Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination
Tóm tắt
In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in
$${\mathbb {R}}^3$$
as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017).
Tài liệu tham khảo
Albi, G., Bellomo, N., Fermo, L., Kim, J., Pareschi, L., Poyato, D., Soler, J., et al.: Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives. Math. Models Methods Appl. Sci. 29(10), 1901–2005 (2019)
Ball, J.M.: Mathematics and liquid crystals. Mol. Cryst. Liq. Cryst. 647(1), 1–27 (2017)
Ball, J.M., Majumdar, A.: Nematic liquid crystals: from Maier–Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11 (2010)
Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(3), 511 (1954)
Bolley, F., Canizo, J.A., Carrillo, J.A.: Stochastic mean-field limit: non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21(11), 2179–2210 (2011)
Bolley, F., Cañizo, J.A., Carrillo, J.A.: Mean-field limit for the stochastic Vicsek model. Appl. Math. Lett. 25(3), 339–343 (2012)
Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33(5), 651–666 (1980)
Carrillo, J.A., Choi, Y.-P., Perez, S.P.: A review on attractive-repulsive hydrodynamics for consensus in collective behavior. In: Bellomo, N., Degond, P., Tadmor, E. (eds.) Active Particles, vol. 1, pp. 259–298. Springer, Berlin (2017)
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, vol. 106. Springer, Berlin (2013)
Chuang, Y.-L., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.S.: State transitions and the continuum limit for a 2d interacting, self-propelled particle system. Phys. D 232(1), 33–47 (2007)
Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)
Degond, P.: Macroscopic limits of the Boltzmann equation: a review. In: Degond, P., Pareschi, L., Russo, G. (eds.) Modeling and Computational Methods for Kinetic Equations, pp. 3–57. Springer, Berlin (2004)
Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18(supp01), 1193–1215 (2008)
Degond, P., Motsch, S.: A macroscopic model for a system of swarming agents using curvature control. J. Stat. Phys. 143(4), 685–714 (2011)
Degond, P., Navoret, L.: A multi-layer model for self-propelled disks interacting through alignment and volume exclusion. Math. Models Methods Appl. Sci. 25(13), 2439–2475 (2015)
Degond, P., Frouvelle, A., Liu, J.-G.: Macroscopic limits and phase transition in a system of self-propelled particles. J. Nonlinear Sci. 23(3), 427–456 (2013)
Degond, P., Frouvelle, A., Liu, J.-G.: Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics. Arch. Ration. Mech. Anal. 216(1), 63–115 (2015)
Degond, P., Frouvelle, A., Merino-Aceituno, S.: A new flocking model through body attitude coordination. Math. Models Methods Appl. Sci. 27(06), 1005–1049 (2017)
Degond, P., Frouvelle, A., Merino-Aceituno, S., Trescases, A.: Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations. arXiv preprint arXiv:1810.06903 (2018a)
Degond, P., Frouvelle, A., Merino-Aceituno, S., Trescases, A.: Quaternions in collective dynamics. Multiscale Model. Simul. 16(1), 28–77 (2018b)
Degond, P., Frouvelle, A., Merino-Aceituno, S., Trescases, A.: Hyperbolicity of SOHB Models (2020)
Diez, A.: Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles. arXiv preprint arXiv:1908.00293 (2019)
Dimarco, G., Motsch, S.: Self-alignment driven by jump processes: macroscopic limit and numerical investigation. Math. Models Methods Appl. Sci. 26(07), 1385–1410 (2016)
Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4(1), 1 (2018)
Figalli, A., Kang, M.-J., Morales, J.: Global well-posedness of the spatially homogeneous Kolmogorov–Vicsek model as a gradient flow. Arch. Ration. Mech. Anal. 227(3), 869–896 (2018)
Gallagher, I., Saint-Raymond, L., Texier, B.: From Newton to Boltzmann: hard spheres and short-range potentials. Zürich lectures in advanced mathematics. European Mathematical Society (2013). ISBN: 9783037191293
Gamba, I.M., Kang, M.-J.: Global weak solutions for Kolmogorov–Vicsek type equations with orientational interactions. Arch. Ration. Mech. Anal. 222(1), 317–342 (2016)
Giacomin, G., Pakdaman, K., Pellegrin, X.: Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators. Nonlinearity 25(5), 1247 (2012a)
Giacomin, G., Pakdaman, K., Pellegrin, X., Poquet, C.: Transitions in active rotator systems: invariant hyperbolic manifold approach. SIAM J. Math. Anal. 44(6), 4165–4194 (2012b)
Golse, F., Saint-Raymond, L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004)
Guo, Y., Jang, J.: Global Hilbert expansion for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 299(2), 469–501 (2010)
Ha, S.-Y., Liu, J.-G., et al.: A simple proof of the Cucker–Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7(2), 297–325 (2009)
Han, J., Luo, Y., Wang, W., Zhang, P., Zhang, Z.: From microscopic theory to macroscopic theory: a systematic study on modeling for liquid crystals. Arch. Ration. Mech. Anal. 215(3), 741–809 (2015)
Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. Springer, Berlin (2010)
Haraux, A.: Some applications of the Łojasiewicz gradient inequality. Commun. Pure Appl. Anal. 6, 2417–2427 (2012)
Hauray, M., Jabin, P.-E.: N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183(3), 489–524 (2007)
Hemelrijk, C.K., Hildenbrandt, H.: Schools of fish and flocks of birds: their shape and internal structure by self-organization. Interface Focus 2(6), 726–737 (2012)
Hemelrijk, C.K., Hildenbrandt, H., Reinders, J., Stamhuis, E.J.: Emergence of oblong school shape: models and empirical data of fish. Ethology 116(11), 1099–1112 (2010)
Hildenbrandt, H., Carere, C., Hemelrijk, C.K.: Self-organized aerial displays of thousands of starlings: a model. Behav. Ecol. 21(6), 1349–1359 (2010)
Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, London (2012)
Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math. 76(3), 620–630 (1954)
Jabin, P.-E.: A review of the mean field limits for Vlasov equations. Kinet. Relat. Models 7(4), 661–711 (2014)
Jiang, N., Xiong, L., Zhang, T.-F.: Hydrodynamic limits of the kinetic self-organized models. SIAM J. Math. Anal. 48(5), 3383–3411 (2016)
Jiang, N., Luo, Y.-L., Zhang, T.-F.: Coupled self-organized hydrodynamics and Navier–Stokes models: local well-posedness and the limit from the self-organized kinetic-fluid models. arXiv preprint arXiv:1712.10134 (2017)
Jourdain, B., Méléard, S.: Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. Henri Poincaré Probab. Stat. 34(6), 727–766 (1998)
Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3, pp. 171–197. University of California Press Berkeley and Los Angeles, CA (1956)
Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications, pp. 1–111. Springer, Berlin (1975)
Lax, P.D.: Linear Algebra and Its Applications, 2nd edn. Wiley, New York (2007)
Łojasiewicz, S.: Sur les trajectoires du gradient d’une fonction analytique. Seminari di geometria, Univ. Stud. Bologna, Bologna 1982–1983, 115–117 (1984)
Marchetti, M.C., Joanny, J.F., Ramaswamy, S., Liverpool, T.B., Prost, J., Rao, M., Simha, R.A.: Hydrodynamics of soft active matter. Rev. Mod. Phys. 85(3), 1143–1189 (2013)
Mischler, S., Mouhot, C.: Kac’s program in kinetic theory. Invent. Math. 193(1), 1–147 (2013)
Morales, J., Poyato, D.: On the trend to global equilibrium for Kuramoto Oscillators. arXiv preprint arXiv:1908.07657 (2019)
Motsch, S., Tadmor, E.: A new model for self-organized dynamics and its flocking behavior. J. Stat. Phys. 144(5), 923 (2011)
Oelschläger, K.: A law of large numbers for moderately interacting diffusion processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 69(2), 279–322 (1985)
Perko, L.: Differential Equations and Dynamical Systems, vol. 7. Springer, Berlin (2013)
Perthame, B.: Global existence to the BGK model of Boltzmann equation. J. Differ. Equ. 82, 191–205 (1989)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, vol. 37. Springer, Berin (2010)
Saint-Raymond, L.: From the BGK model to the Navier–Stokes equations. Ann. Sci. Éc. Norm. Supér. 36(4), 271–317 (2003)
Salamin, E.: Application of quaternions to computation with rotations. Technical report, Working Paper (1979)
Sznitman, A.-S.: Topics in propagation of chaos. In: Hennequin, P.-L. (ed.) Éc. Été Probab. St.-Flour XIX—1989, pp. 165–251. Springer, Berlin (1991)
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226 (1995)
Wang, H., Hoffman, P., et al.: A unified view on the rotational symmetry of equilibiria of nematic polymers, dipolar nematic polymers, and polymers in higher dimensional space. Commun. Math. Sci. 6(4), 949–974 (2008)
Zhang, T.-F., Jiang, N.: A local existence of viscous self-organized hydrodynamic model. Nonlinear Anal. Real World Appl. 34, 495–506 (2017)
Zhou, H., Wang, H.: Stability of equilibria of nematic liquid crystalline polymers. Acta Math. Sci. 31(6), 2289–2304 (2011)