Self-propelled Vicsek particles at low speed and low density

TL;DR

This study investigates the behavior of the Vicsek model at very low speeds and densities through numerical simulations, revealing new power-law relationships for critical noise and relaxation times in different noise types and dimensions.

Contribution

It provides new insights into the critical noise dependence on speed and density at low parameters, especially highlighting differences between scalar and vector noise cases.

Findings

Critical noise follows a power law in density and speed for scalar noise.

The density exponent in 2D matches previous results, but the speed exponent differs at low speeds.

Relaxation time scales with a power law in both 2D and 3D, with exponents depending on noise levels.

Abstract

We study through numerical simulation the Vicsek model for very low speeds and densities. We consider scalar noise in 2-d and 3-d, and vector noise in 3-d. We focus on the behavior of the critical noise with density and speed, trying to clarify seemingly contradictory earlier results. We find that, for scalar noise, the critical noise is a power law both in density and speed, but although we confirm the density exponent in 2-d, we find a speed exponent different from earlier reports (we consider lower speeds than previous studies). On the other hand, for the vector noise case we find that the dependence of the critical noise cannot be separated as a product of power laws in speed and density. Finally, we study the dependence of the relaxation time with speed and find the same power law in 2-d and 3-d, with and exponent that depends on whether the noise is above or below the critical value.