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Self-organized hydrodynamics with density-dependent velocity
1. | Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom |
2. | Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom |
3. | Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, 52062, Germany |
Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is $(ρ v(ρ))'≥q 0$, i.e. a nondecreasing mass flux $ρ v(ρ)$ with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.
References:
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Excluded-volume effects in the diffusion of hard spheres, Phys. Rev. E, 85 (2012), 011103.
doi: 10.1137/100783674. |
[2] |
M. Burger, M. Di Francesco, J. Pietschmann and B. Schlake,
Nonlinear Cross-Diffusion with Size Exclusion, SIAM Journal on Mathematical Analysis, 42 (2010), 2842-2871.
doi: 10.1137/100783674. |
[3] |
M. E. Cates and J. Tailleur,
When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation, Europhysics Letters, 101 (2013), 20010.
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P. M. Chaikin and T. C. Lubensky,
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H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud,
Collective motion of self-propelled particles interacting without cohesion, Phys. Rev., E77 (2008), 046113.
|
[6] |
H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud,
Modeling collective motion: Variations on the Vicsek model, Eur. Phys. J. B, 64 (2008), 451-456.
|
[7] |
A. Creppy, F. Plouraboué, O. Praud, H. Druart, S. Cazin, H. Yu and P. Degond,
Symmetry-breaking phase-transitions in highly concentrated semen, Interface, 13 (2016), 20160575.
doi: 10.1098/rsif.2016.0575. |
[8] |
A. Czirok and T. Vicsek,
Collective behavior of interacting self-propelled particles, Physica A, 281 (2000), 17-29.
doi: 10.1016/S0378-4371(00)00013-3. |
[9] |
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. |
[10] |
P. Degond, G. Dimarco, T. B. N. Mac and N. Wang,
Macroscopic models of collective motion with repulsion, Communications in Mathematical Sciences, 13 (2015), 1615-1638.
doi: 10.4310/CMS.2015.v13.n6.a12. |
[11] |
P. Degond and J. Hua,
Self-organized hydrodynamics with congestion and path formation in crowds, Journal of Computational Physics, 237 (2013), 299-319.
doi: 10.1016/j.jcp.2012.11.033. |
[12] |
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. |
[13] |
P. Degond, P. Peyrard, G. Russo and P. Villedieu,
Polynomial upwind schemes for hyperbolic systems, Comptes Rendus de l'Académie des Sciences -Series Ⅰ -Mathematics, 328 (1999), 479-483.
doi: 10.1016/S0764-4442(99)80194-3. |
[14] |
P. Degond and H. Yu, Self-Organized Hydrodynamics models with repulsion force in an annular domain, In preparation, 2016. |
[15] |
F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo and J. Tailleur,
Pattern formation in self-propelled particles with density-dependent motility, Phys. Rev. Lett., 108 (2012), 248101.
doi: 10.1103/PhysRevLett.108.248101. |
[16] |
Y. Fily, A. Baskaran and M. F. Hagan,
Dynamics of self-propelled particles under strong confinement, Soft Mater, 10 (2014), 5609-5617.
doi: 10.1039/C4SM00975D. |
[17] |
Y. Fily and M. C. Marchetti,
Athermal phase separation of self-propelled particles with no allignment, Phys. Rev. Lett., 108 (2012), 235702.
|
[18] |
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. |
[19] |
S. Henkes, Y. Fily and M. C. Marchetti,
Active jamming: Self-propelled soft particles at high density, Phys. Rev. E, 84 (2011), 040301.
doi: 10.1103/PhysRevE.84.040301. |
[20] |
S. Motsch and L. Navoret,
Numerical simulations of a nonconvervative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul., 9 (2011), 1253-1275.
doi: 10.1137/100794067. |
[21] |
J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P. M. Chaikin,
Living crystals of light-activated colloidal surfers, Science, 339 (2013), 936-940.
doi: 10.1126/science.1230020. |
[22] |
A. Peshkov, S. Ngo, E. Bertin, H. Chaté and F. Ginelli,
Continuous theory of active matter systems with metric-free interactions, Phys. Rev. Lett., 109 (2012), 098101.
doi: 10.1103/PhysRevLett.109.098101. |
[23] |
G. S. Redner, M. F. Hagan and A. Baskaran,
Structure and Dynamics of a Phase-Separating Active Colloidal Fluid, Phys. Rev. Lett., 110 (2013), 055701.
|
[24] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous Media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
[25] |
A. Seyfried, B. Steffen, M. Klingsch and M. Boltes,
The fundamental diagram of pedestrian movement revisited, Journal of Statistical Mechanics: Theory and Experiment, 10 (2005), P10002.
doi: 10.1088/1742-5468/2005/10/P10002. |
[26] |
B. Szabó, M. E. Szöllösi, B. Gönci, Zs. Jurányi, D. Selmeczi and T. Vicsek,
Phase transition in the collective migration of tissue cells: Experiment and model, Phys. Rev. E, 74 (2006), 061908.
|
[27] |
J. Tailleur and M. E. Cates,
Statistical mechanics of interacting run-and-tumble bacteria, Phys. Rev. Lett., 100 (2008), 218103.
doi: 10.1103/PhysRevLett.100.218103. |
[28] |
J. Toner, Y. Tu and S. Ramaswamy,
Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244.
doi: 10.1016/j.aop.2005.04.011. |
[29] |
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. |
show all references
This paper entilted "Self-organized hydrodynamics with density-dependent velocity" is licensed under a Creative Commons Attribution 3.0 Unported License. See http://creativecommons.org/licenses/by/3.0/.
References:
[1] |
M. Bruan and S. J. Chapman,
Excluded-volume effects in the diffusion of hard spheres, Phys. Rev. E, 85 (2012), 011103.
doi: 10.1137/100783674. |
[2] |
M. Burger, M. Di Francesco, J. Pietschmann and B. Schlake,
Nonlinear Cross-Diffusion with Size Exclusion, SIAM Journal on Mathematical Analysis, 42 (2010), 2842-2871.
doi: 10.1137/100783674. |
[3] |
M. E. Cates and J. Tailleur,
When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation, Europhysics Letters, 101 (2013), 20010.
doi: 10.1209/0295-5075/101/20010. |
[4] |
P. M. Chaikin and T. C. Lubensky,
Principles of Condensed Matter Physics Cambridge University Press, 1995.
doi: 10.1017/CBO9780511813467. |
[5] |
H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud,
Collective motion of self-propelled particles interacting without cohesion, Phys. Rev., E77 (2008), 046113.
|
[6] |
H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud,
Modeling collective motion: Variations on the Vicsek model, Eur. Phys. J. B, 64 (2008), 451-456.
|
[7] |
A. Creppy, F. Plouraboué, O. Praud, H. Druart, S. Cazin, H. Yu and P. Degond,
Symmetry-breaking phase-transitions in highly concentrated semen, Interface, 13 (2016), 20160575.
doi: 10.1098/rsif.2016.0575. |
[8] |
A. Czirok and T. Vicsek,
Collective behavior of interacting self-propelled particles, Physica A, 281 (2000), 17-29.
doi: 10.1016/S0378-4371(00)00013-3. |
[9] |
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. |
[10] |
P. Degond, G. Dimarco, T. B. N. Mac and N. Wang,
Macroscopic models of collective motion with repulsion, Communications in Mathematical Sciences, 13 (2015), 1615-1638.
doi: 10.4310/CMS.2015.v13.n6.a12. |
[11] |
P. Degond and J. Hua,
Self-organized hydrodynamics with congestion and path formation in crowds, Journal of Computational Physics, 237 (2013), 299-319.
doi: 10.1016/j.jcp.2012.11.033. |
[12] |
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. |
[13] |
P. Degond, P. Peyrard, G. Russo and P. Villedieu,
Polynomial upwind schemes for hyperbolic systems, Comptes Rendus de l'Académie des Sciences -Series Ⅰ -Mathematics, 328 (1999), 479-483.
doi: 10.1016/S0764-4442(99)80194-3. |
[14] |
P. Degond and H. Yu, Self-Organized Hydrodynamics models with repulsion force in an annular domain, In preparation, 2016. |
[15] |
F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo and J. Tailleur,
Pattern formation in self-propelled particles with density-dependent motility, Phys. Rev. Lett., 108 (2012), 248101.
doi: 10.1103/PhysRevLett.108.248101. |
[16] |
Y. Fily, A. Baskaran and M. F. Hagan,
Dynamics of self-propelled particles under strong confinement, Soft Mater, 10 (2014), 5609-5617.
doi: 10.1039/C4SM00975D. |
[17] |
Y. Fily and M. C. Marchetti,
Athermal phase separation of self-propelled particles with no allignment, Phys. Rev. Lett., 108 (2012), 235702.
|
[18] |
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. |
[19] |
S. Henkes, Y. Fily and M. C. Marchetti,
Active jamming: Self-propelled soft particles at high density, Phys. Rev. E, 84 (2011), 040301.
doi: 10.1103/PhysRevE.84.040301. |
[20] |
S. Motsch and L. Navoret,
Numerical simulations of a nonconvervative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul., 9 (2011), 1253-1275.
doi: 10.1137/100794067. |
[21] |
J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine and P. M. Chaikin,
Living crystals of light-activated colloidal surfers, Science, 339 (2013), 936-940.
doi: 10.1126/science.1230020. |
[22] |
A. Peshkov, S. Ngo, E. Bertin, H. Chaté and F. Ginelli,
Continuous theory of active matter systems with metric-free interactions, Phys. Rev. Lett., 109 (2012), 098101.
doi: 10.1103/PhysRevLett.109.098101. |
[23] |
G. S. Redner, M. F. Hagan and A. Baskaran,
Structure and Dynamics of a Phase-Separating Active Colloidal Fluid, Phys. Rev. Lett., 110 (2013), 055701.
|
[24] |
A. Schadschneider and A. Seyfried,
Empirical results for pedestrian dynamics and their implications for modeling, Networks and Heterogeneous Media, 6 (2011), 545-560.
doi: 10.3934/nhm.2011.6.545. |
[25] |
A. Seyfried, B. Steffen, M. Klingsch and M. Boltes,
The fundamental diagram of pedestrian movement revisited, Journal of Statistical Mechanics: Theory and Experiment, 10 (2005), P10002.
doi: 10.1088/1742-5468/2005/10/P10002. |
[26] |
B. Szabó, M. E. Szöllösi, B. Gönci, Zs. Jurányi, D. Selmeczi and T. Vicsek,
Phase transition in the collective migration of tissue cells: Experiment and model, Phys. Rev. E, 74 (2006), 061908.
|
[27] |
J. Tailleur and M. E. Cates,
Statistical mechanics of interacting run-and-tumble bacteria, Phys. Rev. Lett., 100 (2008), 218103.
doi: 10.1103/PhysRevLett.100.218103. |
[28] |
J. Toner, Y. Tu and S. Ramaswamy,
Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244.
doi: 10.1016/j.aop.2005.04.011. |
[29] |
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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