A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms

TL;DR

This paper develops a mean-field theory to analyze phase transitions in two-dimensional self-propelled particles with velocity alignment, revealing critical noise thresholds and confirming predictions with simulations.

Contribution

It introduces a mean-field approach for self-propelled particles with velocity alignment, analyzing ferromagnetic and liquid-crystal cases, and validates predictions with simulations.

Findings

Second order phase transition at critical noise strength

Scaling exponent of 1/2 for order parameters

Critical noise amplitude in LC case is smaller than in F case

Abstract

A mean-field approach (MFA) is proposed for the analysis of orientational order in a two-dimensional system of stochastic self-propelled particles interacting by local velocity alignment mechanism. The treatment is applied to the cases of ferromagnetic (F) and liquid-crystal (LC) alignment. In both cases, MFA yields a second order phase transition for a critical noise strength and a scaling exponent of 1/2 for the respective order parameters. We find that the critical noise amplitude $\eta_c$ at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. $\eta^{LC}_{C}<\eta^{F}_{C}$. A comparison with simulations of individual-based models with F- resp. LC-alignment shows that the predictions about the critical behavior and the qualitative relation between the respective critical noise amplitudes are correct.