The Memory Function of the Generalized Diffusion Equation of Active Motion

TL;DR

This paper develops an exact generalized diffusion equation with a memory function to describe the complex statistical motion of active particles in three dimensions, incorporating non-local temporal and spatial effects.

Contribution

It derives an exact form of the memory function that links the probability density's rate of change to its Laplacian over all previous times and positions, extending standard models of active particle motion.

Findings

Provides an exact generalized diffusion equation for active particles.

Identifies the memory function linking past and present particle distributions.

Enhances understanding of non-Markovian effects in active motion.

Abstract

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.