March  2017, 10(1): 193-213. doi: 10.3934/krm.2017008

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

* Corresponding author: Pierre Degond

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/.

Received  February 2016 Revised  May 2016 Published  November 2016

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.

Citation: Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008
References:
[1]

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M. BurgerM. Di FrancescoJ. Pietschmann and B. Schlake, Nonlinear Cross-Diffusion with Size Exclusion, SIAM Journal on Mathematical Analysis, 42 (2010), 2842-2871.  doi: 10.1137/100783674.  Google Scholar

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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.  Google Scholar

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H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: Variations on the Vicsek model, Eur. Phys. J. B, 64 (2008), 451-456.   Google Scholar

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A. CreppyF. PlourabouéO. PraudH. DruartS. CazinH. Yu and P. Degond, Symmetry-breaking phase-transitions in highly concentrated semen, Interface, 13 (2016), 20160575.  doi: 10.1098/rsif.2016.0575.  Google Scholar

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P. DegondJ-G. LiuS. 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.  Google Scholar

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P. DegondG. DimarcoT. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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P. Degond and H. Yu, Self-Organized Hydrodynamics models with repulsion force in an annular domain, In preparation, 2016. Google Scholar

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F. D. C. FarrellM. C. MarchettiD. 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.  Google Scholar

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Y. FilyA. Baskaran and M. F. Hagan, Dynamics of self-propelled particles under strong confinement, Soft Mater, 10 (2014), 5609-5617.  doi: 10.1039/C4SM00975D.  Google Scholar

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Y. Fily and M. C. Marchetti, Athermal phase separation of self-propelled particles with no allignment, Phys. Rev. Lett., 108 (2012), 235702.   Google Scholar

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S. HenkesY. 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.  Google Scholar

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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.  Google Scholar

[21]

J. PalacciS. SacannaA. P. SteinbergD. J. Pine and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science, 339 (2013), 936-940.  doi: 10.1126/science.1230020.  Google Scholar

[22]

A. PeshkovS. NgoE. BertinH. 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.  Google Scholar

[23]

G. S. RednerM. F. Hagan and A. Baskaran, Structure and Dynamics of a Phase-Separating Active Colloidal Fluid, Phys. Rev. Lett., 110 (2013), 055701.   Google Scholar

[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.  Google Scholar

[25]

A. SeyfriedB. SteffenM. 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.  Google Scholar

[26]

B. SzabóM. E. SzöllösiB. GönciZs. JurányiD. Selmeczi and T. Vicsek, Phase transition in the collective migration of tissue cells: Experiment and model, Phys. Rev. E, 74 (2006), 061908.   Google Scholar

[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.  Google Scholar

[28]

J. TonerY. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244.  doi: 10.1016/j.aop.2005.04.011.  Google Scholar

[29]

T. VicsekA. CzirókE. Ben-JacobI. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

M. BurgerM. Di FrancescoJ. Pietschmann and B. Schlake, Nonlinear Cross-Diffusion with Size Exclusion, SIAM Journal on Mathematical Analysis, 42 (2010), 2842-2871.  doi: 10.1137/100783674.  Google Scholar

[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.  Google Scholar

[4]

P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge University Press, 1995. doi: 10.1017/CBO9780511813467.  Google Scholar

[5]

H. ChatéF. GinelliG. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev., E77 (2008), 046113.   Google Scholar

[6]

H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: Variations on the Vicsek model, Eur. Phys. J. B, 64 (2008), 451-456.   Google Scholar

[7]

A. CreppyF. PlourabouéO. PraudH. DruartS. CazinH. Yu and P. Degond, Symmetry-breaking phase-transitions in highly concentrated semen, Interface, 13 (2016), 20160575.  doi: 10.1098/rsif.2016.0575.  Google Scholar

[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.  Google Scholar

[9]

P. DegondJ-G. LiuS. 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.  Google Scholar

[10]

P. DegondG. DimarcoT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. DegondP. PeyrardG. 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.  Google Scholar

[14]

P. Degond and H. Yu, Self-Organized Hydrodynamics models with repulsion force in an annular domain, In preparation, 2016. Google Scholar

[15]

F. D. C. FarrellM. C. MarchettiD. 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.  Google Scholar

[16]

Y. FilyA. Baskaran and M. F. Hagan, Dynamics of self-propelled particles under strong confinement, Soft Mater, 10 (2014), 5609-5617.  doi: 10.1039/C4SM00975D.  Google Scholar

[17]

Y. Fily and M. C. Marchetti, Athermal phase separation of self-propelled particles with no allignment, Phys. Rev. Lett., 108 (2012), 235702.   Google Scholar

[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.  Google Scholar

[19]

S. HenkesY. 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.  Google Scholar

[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.  Google Scholar

[21]

J. PalacciS. SacannaA. P. SteinbergD. J. Pine and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science, 339 (2013), 936-940.  doi: 10.1126/science.1230020.  Google Scholar

[22]

A. PeshkovS. NgoE. BertinH. 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.  Google Scholar

[23]

G. S. RednerM. F. Hagan and A. Baskaran, Structure and Dynamics of a Phase-Separating Active Colloidal Fluid, Phys. Rev. Lett., 110 (2013), 055701.   Google Scholar

[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.  Google Scholar

[25]

A. SeyfriedB. SteffenM. 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.  Google Scholar

[26]

B. SzabóM. E. SzöllösiB. GönciZs. JurányiD. Selmeczi and T. Vicsek, Phase transition in the collective migration of tissue cells: Experiment and model, Phys. Rev. E, 74 (2006), 061908.   Google Scholar

[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.  Google Scholar

[28]

J. TonerY. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244.  doi: 10.1016/j.aop.2005.04.011.  Google Scholar

[29]

T. VicsekA. CzirókE. Ben-JacobI. 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.  Google Scholar

Figure 1.  The accuracy test at $t=1$ shows that the splitting scheme is of first order. The initial data is given by $\rho_0 = \rho_s(1+0.1\sin(\pi x)), \theta_0 = \theta_s(1+0.1\sin(\pi x))$ with $(\rho_s, \theta_s) = (0.01, \frac{\pi}{4})$ and the mesh sizes are iteratively $N_x = N_y = 32, 64,128,256$
Figure 2.  Comparison between the particle (left) and SOH models (right). For the particle model, $N = 10^5, \nu = 100, D = 10, R_1 = R_2 = 0.1$. The result is the average of 40 simulations. For the SOH model, $N_x = N_y = 100$
Figure 3.  Stability test of the SOH model with $\gamma = 0$ and $\sigma = 0.1$. The profiles show the RMSF of $\rho$ and $\theta$ in $\log$-scale with respect to time $t$. The numerical solutions evolves from the steady states, and locally high concentrations develop
Figure 4.  Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.02$. RMSF of the density and angle are decreasing with time
Figure 5.  Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.005$. The RMSF of the numerical solutions grows gradually and high local concentrations develop. The linear scaling of the $\log$ of RMSF implies an exponential growth of the perturbation as a function of time $t$
Figure 6.  Growth rate of the perturbation $\rho_\sigma$. The parameters are $c_1 = 0.975, c_2 = 0.925, d = 0.05$ and $k_0 = 0.125$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. The steady state for the density is fixed at $\rho_s = 0.01$ and the final time is $t = 1$. (a) is computed using the fomula (11). In order to obtain (b), we compute the numerical solutions of the SOH model and perform a simple linear regression on the Discrete Fourier transform of the perturbed part, i.e. $\rho_\sigma = \rho - \rho_s$. The growth rate is interpreted as the slope of the function $t \to \hat\rho_\sigma(\xi, t)$
Figure 7.  Growth rate of the perturbation $\rho_\sigma$ given by the particle model with $N = 10^5$. The parameters are $\nu = 100, D = 5, R_1 = R_2 = 0.1$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. For each $\theta_s$, the growth rate is computed using the average of $10$ simulations, in order to reduce the effects of noise
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