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

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

[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. 

[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.

[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. 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.

[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.

[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. 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.

[14]

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

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 

[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. 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.

[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. 

[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. 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.

[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.

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. 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.

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

[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. 

[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.

[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. 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.

[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.

[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. 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.

[14]

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

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 

[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. 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.

[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. 

[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. 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.

[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.

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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