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

2016/07/31 by Moon-Jin Kang, Kang, Moon-Jin, Javier Morales +1
Environmental Science · Mathematics · #Analysis of PDEs (math.AP) #Ecosystem dynamics and resilience #FOS: Mathematics #Mathematical Biology Tumor Growth #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1608.00185

openalex publication_date 2016/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 the plane at a constant speed, under space-homogeneity. In \citeF-K-M, Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior has been 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.

Citations

• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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