Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles

Michele Gianfelice; Enza Orlandi; Michele Gianfelice; Enza Orlandi

doi:10.3934/nhm.2014.9.269

Networks and Heterogeneous Media

2014, Volume 9, Issue 2: 269-297. doi: 10.3934/nhm.2014.9.269

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Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles

Michele Gianfelice ^{1
,},
Enza Orlandi ^{2
,}

1.
Dipartimento di Matematica, Università della Calabria, Campus di Arcavacata, Ponte P. Bucci - cubo 30B, 87036 Arcavacata di Rende (CS)
2.
Dipartimento di Matematica, Università di Roma Tre, L.go S.Murialdo 1, 00146 Roma

Received: 01 November 2012 Revised: 01 February 2014
Primary: 35F20, 35Q83, 35B40; Secondary: 82C22, 35Q82, 82C40, 92D50, 91B80.

We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one [29]. 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. The weights are given in terms of the interaction rate function. The interaction is not of mean field type and the system is non-Hamiltonian. 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 nonlinear kinetic equation of Vlasov type, 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 in time.
- Flocking,
- swarming,
- self-propelled particle systems,
- agents network,
- kinetic limit
Citation: Michele Gianfelice, Enza Orlandi. Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles[J]. Networks and Heterogeneous Media, 2014, 9(2): 269-297. doi: 10.3934/nhm.2014.9.269

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Abstract

We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one [29]. 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. The weights are given in terms of the interaction rate function. The interaction is not of mean field type and the system is non-Hamiltonian. 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 nonlinear kinetic equation of Vlasov type, 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 in time.

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