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A continuum model for nematic alignment of self-propelled particles
1. | Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom |
2. | Faculty of Mathematics, University of Vienna, Austria, Current affiliation: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA |
3. | Université de Toulouse; UPS, INSA, UT1, UTM, CNRS; Institut de Mathématiques de Toulouse, UMR 5219, France, and Department of Mathematics, Imperial College London, United Kingdom, Current affiliation: Inst. für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, 52056, Germany |
A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.
References:
[1] |
I. Aoki,
A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fish., 48 (1982), 1081-1088.
doi: 10.2331/suisan.48.1081. |
[2] |
J. P. Arcede and E. A. Cabral,
An equivalent definition for the backwards Itô integral, Thai J. Math., 9 (2011), 619-630.
|
[3] |
A. Barbaro and P. Degond,
Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278.
doi: 10.3934/dcdsb.2014.19.1249. |
[4] |
A. Baskaran and M. Marchetti,
Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9pp.
doi: 10.1103/PhysRevE.77.011920. |
[5] |
A. Baskaran and C. M. Marchetti,
Nonequilibrium statistical mechanics of self-propelled hard rods, J. Stat. Mech.: Theory Exp., 2010 (2010), P04019.
doi: 10.1088/1742-5468/2010/04/P04019. |
[6] |
E. Ben-Jacob, I. Cohen and H. Levine,
Cooperative self-organization of microorganisms, Adv. in Phys., 49 (2000), 395-554.
doi: 10.1080/000187300405228. |
[7] |
E. Bertin, M. Droz and G. Grégoire,
Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001.
doi: 10.1088/1751-8113/42/44/445001. |
[8] |
U. Börner, A. Deutsch, H. Reichenbach and M. Bär,
Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Phys. Rev. Lett., 89 (2002), 078101.
|
[9] |
J. Buhl, D. Sumpter, I. Couzin, J. Hale, E. Despland, E. Miller and S. Simpson,
From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.
doi: 10.1126/science.1125142. |
[10] |
J. Carrillo, M. 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. |
[11] |
A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale,
Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. USA, 107 (2010), 11865-11870.
doi: 10.1073/pnas.1005766107. |
[12] |
H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud,
Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.
|
[13] |
Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes,
State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[14] |
I. Couzin, J. Krause, R. James, G. Ruxton and N. Franks,
Collective memory and spatial sorting in animal groups, J. Theoret. Biol., 218 (2002), 1-11.
doi: 10.1006/jtbi.2002.3065. |
[15] |
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2 edition, Oxford University Press, United Kingdom, 1993.
![]() |
[16] |
P. Degond, F. Delebecque and D. Peurichard,
Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.
doi: 10.1142/S0218202516400030. |
[17] |
P. Degond, G. Dimarco and T. B. N. Mac,
Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles, Math. Models Methods Appl. Sci., 24 (2014), 277-325.
doi: 10.1142/S0218202513400095. |
[18] |
P. Degond, G. Dimarco, T. B. N. Mac and N. Wang,
Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.
doi: 10.4310/CMS.2015.v13.n6.a12. |
[19] |
P. Degond, A. Frouvelle and J.-G. Liu,
Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456.
doi: 10.1007/s00332-012-9157-y. |
[20] |
P. Degond, A. Frouvelle and J.-G. Liu,
Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.
doi: 10.1007/s00205-014-0800-7. |
[21] |
P. Degond and J.-G. Liu,
Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation, Math. Models Methods Appl. Sci., 22 (2012), 114001, 18pp.
doi: 10.1142/S021820251140001X. |
[22] |
P. Degond, J.-G. Liu, S. Motsch and V. Panferov,
Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.
doi: 10.4310/MAA.2013.v20.n2.a1. |
[23] |
P. Degond, J.-G. Liu and C. Ringhofer,
Evolution of wealth in a nonconservative economy driven by local Nash equilibria, Philos. Trans. A, 372 (2014), 20130394, 15pp.
doi: 10.1098/rsta.2013.0394. |
[24] |
P. Degond, A. Manhart and H. Yu,
An age-structured continuum model for myxobacteria, In preparation, 372 (2017).
|
[25] |
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. |
[26] |
P. Degond and L. Navoret,
A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Math. Models Methods Appl. Sci., 25 (2015), 2439-2475.
doi: 10.1142/S021820251540014X. |
[27] |
P. Degond and H. Yu,
Self-organized hydrodynamics in an annular domain: Modal analysis and nonlinear effects, Math. Models Methods Appl. Sci., 25 (2015), 495-519.
doi: 10.1142/S0218202515400047. |
[28] |
P. Dhar, T. Fischer, Y. Wang, T. Mallouk, W. Paxton and A. Sen,
Autonomously moving nanorods at a viscous interface, Nano Lett., 6 (2006), 66-72.
doi: 10.1021/nl052027s. |
[29] |
M. Dworkin,
Recent advances in the social and developmental biology of the myxobacteria, Microbiol. Rev., 60 (1996), 70-102.
|
[30] |
A. Frouvelle,
A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40pp.
doi: 10.1142/S021820251250011X. |
[31] |
F. Ginelli, F. Peruani, M. Bär and H. Chaté,
Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502.
doi: 10.1103/PhysRevLett.104.184502. |
[32] |
D. Helbing, A. Johansson and H. Al-Abideen,
Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
doi: 10.1103/PhysRevE.75.046109. |
[33] |
O. Igoshin, A. Mogilner, R. Welch, D. Kaiser and G. Oster,
Pattern formation and traveling waves in myxobacteria: Theory and modeling, Proc. Natl. Acad. Sci. USA, 98 (2001), 14913-14918.
doi: 10.1073/pnas.221579598. |
[34] |
O. Igoshin, R. Welch, D. Kaiser and G. Oster,
Waves and aggregation patterns in myxobacteria, Proc. Natl. Acad. Sci. USA, 101 (2004), 4256-4261.
doi: 10.1073/pnas.0400704101. |
[35] |
D. Kaiser,
Coupling cell movement to multicellular development in myxobacteria, Nat. Rev. Microbiol., 1 (2003), 45-54.
doi: 10.1038/nrmicro733. |
[36] |
P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky,
Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhys. Lett., 54 (2001), 28.
|
[37] |
P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky,
Brownian walker in a confined geometry leading to a space-dependent diffusion coefficient, Physica A, 304 (2002), 65.
|
[38] |
A. Lau and T. Lubensky,
State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E, 76 (2007), 011123, 17pp.
doi: 10.1103/PhysRevE.76.011123. |
[39] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[40] |
M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz,
The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS ONE, 5 (2010), 1-7.
|
[41] |
N. Jiang, L. Xiong and T.-F. Zhang,
Hydrodynamic Limits of the Kinetic Self-Organized Models, SIAM J. Math. Anal., 48 (2016), 3383-3411.
doi: 10.1137/15M1035665. |
[42] |
V. Narayan, S. Ramaswamy and N. Menon,
Long-lived giant number fluctuations in a swarming granular nematic, Science, 317 (2007), 105-108.
doi: 10.1126/science.1140414. |
[43] |
S. Ngo, F. Ginelli and H. Chaté,
Competing ferromagnetic and nematic alignment in self-propelled polar particles, Phys. Rev. E, 86 (2012), 050101(R).
doi: 10.1103/PhysRevE.86.050101. |
[44] |
J. Parrish and W. Hamner, Animal Groups in Three Dimensions: How Species Aggregate, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511601156.![]() ![]() ![]() |
[45] |
F. Peruani, F. Ginelli, M. Bär and H. Chaté,
Polar vs. apolar alignment in systems of polar self-propelled particles, J. Phys. Conf. Ser., 297 (2011), 012014.
doi: 10.1088/1742-6596/297/1/012014. |
[46] |
C. Reynold,
Flocks, herds, and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34.
doi: 10.1145/37401.37406. |
[47] |
A. Sokolov, I. Aranson, J. Kessler and R. Goldstein,
Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.
doi: 10.1103/PhysRevLett.98.158102. |
[48] |
J. Toner and Y. Tu,
Long-range order in a two-dimensional dynamical xy model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329.
|
[49] |
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. |
[50] |
T. Vicsek and A. Zafeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
[51] |
R. Welch and D. Kaiser,
Cell behavior in traveling wave patterns of myxobacteria, Proc. Natl. Acad. Sci. USA, 98 (2001), 14907-14912.
doi: 10.1073/pnas.261574598. |
[52] |
Y. Wu, A. Kaiser, Y. Jiang and M. Alber,
Periodic reversal of direction allows myxobacteria to swarm, Proc. Natl. Acad. Sci. USA, 106 (2008), 1222-1227.
doi: 10.1073/pnas.0811662106. |
[53] |
H.-P. Zhang, A. Be'er, E.-L. Florin and H. Swinney,
Collective motion and density fluctuations in bacterial colonies, Proc. Natl. Acad. Sci. USA, 107 (2010), 13626-13630.
doi: 10.1073/pnas.1001651107. |
show all references
References:
[1] |
I. Aoki,
A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fish., 48 (1982), 1081-1088.
doi: 10.2331/suisan.48.1081. |
[2] |
J. P. Arcede and E. A. Cabral,
An equivalent definition for the backwards Itô integral, Thai J. Math., 9 (2011), 619-630.
|
[3] |
A. Barbaro and P. Degond,
Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278.
doi: 10.3934/dcdsb.2014.19.1249. |
[4] |
A. Baskaran and M. Marchetti,
Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9pp.
doi: 10.1103/PhysRevE.77.011920. |
[5] |
A. Baskaran and C. M. Marchetti,
Nonequilibrium statistical mechanics of self-propelled hard rods, J. Stat. Mech.: Theory Exp., 2010 (2010), P04019.
doi: 10.1088/1742-5468/2010/04/P04019. |
[6] |
E. Ben-Jacob, I. Cohen and H. Levine,
Cooperative self-organization of microorganisms, Adv. in Phys., 49 (2000), 395-554.
doi: 10.1080/000187300405228. |
[7] |
E. Bertin, M. Droz and G. Grégoire,
Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001.
doi: 10.1088/1751-8113/42/44/445001. |
[8] |
U. Börner, A. Deutsch, H. Reichenbach and M. Bär,
Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Phys. Rev. Lett., 89 (2002), 078101.
|
[9] |
J. Buhl, D. Sumpter, I. Couzin, J. Hale, E. Despland, E. Miller and S. Simpson,
From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.
doi: 10.1126/science.1125142. |
[10] |
J. Carrillo, M. 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. |
[11] |
A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale,
Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. USA, 107 (2010), 11865-11870.
doi: 10.1073/pnas.1005766107. |
[12] |
H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud,
Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.
|
[13] |
Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes,
State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[14] |
I. Couzin, J. Krause, R. James, G. Ruxton and N. Franks,
Collective memory and spatial sorting in animal groups, J. Theoret. Biol., 218 (2002), 1-11.
doi: 10.1006/jtbi.2002.3065. |
[15] |
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2 edition, Oxford University Press, United Kingdom, 1993.
![]() |
[16] |
P. Degond, F. Delebecque and D. Peurichard,
Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.
doi: 10.1142/S0218202516400030. |
[17] |
P. Degond, G. Dimarco and T. B. N. Mac,
Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles, Math. Models Methods Appl. Sci., 24 (2014), 277-325.
doi: 10.1142/S0218202513400095. |
[18] |
P. Degond, G. Dimarco, T. B. N. Mac and N. Wang,
Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.
doi: 10.4310/CMS.2015.v13.n6.a12. |
[19] |
P. Degond, A. Frouvelle and J.-G. Liu,
Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456.
doi: 10.1007/s00332-012-9157-y. |
[20] |
P. Degond, A. Frouvelle and J.-G. Liu,
Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.
doi: 10.1007/s00205-014-0800-7. |
[21] |
P. Degond and J.-G. Liu,
Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation, Math. Models Methods Appl. Sci., 22 (2012), 114001, 18pp.
doi: 10.1142/S021820251140001X. |
[22] |
P. Degond, J.-G. Liu, S. Motsch and V. Panferov,
Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.
doi: 10.4310/MAA.2013.v20.n2.a1. |
[23] |
P. Degond, J.-G. Liu and C. Ringhofer,
Evolution of wealth in a nonconservative economy driven by local Nash equilibria, Philos. Trans. A, 372 (2014), 20130394, 15pp.
doi: 10.1098/rsta.2013.0394. |
[24] |
P. Degond, A. Manhart and H. Yu,
An age-structured continuum model for myxobacteria, In preparation, 372 (2017).
|
[25] |
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. |
[26] |
P. Degond and L. Navoret,
A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Math. Models Methods Appl. Sci., 25 (2015), 2439-2475.
doi: 10.1142/S021820251540014X. |
[27] |
P. Degond and H. Yu,
Self-organized hydrodynamics in an annular domain: Modal analysis and nonlinear effects, Math. Models Methods Appl. Sci., 25 (2015), 495-519.
doi: 10.1142/S0218202515400047. |
[28] |
P. Dhar, T. Fischer, Y. Wang, T. Mallouk, W. Paxton and A. Sen,
Autonomously moving nanorods at a viscous interface, Nano Lett., 6 (2006), 66-72.
doi: 10.1021/nl052027s. |
[29] |
M. Dworkin,
Recent advances in the social and developmental biology of the myxobacteria, Microbiol. Rev., 60 (1996), 70-102.
|
[30] |
A. Frouvelle,
A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40pp.
doi: 10.1142/S021820251250011X. |
[31] |
F. Ginelli, F. Peruani, M. Bär and H. Chaté,
Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502.
doi: 10.1103/PhysRevLett.104.184502. |
[32] |
D. Helbing, A. Johansson and H. Al-Abideen,
Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.
doi: 10.1103/PhysRevE.75.046109. |
[33] |
O. Igoshin, A. Mogilner, R. Welch, D. Kaiser and G. Oster,
Pattern formation and traveling waves in myxobacteria: Theory and modeling, Proc. Natl. Acad. Sci. USA, 98 (2001), 14913-14918.
doi: 10.1073/pnas.221579598. |
[34] |
O. Igoshin, R. Welch, D. Kaiser and G. Oster,
Waves and aggregation patterns in myxobacteria, Proc. Natl. Acad. Sci. USA, 101 (2004), 4256-4261.
doi: 10.1073/pnas.0400704101. |
[35] |
D. Kaiser,
Coupling cell movement to multicellular development in myxobacteria, Nat. Rev. Microbiol., 1 (2003), 45-54.
doi: 10.1038/nrmicro733. |
[36] |
P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky,
Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhys. Lett., 54 (2001), 28.
|
[37] |
P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky,
Brownian walker in a confined geometry leading to a space-dependent diffusion coefficient, Physica A, 304 (2002), 65.
|
[38] |
A. Lau and T. Lubensky,
State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E, 76 (2007), 011123, 17pp.
doi: 10.1103/PhysRevE.76.011123. |
[39] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[40] |
M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz,
The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS ONE, 5 (2010), 1-7.
|
[41] |
N. Jiang, L. Xiong and T.-F. Zhang,
Hydrodynamic Limits of the Kinetic Self-Organized Models, SIAM J. Math. Anal., 48 (2016), 3383-3411.
doi: 10.1137/15M1035665. |
[42] |
V. Narayan, S. Ramaswamy and N. Menon,
Long-lived giant number fluctuations in a swarming granular nematic, Science, 317 (2007), 105-108.
doi: 10.1126/science.1140414. |
[43] |
S. Ngo, F. Ginelli and H. Chaté,
Competing ferromagnetic and nematic alignment in self-propelled polar particles, Phys. Rev. E, 86 (2012), 050101(R).
doi: 10.1103/PhysRevE.86.050101. |
[44] |
J. Parrish and W. Hamner, Animal Groups in Three Dimensions: How Species Aggregate, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511601156.![]() ![]() ![]() |
[45] |
F. Peruani, F. Ginelli, M. Bär and H. Chaté,
Polar vs. apolar alignment in systems of polar self-propelled particles, J. Phys. Conf. Ser., 297 (2011), 012014.
doi: 10.1088/1742-6596/297/1/012014. |
[46] |
C. Reynold,
Flocks, herds, and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34.
doi: 10.1145/37401.37406. |
[47] |
A. Sokolov, I. Aranson, J. Kessler and R. Goldstein,
Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.
doi: 10.1103/PhysRevLett.98.158102. |
[48] |
J. Toner and Y. Tu,
Long-range order in a two-dimensional dynamical xy model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329.
|
[49] |
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. |
[50] |
T. Vicsek and A. Zafeiris,
Collective motion, Phys. Rep., 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
[51] |
R. Welch and D. Kaiser,
Cell behavior in traveling wave patterns of myxobacteria, Proc. Natl. Acad. Sci. USA, 98 (2001), 14907-14912.
doi: 10.1073/pnas.261574598. |
[52] |
Y. Wu, A. Kaiser, Y. Jiang and M. Alber,
Periodic reversal of direction allows myxobacteria to swarm, Proc. Natl. Acad. Sci. USA, 106 (2008), 1222-1227.
doi: 10.1073/pnas.0811662106. |
[53] |
H.-P. Zhang, A. Be'er, E.-L. Florin and H. Swinney,
Collective motion and density fluctuations in bacterial colonies, Proc. Natl. Acad. Sci. USA, 107 (2010), 13626-13630.
doi: 10.1073/pnas.1001651107. |

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