Limits Langevin equation, linear

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. 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

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