Nonlinear Langevin equation

Finally consider the fully nonlinear Langevin equation... 

Conclusion. For internal noise one cannot just postulate a nonlinear Langevin equation or a Fokker-Planck equation and then hope to determine its coefficients from macroscopic data. ) The more fundamental approach of the next chapter is indispensable. ... 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

This is a nonlinear Langevin equation of the first order. It contains a multiplicative noise term. The noise A(t) may be represented, according to van Kampen [58], by a random sequence of delta functions. Thus each delta function jump in A(t) causes a jump in (r). Hence, the value of at the time the delta function arrives is indeterminate and consequently so is g at this time also. A problem arises, as the equation does not indicate which value of one should substitute in g whether the value of before the jump, the value after or a mean of both. 

The treatment which follows is similar to that of Risken [31] as that is extremely clear. Let us first write our nonlinear Langevin equation as an integral equation... 

D.2. The Kramers-Moyal Expansion Coefficients for Nonlinear Langevin Equations... 

It is now desirable to deal with the nonclassical behavior of the kernel in the linear laws in a precise, formal way. Of course, one could simply try to improve the crude method just discussed such an approach is perfectly valid. However, we feel that an alternate procedure, which has almost always been used in the literature, is preferable. Mori s method allows the writing of equations with well-behaved kernels if the proper set of variables is chosen. The kernel in the linear laws is badly behaved due to the influence of the nonlinear variable. If we include the linear and nonlinear variables in the set of variables to which Mori s method is applied, the random forces and the dissipative fluxes (/ will be defined precisely in this section) will be projected orthogonal to all of these variables. The kernels in the resulting equations, the nonlinear Langevin equations, should behave classically. Thus, convolutions involving K will be converted into scalar multiplication by the classical relation. 

For certain cases, it is possible to represent the time derivative of a linear variable exactly, or at least quite reasonably, in terms of a bilinear variable. No expansion or limiting process is involved the bilinear variable just happens to be well suited to express or /k. Under these circumstances, the nonlinear Langevin equation may be deduced from simple physical arguments. The cleanest examples of problems where bilinear variables arise in a fairly obvious way are diffusion problems. In any diffusion problem, self, mutual, or whatever, the linear variable of interest is a concentration, nZ [see Eq. (22)]. The time derivative of a concentration is a momentum density/mass. 

Only bilinear variables with intermediate wave vector kc are to be included in the nonlinear Langevin equation, while the relations just derived contain sums over all wave vectors. Let us split up the sum in Eq. (67) and use the definition of the diffusion flux = ik /t, to obtain... 

Sometimes only a part of the nonlinear Langevin equation can be rigorously expressed in bilinear form. Consider an arbitrary conserved variable. 

We have now eliminated as many variables from consideration as is possible with the use of general agreements it is time to start writing the nonlinear Langevin equation for the remaining variables. To obtain the equations, we must form the product (/n +L°)0 ) The only quantity in this product that is not at least formally available is the inversion of an infinite rank matrix can be a formidable task. Thus, we first attempt to find out if X might be inverted in some simpler manner. 

In attempting to solve the nonlinear Langevin equation [Eq. (49)], one is immediately struck that a solution is not obtainable by ordinary mathematical techniques. Since the nonlinear variables A k+k-A-k are not known as a function of Ak (AA AA) Eq. (49) is simply not a closed equation for the variable of interest, A To circumvent this difficulty, it is necessary to start with the fluctuating form of the nonlinear Langevin equation. The fluctuating form of Eq. (49) is... 

When the solution of the nonlinear Langevin equation was first formulated by Kawasaki, the approximation 2 = was always made. The presence of full correlation functions in phenomenological results for A, as contrasted to zero-order correlation functions in the results of Kawasaki, remained a puzzling point for a few years. Experimental evidence indicated that was a better approximation to 2 than The problem was... 

Needless to say, most interesting phenomena in nonequilibrium are not linear. These phenomena cannot be in the class considered above, but usually they are modeled in terms of nonlinear Langevin equations of the following form (Zwanzig s nonlinear Langevin equation) ... 

It seems that all the successfiilly elucidated dynamical imiversality classes adopt as their minimal models the ones described in terms of nonlinear Langevin equations (33). This indicates the general correctness of Onsager s principle as a fimdamental principle (beyond the linear regime). Just as equilibrium statistical mechanics is a statistical framework based on the principle (of equal... 

To imderstand d5mamics and transport properties, a set of (nonlinear) Langevin equations describing the Edwards model and the solvent velocity field is proposed as a minimal model. Comparison of the outcome of the model with experimental results has revealed that this minimal model is reasonable (47) but closely studied, less than minimal. The importance of the direct chain-chain friction must not be underestimated (48). With this additional elfect, the model seems minimal, but a more careful study may be needed. 

The implication of this equation is better understood by writing down the nonlinear Langevin equation equivalent to it ... 

J. T. Hynes, R. Kapral, M. Weinberg Microscopic theory of Brownian motion Nonlinear Langevin equations. Physica 81 A, 485 (1975)... 

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