible expressions for noise in nonlinear systems. This difficulty led to the more fundamental approach in chapter X. 

Exercise. For the linear Fokker-Planck equation (1.5), the equation (1.7) is a closed equation for the mean value , which can be solved. Show that in this way all moments can be obtained successively. 

This is a linear Fokker-Planck equation. Apart from constants which can be scaled away, it is identical with the equation (IV.3.20) obeyed by the transition probability of the Ornstein-Uhlenbeck process. The stationary solution of (4.6) is the same as the Pl given in (IV.3.10). Thus, in equilibrium V(t) is the Ornstein-Uhlenbeck process. 

Thus our additional approximation for the neighborhood of rf leads to a linear Fokker-Planck equation of the same form as (4.6). The fluctuations in the stationary state are therefore again an Ornstein-Uhlenbeck process. It will be shown in X.4 that (5.6) is a consistent approximation.510... 

In chapter X we shall encounter a linear Fokker-Planck equation of the form (6.4) whose coefficients Aij9 Bare given functions of time. The solution is again Gaussian, and can be obtained in much the same way as before. 

The Gaussian (6.11) with this E and with the averages (6.15) constitutes the solution of the linear Fokker-Planck equation (6.4) with time-dependent coefficients. 0... 

This is a linear Fokker-Planck equation, whose coefficients depend on t through ( ). It has been solved in VIII.6 and the result was that 77 is Gaussian. It therefore suffices to determine the first and second moments of , which contain the most important information anyway. By the usual trick one obtains from (1.1)... 

This is a linear Fokker-Planck equation whose coefficients depend on time through . This approximation was christened in section 1 linear noise approximation . The solution of (4.1) was found in VIII.6 to be a Gaussian510, so that it suffices to determine the first and second moments of On multiplying (4.1) by and 2, respectively, one obtains... 

This is a multivariate linear Fokker-Planck equation of the type solved in VIII.6. We use it to determine the moments of c and i/. 

While the linear noise approximation led to the linear Fokker-Planck equation (4.1) we now see that the higher powers oi Q 1,2 give rise to three modifications. 

This is a necessary and sufficient condition for (4.1) to be transformable into a linear Fokker-Planck equation. 

The equation above is the conventional linear Fokker-Planck equation for stochastic processes with additive noise and the noise strength measured in terms of the temperature T. 

Horsthemke, W. Brenig, I. (1971). Non-linear Fokker-Planck equation as an asymptotic representation of the master equation. Z. Physik, B27, 341-8. 

Fig. 19.2. Plot of the probability distribution in a cross section, tranverse to the ridge, for the Selkov model, (a) results of the Monte Carlo calculation (b) solution of the linearized Fokker-Planck equation. Taken from [3]...

The solution to the linearized Fokker-Planck equation has only a single peak the linearization misses the second peak on the crater 180° opposite the first peak. The Monte Carlo calculation yields two peaks. 

Other comparisons of the results of the linearized Fokker Planck equation and the numerical solutions of the master equation are shown in Fig. 19.3. 

If the derivate moments higher than second order are neglected, Eq. (256) assumes the form of a so-called linear Fokker-Planck equation ... 

For this limit, the linear Fokker-Planck equation given by Eq. (258) assumes the form of the Fokker-Planck equation given by Eqs. (245)-(247) for the case of a single particle with an isotropic friction tensor and in the absence of a force field due to other particles undergoing Brownian motion. 

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