Ha, S-Y; Liu, J-G, *A simple proof of the Cucker-Smale flocking dynamics and mean-field limit*,
Communications in Mathematical Sciences, vol. 7 no. 2
(2009),
pp. 297-325 .
**Abstract:**

*We present a simple proof on the formation of flocking to the Cucker-Smale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity. © 2009 International Press.*