Diffusion in the presence of a local attracting factor: Theory and interdisciplinary applications

In many complex diffusion processes the drift of random walkers is not caused by an external force, as in the case of Brownian motion, but by local variations of fitness perceived by the random walkers. In this paper, a simple but general framework is presented that describes such a type of random motion and may be of relevance in different problems, such as opinion dynamics, cultural spreading, and animal movement. To this aim, we study the problem of a random walker in $d$ dimensions moving in the presence of a local heterogeneous attracting factor expressed in terms of an assigned position-dependent “attractiveness function.” At variance with standard Brownian motion, the attractiveness function introduced here regulates both the advection and diffusion of the random walker, thus providing testable predictions for a specific form of fluctuation-relations. We discuss the relation between the drift-diffusion equation based on the attractiveness function and that describing standard Brownian motion, and we provide some explicit examples illustrating its relevance in different fields, such as animal movement, chemotactic diffusion, and social dynamics.

Figure 1

Scheme of diffusion process with constant space step $L$ in (a) one dimension and (b) two dimensions (Rayleigh problem). Before an actual jump (solid black arrow) from the initial position $x^{\prime }$ (magenta dot) to the arrival position $x$ (black dot), the random walker probes the environment [the function $\kappa \left(x\right)]$ in the possible arrival positions $x^{\prime \prime }$ (green dot and green dashed line, virtual jumps represented by dotted gray arrows). The resulting probability $p\left(x\right|x^{\prime })$ to jump from $x^{\prime }$ to $x$ is the conditional expectation in Eq. (9).