  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 \cite{F-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.
