Tuesday, October 23, 2012

1210.5746 (Michele Gianfelice et al.)

Dynamics And Kinetic Limit For A System Of Noiseless D-Dimensional
Vicsek-Type Particles

Michele Gianfelice, Enza Orlandi
We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles subject to a feedback rule inspired by the original Vicsek's one \cite{VCB-JCS}. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it, which makes the system non-Hamiltonian. The weights are given in terms of the interaction rate function. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponential asymptotic stability. In the kinetic limit we show that the particle density satisfies a Boltzmann-Vlasov equation under suitable conditions on the interaction. We study the qualitative behaviour of the solution and we show that the Boltzmann-Vlasov entropy is strictly decreasing.
View original: http://arxiv.org/abs/1210.5746

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