American Institute of Mathematical Sciences

Advanced
April  2021, 14(2): 323-351. doi: 10.3934/krm.2021007

Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds

Hyunjin Ahn 1, , Seung-Yeal Ha 2,3, and  Woojoo Shim 4,,

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
3. 

School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
4. 

Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Woojoo Shim

Received  September 2020 Revised  December 2020 Published  April 2021 Early access  January 2021

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881). The author would like to thank Prof. Marshall Slemrod for the suggestion of LaSalle type arguments in Corollary 1, Corollary 2 and Theorem 5.2

We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [30] for a CS ensemble, and provide a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the model proposed in [30], our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for isothermal case, our model reduces to the previous CS model in [30] on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit $ d $-sphere $ \mathbb{S}^d $. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit $ d $-sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on $ \mathbb{S}^2 $.

Keywords: Cucker-Smale model, Emergence, Entropy principle, Thermomechanical Cucker-Smale model, Riemannian manifold.
Mathematics Subject Classification: Primary: 34E10; Secondary: 82C22.
Citation: Hyunjin Ahn, Seung-Yeal Ha, Woojoo Shim. Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds. Kinetic and Related Models, 2021, 14 (2) : 323-351. doi: 10.3934/krm.2021007
References:
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J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. 

[2]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

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A. Aydo$\breve{g}$du, S. T. McQuade and N. Pouradier Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.  doi: 10.3934/nhm.2017021.

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I. Barbalat, Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

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J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

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J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.  doi: 10.1007/s10955-019-02442-w.

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H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456. 

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Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.

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Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.  doi: 10.1007/s00021-019-0466-x.

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Y.-P. Choi, S.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

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L. DeVille, Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

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P. Degond, A. Frouvelle and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.  doi: 10.1142/S0218202517400085.

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J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp. doi: 10.1063/1.5084770.

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J.-G. Dong, S.-Y. Ha and D. Kim, On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.  doi: 10.1137/18M1195462.

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J.-G. Dong, S.-Y. Ha and D. Kim, Emergence of mono-cluster flocking in the thermomechanical Cucker-Smale model under switching topologies, Analysis and Applications, (2020), 1-38. doi: 10.1142/S0219530520500025.

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J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

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R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications.

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A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479. doi: 10.1007/978-3-030-15096-9_16.

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R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.

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S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp. doi: 10.1063/1.5066255.

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S.-Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996.

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S.-Y. Ha and D. Kim, Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.  doi: 10.1007/s10955-019-02270-y.

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S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[25]

S.-Y. Ha, J. Kim, J. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.

[26]

S.-Y. Ha, J. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[27]

S.-Y. Ha, J. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[28]

S.-Y. Ha, D. Ko and S. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.

[29]

S.-Y. Ha, D. Ko and S. Ryoo, On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.  doi: 10.1007/s10955-018-2091-0.

[30]

S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020). doi: 10.1109/TAC.2020.3014096.

[31]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[32]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal, 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

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J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011. doi: 10.1007/978-3-642-21298-7.

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M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.

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J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019). doi: 10.1109/TAC.2020.3030849.

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J. Markdahl, J. Thunberg and J. Gonçalves, Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

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S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[40]

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.

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[42]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.

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C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34.  doi: 10.1145/280811.281008.

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L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp. doi: 10.1063/1.5021701.

[46]

A. Sarlette, S. Bonnabel and R. Sepulchre, Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.

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A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM J. Control Optim., 48 (2009), 56-76.  doi: 10.1137/060673400.

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R. Tron, B. Afsari and R. Vidal, Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Contr., 58 (2013), 921-934.  doi: 10.1109/TAC.2012.2225533.

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C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004) 152-174. doi: 10.1137/S0036139903437424.

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J. Toner and Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

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T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

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show all references

References:
[1]

J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. 

[2]

G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[3]

A. Aydo$\breve{g}$du, S. T. McQuade and N. Pouradier Duteil, Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.  doi: 10.3934/nhm.2017021.

[4]

I. Barbalat, Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. 

[5]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[6]

J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.  doi: 10.1007/s10955-019-02442-w.

[7]

H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456. 

[8]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.

[9]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.  doi: 10.1007/s00021-019-0466-x.

[10]

Y.-P. Choi, S.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[11]

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

[12]

L. DeVille, Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

[13]

P. Degond, A. Frouvelle and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.  doi: 10.1142/S0218202517400085.

[14]

J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp. doi: 10.1063/1.5084770.

[15]

J.-G. Dong, S.-Y. Ha and D. Kim, On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.  doi: 10.1137/18M1195462.

[16]

J.-G. Dong, S.-Y. Ha and D. Kim, Emergence of mono-cluster flocking in the thermomechanical Cucker-Smale model under switching topologies, Analysis and Applications, (2020), 1-38. doi: 10.1142/S0219530520500025.

[17]

J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.  doi: 10.1016/j.jde.2018.08.034.

[18]

R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications.

[19]

A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479. doi: 10.1007/978-3-030-15096-9_16.

[20]

R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.

[21]

S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp. doi: 10.1063/1.5066255.

[22]

S.-Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.  doi: 10.1137/18M1205996.

[23]

S.-Y. Ha and D. Kim, Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.  doi: 10.1007/s10955-019-02270-y.

[24]

S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[25]

S.-Y. Ha, J. Kim, J. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.

[26]

S.-Y. Ha, J. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[27]

S.-Y. Ha, J. Kim and T. Ruggeri, From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.  doi: 10.1007/s00205-019-01452-y.

[28]

S.-Y. Ha, D. Ko and S. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.  doi: 10.1007/s10955-017-1797-8.

[29]

S.-Y. Ha, D. Ko and S. Ryoo, On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.  doi: 10.1007/s10955-018-2091-0.

[30]

S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020). doi: 10.1109/TAC.2020.3014096.

[31]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[32]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal, 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

[33] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013.  doi: 10.1016/B978-0-12-382010-5.00001-4.
[34]

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

[35]

J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011. doi: 10.1007/978-3-642-21298-7.

[36]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.

[37]

J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019). doi: 10.1109/TAC.2020.3030849.

[38]

J. Markdahl, J. Thunberg and J. Gonçalves, Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

[39]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[40]

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.

[41]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060-5066. doi: 10.1109/CDC.2006.376811.

[42]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.

[43] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[44]

C. W. Reynolds, Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34.  doi: 10.1145/280811.281008.

[45]

L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp. doi: 10.1063/1.5021701.

[46]

A. Sarlette, S. Bonnabel and R. Sepulchre, Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.

[47]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM J. Control Optim., 48 (2009), 56-76.  doi: 10.1137/060673400.

[48]

R. Tron, B. Afsari and R. Vidal, Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Contr., 58 (2013), 921-934.  doi: 10.1109/TAC.2012.2225533.

[49]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004) 152-174. doi: 10.1137/S0036139903437424.

[50]

J. Toner and Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

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T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

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Figure 1.  Configurations of TCS particles at time $ t = 0 $ and $ t = 200 $
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Figure 2.  Temporal evolutions of $ T_i $'s and $ \ln \mathcal{E} $
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Figure 3.  TCS particles at $ t = 0 $ and $ t\geq 50 $
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Figure 4.  Exponential convergence of $ \mathcal{E} $ to zero
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