Langevin equation dissipative nonlinearity

Let us consider dissipative nonlinearity up to the quartic order contribution, y(V) = Yo + y2F2 + Y4L4. According to the previous result (128), the stationary PDF for the quadratic case with y2 > 0 and y4 = 0 falls off like / t (V) V - -3, and thus Va C (0,2) the variance (V2) is finite. Higher-order moments such as the fourth-order moment (V4) are, however, still infinite. In contrast, if y4 > 0, the fourth-order moment is finite. We investigate this behavior numerically by solving the Langevin equation (121) compare Ref. 64 for details. 

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. 

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