Nonlinear Fokker-Planck equation

In special cases, such as the quantized oscillator, it may happen that a3 = 0, so that at least to one higher order a slightly nonlinear Fokker-Planck equation is consistent. 

The Landau equation in plasma theory is a nonlinear variant, but there P is a particle density rather than a probability. L.D. Landau, Physik. Z. Sovjetunion 10, 154 (1963) = Collected Papers (D. ter Haar ed., Pergamon, Oxford 1965) p. 163. The same is true for the nonlinear Fokker-Planck equation in M. Shiino, Phys. Rev. A 36, 2393 (1987). 

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. 

Even in the present case, however, it is possible to manufacture a nonlinear Fokker-Planck equation, and a corresponding Lange vin equation, which as far as the linear noise approximation is concerned reproduce the same results as found here. But any features they contain beyond that approximation are spurious. For instance, one cannot conclude from this manufactured Fokker-Planck equation that the stationary distribution is given by (VIII. 1.4). 

It has been shown that the lowest order of the Q-expansion yields the macroscopic equation, and the next order the linear noise approximation, provided that the stability condition (X.3.4) holds. This condition is violated, albeit marginally, when a0() = 0. In this case the O-expansion takes an entirely different form its lowest approximation is a nonlinear Fokker-Planck equation. 

The diffusion approximation (1.5) is the nonlinear Fokker-Planck equation (VIII.2.5). In fact, we have now justified the derivation in VIII.2 by demonstrating that it is actually the first term of a systematic expansion in Q 1 for those master equations that have the property (1.1). Only under that condition is it true that the two coefficients... 

Summary. The special class of master equations characterized by (1.1) will be said to be of diffusion type. For such master equations the -expansion leads to the nonlinear Fokker-Planck equation (1.5), rather than to a macroscopic law with linear noise, as found in the previous chapter for master equations characterized by (X.3.4). The definition of both types presupposes that the transition probabilities have the canonical form (X.2.3), but does not distinguish between discrete and continuous ranges of the stochastic variable. The -expansion leads uniquely to the well-defined equation (1.5) and is therefore immune from the interpretation difficulties of the Ito equation mentioned in IX.4 and IX.5. 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

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,... 

This is a nonlinear Fokker-Planck equation. If it happens that Pe is a constant and therefore cc(x) = y(x) the equation may be written... 

Exercise. Any nonlinear Fokker-Planck equation (1.5) can be transformed into a quasilinear one by a suitable transformation of x. 

Warning. The idea of using a nonlinear Fokker-Planck equation as a general framework for describing fluctuating systems has attracted many authors. Detailed balance, in its extended form, was a useful aid, but the link with the deterministic equation caused difficulties. It may therefore be helpful to emphasize three caveats. 

When a10 = 0, the fl expansion yields as the lowest approximation a nonlinear Fokker-Planck equation ... 

The above results could only be obtained analytically because the corresponding Fokker-Planck equation was linear. In the case of a nonlinear Fokker-Planck equation, all quantities have to be computed numerically. 

Fig. 11.2. Mean first passage time for C, = 50nM (left) and C(, = 80nM (right) computed from the master equation (solid), the fl expansion (dashed) and the nonlinear Fokker-Planck equation (dotted) for di = 0.13pM,d2 = 3pM,d3 = 0.9434pM, 4 = 0.4133pM, ds = 0.24/iM, 02 = 0.4 = 0.2 (pMs), as = 5 (pMs) , N = 25. The dots in the left panel represent the variance of the Cl expansion. The inset in the right panel shows a blow up of the plot for large IPs concentration.

The most important difference between the nonlinear Fokker-Planck equation (11.11) and van Kampen s expansion (11.7) is in the diffusion term. It is constant in van Kampen s expansion - describing additive noise - and linear in (f> and 0 in the nonlinear Fokker-Planck equation thus describing multiplicative noise. As expected intuitively, the results in figure 11.2 show a better agreement between the nonlinear Fokker-Planck equation and... 

We exclude van Kampen s expansion in the above analysis, because its validity requires a single stationary state throughout the stochastic motion [34]. In contrast to a constant Ca concentration, the nonlinear Fokker-Planck equation underestimates the results of the master equation. Nevertheless, the results in figure 11.7, which correspond to the stochastic fraction of the puff frequency, are in the same range as experimentally determined puff periods [21]. 

At a constant Ca concentration, the main difference between van Kampen s expansion and the nonlinear Fokker-Planck equation is in the character of fluctuations. They correspond to additive noise for the D expansion and to multiplicative noise in the latter approach. Although the noise is intrinsically multiplicative, van Kampen s expansion provides a reasonable approximation, which improves with increasing base level and growing IP3 concentration. It opens up the opportunity for further studies since the Q expansion is the only method that yields analytic expressions for the probability density and all higher moments. That distinguishes it from the master equation and the nonlinear Fokker-Planck equation, for which only the first moment is directly accessible. 

It was argued (Horsthemke Brenig, 1971 Blomberg, 1981 Haenggi et al., 1984) that the nonlinear Fokker-Planck equations (derived in a slightly different way) also operate correctly in the critical point. 

Hongler, M. O. (1979a). Exact solutions of a class of nonlinear Fokker-Planck equations, Phys. Letters, 75A (3-4). 

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