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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 |
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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