Dynamics of a spatially homogeneous Vicsek model for oriented particles on a plane

Abstract

We consider a spatially homogeneous Kolmogorov-Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov-Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869-896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality.