Collection of polar self-propelled particles with a modified alignment interaction

Abstract

We study the disorder-to-order transition in a collection of polar self-propelled particles interacting through a distance dependent alignment interaction. Strength of the interaction, $a^d$ ($0<a<1$) decays with metric distance $d$ between particle pair, and the interaction is short range. At $a=1.0$, our model reduces to the famous Vicsek model. For all $a>0$, the system shows a transition from a disordered to an ordered state as a function of noise strength. We calculate the critical noise strength, ${\eta }_c(a)$ for different $a$ and compare it with the mean-field result. Nature of the disorder-to-order transition continuously changes from discontinuous to continuous with decreasing $a$. We numerically estimate tri-critical point $a_{TCP}$ at which the nature of transition changes from discontinuous to continuous. The density phase separation is large for $a$ close to unity, and it decays with decreasing $a$. We also write the coarse-grained hydrodynamic equations of motion for general $a$, and find that the homogeneous ordered state is unstable to small perturbation as $a$ approaches to $1$. The instability in the homogeneous ordered state is consistent with the large density phase separation for $a$ close to unity.

17 pages and 7 figures