The Memory Function of the Generalized Diffusion Equation of Active Motion

An exact description of the statistical motion of active particles in three dimension is presented in the framework of a generalized diffusion equation. Such a generalization contemplates a non-local, in time and space, connecting (memory) function. This couples the rate of change of the probability density of finding the particle at position $\boldsymbol{x}$ at time $t$, with the Laplacian of the probability density at all previous times and to all points in space. Starting from the standard Fokker-Planck-like equation for the probability density of finding an active particle at position $\boldsymbol{x}$ swimming along the direction $\hat{\boldsymbol{v}}$ at time $t$, we derive in this paper, in an exact manner, the connecting function that allows a description of active motion in terms of this generalized diffusion equation.