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 and Heterogeneous Media, 2014, 9 (2) : 269-297. doi: 10.3934/nhm.2014.9.269
References:
[1]

V. I. Arnold, Equations Differentielles Ordinaires, Editions Mir, Moscou, 1974.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, Heidelberg, 1989. doi: 10.1007/978-1-4757-2063-1.

[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), 1-16. doi: 10.3934/krm.2011.4.1.

[4]

E. Ben-Naim, F. Vazquez and S. Redner, On the structure of competitive societies, Eur. Phys. J. B, 49 (2006), 531-538. doi: 10.1140/epjb/e2006-00095-y.

[5]

B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, 184, Springer, 1998. doi: 10.1007/978-1-4612-0619-4.

[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), 2179-2210. doi: 10.1142/S0218202511005702.

[7]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Letters, 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

[8]

E. Bonabeau, M. Dorigo and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, New York, 1999.

[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), 515-539. doi: 10.1142/S0218202511005131.

[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), 363-378. doi: 10.3934/krm.2009.2.363.

[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), 218-236. doi: 10.1137/090757290.

[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), 2292-2299. doi: 10.1109/ROBOT.2007.363661.

[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), 513-516. doi: 10.1038/nature03236.

[14]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[15]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[16]

R. Dobrushin, Vlasov equations, Funktsional. Anal. i Prilozhen., 13 (1979), 48-58; English translation in Functional Anal. Appl., 13 (1979), 115-123.

[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), 104302-1/4. doi: 10.1103/PhysRevLett.96.104302.

[18]

A. Dragulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729. doi: 10.1007/s100510070114.

[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), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[20]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[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), 988-1001. doi: 10.1109/TAC.2003.812781.

[22]

P. Malliavin, Integration and Probability, Graduate Texts in Mathematics, 157, Springer Verlag, Berlin Heidelberg, 1995. doi: 10.1007/978-1-4612-4202-4.

[23]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[24]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Mathematics, 1048, Springer Verlag, Heidelberg, 1984, 60-110. doi: 10.1007/BFb0071878.

[25]

C. Reynolds, Flocks, birds and schools: A distributed behavioural model, Comput. Graph., 21 (1987), 25-34.

[26]

H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics, Springer Verlag, Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[27]

D. W. Strook, An Introduction to Markov Processes, Graduate Texts in Mathematics, 230, Springer Verlag, Berlin Heidelberg, 2005.

[28]

C. Villani, Optimal Transport Old and New, A Series of Comprehensive Studies in Mathematics, 338, Springer Verlag, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-71050-9.

[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), 1226-1229.

[30]

W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences, Harwood Academic Publishers, 2000.

show all references

References:
[1]

V. I. Arnold, Equations Differentielles Ordinaires, Editions Mir, Moscou, 1974.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, Graduate Texts in Mathematics, 60, Springer Verlag, Heidelberg, 1989. doi: 10.1007/978-1-4757-2063-1.

[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), 1-16. doi: 10.3934/krm.2011.4.1.

[4]

E. Ben-Naim, F. Vazquez and S. Redner, On the structure of competitive societies, Eur. Phys. J. B, 49 (2006), 531-538. doi: 10.1140/epjb/e2006-00095-y.

[5]

B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, 184, Springer, 1998. doi: 10.1007/978-1-4612-0619-4.

[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), 2179-2210. doi: 10.1142/S0218202511005702.

[7]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Letters, 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

[8]

E. Bonabeau, M. Dorigo and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, New York, 1999.

[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), 515-539. doi: 10.1142/S0218202511005131.

[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), 363-378. doi: 10.3934/krm.2009.2.363.

[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), 218-236. doi: 10.1137/090757290.

[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), 2292-2299. doi: 10.1109/ROBOT.2007.363661.

[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), 513-516. doi: 10.1038/nature03236.

[14]

F. Cucker and S. Smale, Emergent behaviour in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[15]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[16]

R. Dobrushin, Vlasov equations, Funktsional. Anal. i Prilozhen., 13 (1979), 48-58; English translation in Functional Anal. Appl., 13 (1979), 115-123.

[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), 104302-1/4. doi: 10.1103/PhysRevLett.96.104302.

[18]

A. Dragulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729. doi: 10.1007/s100510070114.

[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), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[20]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[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), 988-1001. doi: 10.1109/TAC.2003.812781.

[22]

P. Malliavin, Integration and Probability, Graduate Texts in Mathematics, 157, Springer Verlag, Berlin Heidelberg, 1995. doi: 10.1007/978-1-4612-4202-4.

[23]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[24]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Mathematics, 1048, Springer Verlag, Heidelberg, 1984, 60-110. doi: 10.1007/BFb0071878.

[25]

C. Reynolds, Flocks, birds and schools: A distributed behavioural model, Comput. Graph., 21 (1987), 25-34.

[26]

H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics, Springer Verlag, Heidelberg, 1991. doi: 10.1007/978-3-642-84371-6.

[27]

D. W. Strook, An Introduction to Markov Processes, Graduate Texts in Mathematics, 230, Springer Verlag, Berlin Heidelberg, 2005.

[28]

C. Villani, Optimal Transport Old and New, A Series of Comprehensive Studies in Mathematics, 338, Springer Verlag, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-71050-9.

[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), 1226-1229.

[30]

W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences, Harwood Academic Publishers, 2000.

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