Active particles in one dimension

Abstract

In this thesis, we study the asymptotic behaviour and the dynamics of a one-dimensional active particle model with excluded volume interactions. The model is a version of run-and-tumble motion, where a particle performs both symmetric random walks and active transport. The direction and the speed of the transport are governed by an internal state process. We show that this motion converges to Brownian motion upon diffusive scaling and determine the limiting diffusion coefficient. The internal state converges to a stationary distribution, by which it manifests itself in the diffusion coefficient. Furthermore, we prove that the active particle satisfies the large deviation principle. This allows us to derive an implicit expression for the rate by which the probability of rare events tends to zero. Numerically, we investigate the influence of excluded volume interactions on the diffusion coefficient and the average velocity. We find that the velocity converges exponentially to its theoretical value as the number of particles allowed per position increases. In addition, this exclusion number strongly influences the manner in which the velocity decreases for high particle densities. Predictions for the velocity as a function of particle density based on the model are compared to experimental data of the molecular motor kinesin-II. We find that the model is not adequate for approximating the velocity of molecular motors in crowded environments and extensions in the form of Langmuir kinetics are suggested.