# American Institute of Mathematical Sciences

June  2014, 9(2): 269-297. doi: 10.3934/nhm.2014.9.269

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

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

Received  November 2012 Revised  February 2014 Published  July 2014

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.
Citation: Michele Gianfelice, Enza Orlandi. Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles. Networks & Heterogeneous Media, 2014, 9 (2) : 269-297. doi: 10.3934/nhm.2014.9.269
##### References:
 [1] V. I. Arnold, Equations Differentielles Ordinaires,, Editions Mir, (1974). Google Scholar [2] V. I. Arnold, Mathematical Methods of Classical Mechanics,, 2nd edition, (1989). doi: 10.1007/978-1-4757-2063-1. Google Scholar [3] M. Agueh, R. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type,, Kinetic and Related Models, 4 (2011), 1. doi: 10.3934/krm.2011.4.1. Google Scholar [4] E. Ben-Naim, F. Vazquez and S. Redner, On the structure of competitive societies,, Eur. Phys. J. B, 49 (2006), 531. doi: 10.1140/epjb/e2006-00095-y. Google Scholar [5] B. Bollobás, Modern Graph Theory,, Graduate Texts in Mathematics, (1998). doi: 10.1007/978-1-4612-0619-4. Google Scholar [6] F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces $&$ swarming,, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179. doi: 10.1142/S0218202511005702. Google Scholar [7] F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model,, Appl. Math. Letters, 25 (2012), 339. doi: 10.1016/j.aml.2011.09.011. Google Scholar [8] E. Bonabeau, M. Dorigo and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems,, Oxford University Press, (1999). Google Scholar [9] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Mod. Appl. Sci., 21 (2011), 515. doi: 10.1142/S0218202511005131. Google Scholar [10] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363. Google Scholar [11] J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218. doi: 10.1137/090757290. Google Scholar [12] Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials,, in IEEE International Conference on Robotics and Automation, (2007), 2292. doi: 10.1109/ROBOT.2007.363661. Google Scholar [13] I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin, Effective leadership and decision making in animal groups on the move,, Nature, 433 (2005), 513. doi: 10.1038/nature03236. Google Scholar [14] F. Cucker and S. Smale, Emergent behaviour in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar [15] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar [16] R. Dobrushin, Vlasov equations,, Funktsional. Anal. i Prilozhen., 13 (1979), 48. Google Scholar [17] M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302. Google Scholar [18] A. Dragulescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jour. B, 17 (2000), 723. doi: 10.1007/s100510070114. Google Scholar [19] S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit,, Commun. Math. Sci., 7 (2009), 297. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar [20] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinetic and Related Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar [21] A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules,, IEEE Trans. on Autom. Control., 48 (2003), 988. doi: 10.1109/TAC.2003.812781. Google Scholar [22] P. Malliavin, Integration and Probability,, Graduate Texts in Mathematics, (1995). doi: 10.1007/978-1-4612-4202-4. Google Scholar [23] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar [24] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, in Kinetic Theories and the Boltzmann Equation (Montecatini, (1981), 60. doi: 10.1007/BFb0071878. Google Scholar [25] C. Reynolds, Flocks, birds and schools: A distributed behavioural model,, Comput. Graph., 21 (1987), 25. Google Scholar [26] H. Spohn, Large Scale Dynamics of Interacting Particles,, Texts and Monographs in Physics, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar [27] D. W. Strook, An Introduction to Markov Processes,, Graduate Texts in Mathematics, (2005). Google Scholar [28] C. Villani, Optimal Transport Old and New,, A Series of Comprehensive Studies in Mathematics, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [29] T.Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. Google Scholar [30] W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences,, Harwood Academic Publishers, (2000). Google Scholar

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##### References:
 [1] V. I. Arnold, Equations Differentielles Ordinaires,, Editions Mir, (1974). Google Scholar [2] V. I. Arnold, Mathematical Methods of Classical Mechanics,, 2nd edition, (1989). doi: 10.1007/978-1-4757-2063-1. Google Scholar [3] M. Agueh, R. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type,, Kinetic and Related Models, 4 (2011), 1. doi: 10.3934/krm.2011.4.1. Google Scholar [4] E. Ben-Naim, F. Vazquez and S. Redner, On the structure of competitive societies,, Eur. Phys. J. B, 49 (2006), 531. doi: 10.1140/epjb/e2006-00095-y. Google Scholar [5] B. Bollobás, Modern Graph Theory,, Graduate Texts in Mathematics, (1998). doi: 10.1007/978-1-4612-0619-4. Google Scholar [6] F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces $&$ swarming,, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179. doi: 10.1142/S0218202511005702. Google Scholar [7] F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model,, Appl. Math. Letters, 25 (2012), 339. doi: 10.1016/j.aml.2011.09.011. Google Scholar [8] E. Bonabeau, M. Dorigo and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems,, Oxford University Press, (1999). Google Scholar [9] J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Mod. Appl. Sci., 21 (2011), 515. doi: 10.1142/S0218202511005131. Google Scholar [10] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363. Google Scholar [11] J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218. doi: 10.1137/090757290. Google Scholar [12] Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials,, in IEEE International Conference on Robotics and Automation, (2007), 2292. doi: 10.1109/ROBOT.2007.363661. Google Scholar [13] I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin, Effective leadership and decision making in animal groups on the move,, Nature, 433 (2005), 513. doi: 10.1038/nature03236. Google Scholar [14] F. Cucker and S. Smale, Emergent behaviour in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar [15] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar [16] R. Dobrushin, Vlasov equations,, Funktsional. Anal. i Prilozhen., 13 (1979), 48. Google Scholar [17] M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302. Google Scholar [18] A. Dragulescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jour. B, 17 (2000), 723. doi: 10.1007/s100510070114. Google Scholar [19] S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit,, Commun. Math. Sci., 7 (2009), 297. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar [20] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinetic and Related Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar [21] A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules,, IEEE Trans. on Autom. Control., 48 (2003), 988. doi: 10.1109/TAC.2003.812781. Google Scholar [22] P. Malliavin, Integration and Probability,, Graduate Texts in Mathematics, (1995). doi: 10.1007/978-1-4612-4202-4. Google Scholar [23] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar [24] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, in Kinetic Theories and the Boltzmann Equation (Montecatini, (1981), 60. doi: 10.1007/BFb0071878. Google Scholar [25] C. Reynolds, Flocks, birds and schools: A distributed behavioural model,, Comput. Graph., 21 (1987), 25. Google Scholar [26] H. Spohn, Large Scale Dynamics of Interacting Particles,, Texts and Monographs in Physics, (1991). doi: 10.1007/978-3-642-84371-6. Google Scholar [27] D. W. Strook, An Introduction to Markov Processes,, Graduate Texts in Mathematics, (2005). Google Scholar [28] C. Villani, Optimal Transport Old and New,, A Series of Comprehensive Studies in Mathematics, (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [29] T.Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. Google Scholar [30] W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences,, Harwood Academic Publishers, (2000). Google Scholar
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