Colored-noise Fokker-Planck equation for the shear-induced self-diffusion process of non-Brownian particles

In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive.