# American Institute of Mathematical Sciences

August  2019, 12(4): 791-827. doi: 10.3934/krm.2019031

## A non-linear kinetic model of self-propelled particles with multiple equilibria

 1 Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy 2 Scuola Normale Superiore of Pisa, Piazza dei Cavalieri, 7 56126 Pisa, Italy 3 Heriot Watt University, Mathematics Department, Edinburgh, EH14 4AS, UK 4 Imperial College London, South Kensington Campus, London SW7 2AZ, UK

* Corresponding author: M. Ottobre, m.ottobre@hw.ac.uk

Received  July 2018 Published  May 2019

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density ft, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only hypoelliptic) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate N-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as $N \to \infty$, to the solution of the PDE.

Citation: Paolo Buttà, Franco Flandoli, Michela Ottobre, Boguslaw Zegarlinski. A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic and Related Models, 2019, 12 (4) : 791-827. doi: 10.3934/krm.2019031
##### References:
 [1] A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246. [2] D. G Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations, 3 (1967), 1-14.  doi: 10.1016/0022-0396(67)90002-2. [3] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, 2014. doi: 10.1007/978-3-319-00227-9. [4] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105. [5] A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo and P. Degond, Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type, Multiscale Model. Simul. (SIAM), 14 (2016), 1063-1088.  doi: 10.1137/15M1043637. [6] J. Barre, D. Crisan and T. Goudon, Two-dimensional pseudo-gravity model: Particles motion in a non-potential singular force field, Trans. Amer. Math. Soc., 371 (2019), 2923-2962.  doi: 10.1090/tran/7638. [7] J. Barre, P. Degond and E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Model. Simul. (SIAM), 15 (2017), 1294-1323.  doi: 10.1137/16M1085310. [8] D. Benedetto, E. Caglioti and M. Pulvirenti, A Kinetic equation for granular media, ESAIM Math. Mod. Num. Anal., 31 (1997), 615-641.  doi: 10.1051/m2an/1997310506151. [9] D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non Maxwellian steady distribution for one-dimensional granular media, Jour. Stat. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560. [10] F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly self-consistent Vlasov-Fokker-Planck equation, ESAIM Math. Mod. Num. Anal., 44 (2010), 867-884.  doi: 10.1051/m2an/2010045. [11] M. Bossy, J. F. Jabir and D. Talay, On conditional McKean Lagrangian stochastic models, Probab. Theory Relat. Fields, 151 (2011), 319-351.  doi: 10.1007/s00440-010-0301-z. [12] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.  doi: 10.1126/science.1125142. [13] P. Buttà and J. L. Lebowitz, Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces, J. Statist. Phys., 94 (1999), 653-694.  doi: 10.1023/A:1004512807858. [14] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in Biological Systems, Princeton University Press, 2003. [15] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376. [16] J. A. Carrillo, R. Eftimie and Ho ffmann, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kinet. Relat. Models, 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413. [17] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363. [18] T. Cass, D. Crisan, P. Dobson and M. Ottobre, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, preprint, arXiv: 1805.01350. [19] N. Champagnat and S. Málárd, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., 55 (2007), 147-188.  doi: 10.1007/s00285-007-0072-z. [20] 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. [21] F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [22] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [23] A. Czirók, A. L. Barabási and T. Vicsek, Collective Motion of Self-Propelled Particles: Kinetic Phase Transition in One Dimension, Phys. Rev. Lett., 82 (1999), 209. [24] A. Czirók and T. Vicsek, Collective behavior of interacting self-propelled particle., Physica A, 281 (2000), 17-29. [25] G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, 115–122, Progr. Probab., 59, Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-8458-6_7. [26] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Ann. Sci. cole Norm. Sup. (4), 19 (1986), 519-542.  doi: 10.24033/asens.1516. [27] P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1. [28] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005. [29] R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review, J. Math. Biol., 65 (2012), 35-75.  doi: 10.1007/s00285-011-0452-2. [30] R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proceedings of the National Academy of Sciences, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104. [31] F. Flandoli, E. Priola and G. Zanco, A mean-field model with discontinuous coefficients for neurons with spatial interaction, preprint, arXiv: 1708.04156. [32] F. Flandoli and V. Capasso, On the mean field approximation of a stochastic model of tumor-induced angiogenesis, preprint, arXiv: 1708.03830. [33] J. Garnier, G. Papanicolaou and T.-W. Yang, Mean field model for collective motion bistability, Discr. Cont. Dyn. Syst. B, 24 (2019), 851-879.  doi: 10.3934/dcdsb.2018210. [34] M. Gianfelice and E. Orlandi, Dynamics and kinetic limit for a system of noiseless d-dimensional Vicsek-type particles, Networks and Heterogeneous Media, 9 (2014), 269-297.  doi: 10.3934/nhm.2014.9.269. [35] S. N. Gomes and G. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Sci, 28 (2018), 905-941.  doi: 10.1007/s00332-017-9433-y. [36] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [37] B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, Springer, 2005. doi: 10.1007/b104762. [38] F. Herau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 224 (2007), 95-118.  doi: 10.1016/j.jfa.2006.11.013. [39] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081. [40] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer, Berlin, 1999 doi: 10.1007/978-3-662-03752-2. [41] V. Kontis, M. Ottobre and B. Zegarlinski, Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing, J. Funct. Anal., 270 (2016), 3173-3223.  doi: 10.1016/j.jfa.2016.02.005. [42] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198.  doi: 10.1090/S0002-9947-09-04738-2. [43] H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2000), 017101. doi: 10.1103/PhysRevE.63.017101. [44] R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in biological aggregates, Bull Math Biol., 71 (2009), 352-382.  doi: 10.1007/s11538-008-9365-7. [45] P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1-29. [46] 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. [47] C. Nee, Sharp Gradient Bounds for the Diffusion Semigroup, PhD Thesis, Imperial College London, 2011. [48] S. Olla and S. R. S. Varadhan, Scaling limit for interacting Ornstein-Uhlenbeck processes, Comm. Math. Phys., 135 (1991), 355-378.  doi: 10.1007/BF02098047. [49] M. Ottobre, G. A. Pavliotis and K. Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems, J. Funct. Anal., 262 (2012), 4000-4039.  doi: 10.1016/j.jfa.2012.02.008. [50] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 284 (1999), 99-101.  doi: 10.1126/science.284.5411.99. [51] P. J. Smith, A recursive formulation of the old problem of obtaining moments from cumulants and vice versa, Amer. Statist., 49 (1995), 217-218.  doi: 10.2307/2684642. [52] D. W. 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##### References:
 [1] A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246. [2] D. G Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations, 3 (1967), 1-14.  doi: 10.1016/0022-0396(67)90002-2. [3] D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, 2014. doi: 10.1007/978-3-319-00227-9. [4] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105. [5] A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo and P. Degond, Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type, Multiscale Model. Simul. (SIAM), 14 (2016), 1063-1088.  doi: 10.1137/15M1043637. [6] J. Barre, D. Crisan and T. Goudon, Two-dimensional pseudo-gravity model: Particles motion in a non-potential singular force field, Trans. Amer. Math. Soc., 371 (2019), 2923-2962.  doi: 10.1090/tran/7638. [7] J. Barre, P. Degond and E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Model. Simul. (SIAM), 15 (2017), 1294-1323.  doi: 10.1137/16M1085310. [8] D. Benedetto, E. Caglioti and M. Pulvirenti, A Kinetic equation for granular media, ESAIM Math. Mod. Num. Anal., 31 (1997), 615-641.  doi: 10.1051/m2an/1997310506151. [9] D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non Maxwellian steady distribution for one-dimensional granular media, Jour. Stat. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560. [10] F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly self-consistent Vlasov-Fokker-Planck equation, ESAIM Math. Mod. Num. Anal., 44 (2010), 867-884.  doi: 10.1051/m2an/2010045. [11] M. Bossy, J. F. Jabir and D. Talay, On conditional McKean Lagrangian stochastic models, Probab. Theory Relat. Fields, 151 (2011), 319-351.  doi: 10.1007/s00440-010-0301-z. [12] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.  doi: 10.1126/science.1125142. [13] P. Buttà and J. L. Lebowitz, Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces, J. Statist. Phys., 94 (1999), 653-694.  doi: 10.1023/A:1004512807858. [14] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-organization in Biological Systems, Princeton University Press, 2003. [15] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376. [16] J. A. Carrillo, R. Eftimie and Ho ffmann, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kinet. Relat. Models, 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413. [17] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363. [18] T. Cass, D. Crisan, P. Dobson and M. Ottobre, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, preprint, arXiv: 1805.01350. [19] N. Champagnat and S. Málárd, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., 55 (2007), 147-188.  doi: 10.1007/s00285-007-0072-z. [20] 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. [21] F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [22] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [23] A. Czirók, A. L. Barabási and T. Vicsek, Collective Motion of Self-Propelled Particles: Kinetic Phase Transition in One Dimension, Phys. Rev. Lett., 82 (1999), 209. [24] A. Czirók and T. Vicsek, Collective behavior of interacting self-propelled particle., Physica A, 281 (2000), 17-29. [25] G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, 115–122, Progr. Probab., 59, Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-8458-6_7. [26] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Ann. Sci. cole Norm. Sup. (4), 19 (1986), 519-542.  doi: 10.24033/asens.1516. [27] P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1. [28] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005. [29] R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review, J. Math. Biol., 65 (2012), 35-75.  doi: 10.1007/s00285-011-0452-2. [30] R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proceedings of the National Academy of Sciences, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104. [31] F. Flandoli, E. Priola and G. Zanco, A mean-field model with discontinuous coefficients for neurons with spatial interaction, preprint, arXiv: 1708.04156. [32] F. Flandoli and V. Capasso, On the mean field approximation of a stochastic model of tumor-induced angiogenesis, preprint, arXiv: 1708.03830. [33] J. Garnier, G. Papanicolaou and T.-W. Yang, Mean field model for collective motion bistability, Discr. Cont. Dyn. Syst. B, 24 (2019), 851-879.  doi: 10.3934/dcdsb.2018210. [34] M. Gianfelice and E. Orlandi, Dynamics and kinetic limit for a system of noiseless d-dimensional Vicsek-type particles, Networks and Heterogeneous Media, 9 (2014), 269-297.  doi: 10.3934/nhm.2014.9.269. [35] S. N. Gomes and G. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Sci, 28 (2018), 905-941.  doi: 10.1007/s00332-017-9433-y. [36] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [37] B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, Springer, 2005. doi: 10.1007/b104762. [38] F. Herau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 224 (2007), 95-118.  doi: 10.1016/j.jfa.2006.11.013. [39] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081. [40] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer, Berlin, 1999 doi: 10.1007/978-3-662-03752-2. [41] V. Kontis, M. Ottobre and B. Zegarlinski, Markov semigroups with hypocoercive-type generator in infinite dimensions: Ergodicity and smoothing, J. Funct. Anal., 270 (2016), 3173-3223.  doi: 10.1016/j.jfa.2016.02.005. [42] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198.  doi: 10.1090/S0002-9947-09-04738-2. [43] H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2000), 017101. doi: 10.1103/PhysRevE.63.017101. [44] R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in biological aggregates, Bull Math Biol., 71 (2009), 352-382.  doi: 10.1007/s11538-008-9365-7. [45] P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1-29. [46] S. Motsch and E. 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