Generalized persistence dynamics for active motion

We analyse the statistical physics of self-propelled particles from a general theoretical framework that properly describes the most salient characteristics of active motion in arbitrary spatial dimensions. Such a framework is devised in terms of a Smoluchowski-like equation for the probability density of finding a particle at a given position, that carries the Brownian component of the motion due to thermal fluctuations, and the active component due to the intrinsic persistent motion of the particle. The active probability current not only considers the gradient of the probability density at the current time, as in the standard Fick's law, but also the gradient of the probability density at all previous times weighted by a memory function that entails the main features of active motion. We focus in the consequences when the memory function depends only on time and decays as a power law in the short-time regime, and exponentially in the long-time one. In addition, we found analytical expressions for the Intermediate Scattering Function and the time dependence of the mean-squared displacement and the kurtosis.