Run-and-tumble particles on a line with a fertile site
Abstract
We propose a model of run-and-tumble particles (RTPs) on a line with a fertile site at the origin. After going through the fertile site, a run-and-tumble particle gives rise to new particles until it flips direction. The process of creation of new particles is modelled by a fertility function (of the distance to the fertile site), multiplied by a fertility rate. If the initial conditions correspond to a single RTP with even probability density, the system is parity-invariant. The equations of motion can be solved in the Laplace domain, in terms of the density of right-movers at the origin. At large time, this density is shown to grow exponentially, at a rate that depends only on the fertility function and fertility rate. Moreover, the total density of RTPs (divided by the density of right-movers at the origin), reaches a stationary state that does not depend on the initial conditions, and presents a local minimum at the fertile site.
Keywords
Cite
@article{arxiv.2012.08276,
title = {Run-and-tumble particles on a line with a fertile site},
author = {Pascal Grange and Xueqi Yao},
journal= {arXiv preprint arXiv:2012.08276},
year = {2021}
}
Comments
28 pages, LaTeX; v2: typos corrected; v3: Eq. 81 corrected, more typos corrected; v4: more typos corrected, references added; v5: more typos corrected; v6: more typos corrected, reference added, discussion expanded, accepted version