Emergent flocking dynamics of the discrete thermodynamic Cucker-Smale model
Authors:
Seung-Yeal Ha, Doheon Kim and Zhuchun Li
Journal:
Quart. Appl. Math. 78 (2020), 589-615
MSC (2010):
Primary 92D25
DOI:
https://doi.org/10.1090/qam/1565
Published electronically:
December 20, 2019
MathSciNet review:
4148820
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Abstract: We present two sufficient frameworks for the emergent dynamics to the discrete thermodynamic Cucker-Smale (TCS) model. Our proposed frameworks are formulated in terms of the initial data and system parameters. The TCS model was first introduced to incorporate the effect of a temperature field in the dynamics of the Cucker-Smale model, and it has been systematically derived from the hydrodynamic model for gas mixture under the spatial homogeneity assumption. The particle model by Cucker and Smale describes the temporal evolution of mechanical observables such as position and velocity, whereas our TCS model governs the dynamics of position, velocity, and temperature of thermodynamic C-S particles. The TCS model conserves the mass, momentum, and energy, and the total entropy is monotonically increasing so that it is consistent with the principle of thermodynamics.
References
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- Seung-Yeal Ha and Tommaso Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1397–1425. MR 3594359, DOI https://doi.org/10.1007/s00205-016-1062-3
- Seung-Yeal Ha, Kiseop Lee, and Doron Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci. 7 (2009), no. 2, 453–469. MR 2536447
- Seung-Yeal Ha, Sungeun Jung, and Marshall Slemrod, Fast-slow dynamics of planar particle models for flocking and swarming, J. Differential Equations 252 (2012), no. 3, 2563–2579. MR 2860630, DOI https://doi.org/10.1016/j.jde.2011.09.014
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- Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440
- Seung-Yeal Ha and Marshall Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations 22 (2010), no. 2, 325–330. MR 2665438, DOI https://doi.org/10.1007/s10884-009-9142-9
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- Zhuchun Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3683–3702. MR 3190999, DOI https://doi.org/10.3934/dcds.2014.34.3683
- Zhuchun Li and Seung-Yeal Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math. 73 (2015), no. 4, 693–709. MR 3432279, DOI https://doi.org/10.1090/S0033-569X-2015-01401-9
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- Jaemann Park, H. Jin Kim, and Seung-Yeal Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control 55 (2010), no. 11, 2617–2623. MR 2721906, DOI https://doi.org/10.1109/TAC.2010.2061070
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- John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI https://doi.org/10.1103/PhysRevE.58.4828
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI https://doi.org/10.1137/S0036139903437424
- Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), no. 6, 1226–1229. MR 3363421, DOI https://doi.org/10.1103/PhysRevLett.75.1226
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967). 15–42.
References
- Shin Mi Ahn, Heesun Choi, Seung-Yeal Ha, and Ho Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci. 10 (2012), no. 2, 625–643. MR 2901323, DOI https://doi.org/10.4310/CMS.2012.v10.n2.a10
- Shin Mi Ahn and Seung-Yeal Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys. 51 (2010), no. 10, 103301, 17. MR 2761313, DOI https://doi.org/10.1063/1.3496895
- François Bolley, José A. Cañizo, and José A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci. 21 (2011), no. 11, 2179–2210. MR 2860672, DOI https://doi.org/10.1142/S0218202511005702
- J. A. Cañizo, J. A. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci. 21 (2011), no. 3, 515–539. MR 2782723, DOI https://doi.org/10.1142/S0218202511005131
- J. A. Carrillo, M. R. D’Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models 2 (2009), no. 2, 363–378. MR 2507454, DOI https://doi.org/10.3934/krm.2009.2.363
- 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), no. 1, 218–236. MR 2596552, DOI https://doi.org/10.1137/090757290
- José A. Carrillo, Axel Klar, Stephan Martin, and Sudarshan Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci. 20 (2010), no. suppl. 1, 1533–1552. MR 3090592, DOI https://doi.org/10.1142/S0218202510004684
- Junghee Cho, Seung-Yeal Ha, Feimin Huang, Chunyin Jin, and Dongnam Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci. 26 (2016), no. 6, 1191–1218. MR 3484572, DOI https://doi.org/10.1142/S0218202516500287
- Junghee Cho, Seung-Yeal Ha, Feimin Huang, Chunyin Jin, and Dongnam Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl. (Singap.) 14 (2016), no. 1, 39–73. MR 3438646, DOI https://doi.org/10.1142/S0219530515400023
- Young-Pil Choi and Seung-Yeal Ha, A simple proof of the complete consensus of discrete-time dynamical networks with time-varying couplings, Int. J. Numer. Anal. Model. Ser. B 1 (2010), no. 1, 58–69. MR 2837400
- Felipe Cucker and Ernesto Mordecki, Flocking in noisy environments, J. Math. Pures Appl. (9) 89 (2008), no. 3, 278–296 (English, with English and French summaries). MR 2401690, DOI https://doi.org/10.1016/j.matpur.2007.12.002
- Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245, DOI https://doi.org/10.1109/TAC.2007.895842
- Pierre Degond and Sébastien Motsch, Macroscopic limit of self-driven particles with orientation interaction, C. R. Math. Acad. Sci. Paris 345 (2007), no. 10, 555–560 (English, with English and French summaries). MR 2374464, DOI https://doi.org/10.1016/j.crma.2007.10.024
- Pierre Degond and Sébastien Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys. 131 (2008), no. 6, 989–1021. MR 2407377, DOI https://doi.org/10.1007/s10955-008-9529-8
- Pierre Degond and Sébastien Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008), no. suppl., 1193–1215. MR 2438213, DOI https://doi.org/10.1142/S0218202508003005
- Renjun Duan, Massimo Fornasier, and Giuseppe Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys. 300 (2010), no. 1, 95–145. MR 2725184, DOI https://doi.org/10.1007/s00220-010-1110-z
- Seung-Yeal Ha, Taeyoung Ha, and Jong-Ho Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A 43 (2010), no. 31, 315201, 19. MR 2665672, DOI https://doi.org/10.1088/1751-8113/43/31/315201
- Seung-Yeal Ha, Jeongho Kim, and Tommaso Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal. 50 (2018), no. 3, 3092–3121. MR 3814022, DOI https://doi.org/10.1137/17M111064X
- Seung-Yeal Ha and Tommaso Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1397–1425. MR 3594359, DOI https://doi.org/10.1007/s00205-016-1062-3
- Seung-Yeal Ha, Kiseop Lee, and Doron Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci. 7 (2009), no. 2, 453–469. MR 2536447
- Seung-Yeal Ha, Sungeun Jung, and Marshall Slemrod, Fast-slow dynamics of planar particle models for flocking and swarming, J. Differential Equations 252 (2012), no. 3, 2563–2579. MR 2860630, DOI https://doi.org/10.1016/j.jde.2011.09.014
- Seung-Yeal Ha, Corrado Lattanzio, Bruno Rubino, and Marshall Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math. 69 (2011), no. 1, 91–103. MR 2807979, DOI https://doi.org/10.1090/S0033-569X-2010-01200-7
- Seung-Yeal Ha, Zhuchun Li, Marshall Slemrod, and Xiaoping Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math. 72 (2014), no. 4, 689–701. MR 3291822, DOI https://doi.org/10.1090/S0033-569X-2014-01350-5
- Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440
- Seung-Yeal Ha and Marshall Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations 22 (2010), no. 2, 325–330. MR 2665438, DOI https://doi.org/10.1007/s10884-009-9142-9
- Seung-Yeal Ha and Eitan Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415–435. MR 2425606, DOI https://doi.org/10.3934/krm.2008.1.415
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Physics 39 (1975), 420.
- Zhuchun Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3683–3702. MR 3190999, DOI https://doi.org/10.3934/dcds.2014.34.3683
- Zhuchun Li and Seung-Yeal Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math. 73 (2015), no. 4, 693–709. MR 3432279, DOI https://doi.org/10.1090/qam/1401
- Zhuchun Li, Seung-Yeal Ha, and Xiaoping Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci. 24 (2014), no. 7, 1389–1419. MR 3192593, DOI https://doi.org/10.1142/S0218202514500043
- Zhuchun Li and Xiaoping Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math. 70 (2010), no. 8, 3156–3174. MR 2763499, DOI https://doi.org/10.1137/100791774
- Sebastien Motsch and Eitan Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), no. 5, 923–947. MR 2836613, DOI https://doi.org/10.1007/s10955-011-0285-9
- L. Perea, P. Elosegui, and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dynam. 32 (2009), 526–536.
- Jaemann Park, H. Jin Kim, and Seung-Yeal Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control 55 (2010), no. 11, 2617–2623. MR 2721906, DOI https://doi.org/10.1109/TAC.2010.2061070
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. MR 1869044
- Jackie Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math. 68 (2007/08), no. 3, 694–719. MR 2375291, DOI https://doi.org/10.1137/060673254
- John Toner and Yuhai Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324, DOI https://doi.org/10.1103/PhysRevE.58.4828
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI https://doi.org/10.1137/S0036139903437424
- Tamás Vicsek, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), no. 6, 1226–1229. MR 3363421, DOI https://doi.org/10.1103/PhysRevLett.75.1226
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967). 15–42.
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea — and — Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Doheon Kim
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
MR Author ID:
1277907
Email:
doheonkim@kias.re.kr
Zhuchun Li
Affiliation:
School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China
MR Author ID:
912583
ORCID:
0000-0001-5278-1927
Email:
lizhuchun@hit.edu.cn
Keywords:
Collective behaviors,
discrete-time dynamics,
thermodynamic Cucker-Smale model
Received by editor(s):
August 13, 2019
Received by editor(s) in revised form:
October 27, 2019
Published electronically:
December 20, 2019
Additional Notes:
The first author was supported by NRF grant (NRF-2017R1A2B2001864).
The third author was supported by the Heilongjiang Provincial Natural Science Foundation of China (LH2019A012) and the Natural Science Foundation of China (11671109). The third author is the corresponding author.
Article copyright:
© Copyright 2019
Brown University