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February  2019, 24(2): 851-879. doi: 10.3934/dcdsb.2018210

Mean field model for collective motion bistability

1. 

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France

2. 

Department of Mathematics, Stanford University, Stanford, CA 94305, USA

3. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

* Corresponding author

Received  October 2017 Published  June 2018

We consider the Czirók model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collectivemotion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.

Citation: Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210
References:
[1]

G. Ariel and A. Ayali, Locust collective motion and its modeling, PLoS Comput Biol, 11 (2015), e1004522. doi: 10.1371/journal.pcbi.1004522. Google Scholar

[2]

G. Ariel, Y. Ophir, S. Levi, E. Ben-Jacob and A. Ayali, Individual pause-and-go motion is instrumental to the formation and maintenance of swarms of marching locust nymphs, PLoS ONE, 9 (2014), e101636. doi: 10.1371/journal.pone.0101636. Google Scholar

[3]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101. doi: 10.1103/PhysRevE.74.022101. Google Scholar

[4]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001. Google Scholar

[5]

E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, Mesoscopic theory for fluctuating active nematics, New Journal of Physics, 15 (2013), 085032. doi: 10.1088/1367-2630/15/8/085032. Google Scholar

[6]

A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102. doi: 10.1214/10-AOP616. Google Scholar

[7]

J. BuhlD. J. T. SumpterI. D. CouzinJ. J. HaleE. DesplandE. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142. Google Scholar

[8]

J. -B. Caussin, A. Solon, A. Peshkov, H. Chaté, T. Dauxois, J. Tailleur, V. Vitelli, and D. Bartolo, Emergent spatial structures in flocking models: A dynamical system insight, Phys. Rev. Lett., 112 (2014), 148102. doi: 10.1103/PhysRevLett.112.148102. Google Scholar

[9]

H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: variations on the vicsek model, The European Physical Journal B, 64 (2008), 451-456. Google Scholar

[10]

I. D. CouzinJ. KrauseN. 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. Google Scholar

[11]

A. CzirókA.L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Phys. Rev. Lett., 82 (1999), 209-212. Google Scholar

[12]

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), 29-85. doi: 10.1007/BF01010922. Google Scholar

[13]

D. A. Dawson and J. Gärtner, Large deviations from the Mckean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446. Google Scholar

[14]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[15]

P. Degond and H. Yu, Self-Organized Hydrodynamics in an annular domain: modal analysis and nonlinear effects, Mathematical Models and Methods in Applied Sciences, 25 (2015), 495-519. doi: 10.1142/S0218202515400047. Google Scholar

[16]

M. H. DeGroot and M. J. Schervish, Probability and Statistics, Addison-Wesley, Boston, 2012.Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar

[18]

L. Edelstein-KeshetJ. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? an investigation for migratory locusts, Journal of Mathematical Biology, 36 (1998), 515-549. doi: 10.1007/s002850050112. Google Scholar

[19]

C. Escudero, C. A. Yates, J. Buhl, I. D. Couzin, R. Erban, I. G. Kevrekidis and P. K. Maini, Ergodic directional switching in mobile insect groups, Phys. Rev. E, 82 (2010), 011926. doi: 10.1103/PhysRevE.82.011926. Google Scholar

[20]

K.-T. FangC.-X. Ma and P. Winker, Centered L2-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs, Math. Comp., 71 (2002), 275-296. doi: 10.1090/S0025-5718-00-01281-3. Google Scholar

[21]

J. GarnierG. Papanicolaou and T.-W. Yang, Consensus convergence with stochastic effects, Vietnam Journal of Mathematics, 45 (2017), 51-75. doi: 10.1007/s10013-016-0190-2. Google Scholar

[22]

J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. Google Scholar

[23]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[24]

F. J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp., 67 (1998), 299-322. doi: 10.1090/S0025-5718-98-00894-1. Google Scholar

[25]

T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev. E, 83 (2011), 030901. doi: 10.1103/PhysRevE.83.030901. Google Scholar

[26]

S. Mishra, A. Baskaran and M. C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E, 81 (2010), 061916. doi: 10.1103/PhysRevE.81.061916. Google Scholar

[27]

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. Google Scholar

[28]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[29]

O. J. O'Loan and M. R. Evans, Alternating steady state in one-dimensional flocking, Journal of Physics A: Mathematical and General, 32 (1999), L99. doi: 10.1088/0305-4470/32/8/002. Google Scholar

[30]

A. M. ReynoldsG. A. SwordS. J. Simpson and D. R. Reynolds, Predator percolation, insect outbreaks, and phase polyphenism, Current Biology, 19 (2009), 20-24. doi: 10.1016/j.cub.2008.10.070. Google Scholar

[31]

P. RomanczukM. BärW. EbelingB. Lindner and L. Schimansky-Geier, Active brownian particles, The European Physical Journal Special Topics, 202 (2012), 1-162. doi: 10.1140/epjst/e2012-01529-y. Google Scholar

[32]

A. V. Savkin, Coordinated collective motion of groups of autonomous mobile robots: Analysis of Vicsek's model, IEEE Trans. Automat. Control, 49 (2004), 981-983. doi: 10.1109/TAC.2004.829621. Google Scholar

[33]

A. P. Solon and J. Tailleur, Revisiting the flocking transition using active spins, Phys. Rev. Lett., 111 (2013), 078101. doi: 10.1103/PhysRevLett.111.078101. Google Scholar

[34]

A. P. Solon and J. Tailleur, Flocking with discrete symmetry: The 2d active Ising model, Phys. Rev. E, 92 (2015), 042119. doi: 10.1103/PhysRevE.92.042119. Google Scholar

[35]

A. P. Solon, J. -B. Caussin, D. Bartolo, H. Chaté and J. Tailleur, Pattern formation in flocking models: A hydrodynamic description, Phys. Rev. E, 92 (2015), 062111. doi: 10.1103/PhysRevE.92.062111. Google Scholar

[36]

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. Google Scholar

[37]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828. Google Scholar

[38]

C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet, and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput Biol, 8 (2012), e1002642, 11 pp. doi: 10.1371/journal.pcbi.1002642. Google Scholar

[39]

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

[40]

T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. Google Scholar

[41]

C. A. YatesR. ErbanC. EscuderoI. D. CouzinJ. BuhlI. G. KevrekidisP. K. Maini and D. J. T. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106. Google Scholar

show all references

References:
[1]

G. Ariel and A. Ayali, Locust collective motion and its modeling, PLoS Comput Biol, 11 (2015), e1004522. doi: 10.1371/journal.pcbi.1004522. Google Scholar

[2]

G. Ariel, Y. Ophir, S. Levi, E. Ben-Jacob and A. Ayali, Individual pause-and-go motion is instrumental to the formation and maintenance of swarms of marching locust nymphs, PLoS ONE, 9 (2014), e101636. doi: 10.1371/journal.pone.0101636. Google Scholar

[3]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101. doi: 10.1103/PhysRevE.74.022101. Google Scholar

[4]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001. Google Scholar

[5]

E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, Mesoscopic theory for fluctuating active nematics, New Journal of Physics, 15 (2013), 085032. doi: 10.1088/1367-2630/15/8/085032. Google Scholar

[6]

A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102. doi: 10.1214/10-AOP616. Google Scholar

[7]

J. BuhlD. J. T. SumpterI. D. CouzinJ. J. HaleE. DesplandE. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142. Google Scholar

[8]

J. -B. Caussin, A. Solon, A. Peshkov, H. Chaté, T. Dauxois, J. Tailleur, V. Vitelli, and D. Bartolo, Emergent spatial structures in flocking models: A dynamical system insight, Phys. Rev. Lett., 112 (2014), 148102. doi: 10.1103/PhysRevLett.112.148102. Google Scholar

[9]

H. ChatéF. GinelliG. GrégoireF. Peruani and F. Raynaud, Modeling collective motion: variations on the vicsek model, The European Physical Journal B, 64 (2008), 451-456. Google Scholar

[10]

I. D. CouzinJ. KrauseN. 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. Google Scholar

[11]

A. CzirókA.L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Phys. Rev. Lett., 82 (1999), 209-212. Google Scholar

[12]

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), 29-85. doi: 10.1007/BF01010922. Google Scholar

[13]

D. A. Dawson and J. Gärtner, Large deviations from the Mckean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446. Google Scholar

[14]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[15]

P. Degond and H. Yu, Self-Organized Hydrodynamics in an annular domain: modal analysis and nonlinear effects, Mathematical Models and Methods in Applied Sciences, 25 (2015), 495-519. doi: 10.1142/S0218202515400047. Google Scholar

[16]

M. H. DeGroot and M. J. Schervish, Probability and Statistics, Addison-Wesley, Boston, 2012.Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-5320-4. Google Scholar

[18]

L. Edelstein-KeshetJ. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? an investigation for migratory locusts, Journal of Mathematical Biology, 36 (1998), 515-549. doi: 10.1007/s002850050112. Google Scholar

[19]

C. Escudero, C. A. Yates, J. Buhl, I. D. Couzin, R. Erban, I. G. Kevrekidis and P. K. Maini, Ergodic directional switching in mobile insect groups, Phys. Rev. E, 82 (2010), 011926. doi: 10.1103/PhysRevE.82.011926. Google Scholar

[20]

K.-T. FangC.-X. Ma and P. Winker, Centered L2-discrepancy of random sampling and Latin hypercube design, and construction of uniform designs, Math. Comp., 71 (2002), 275-296. doi: 10.1090/S0025-5718-00-01281-3. Google Scholar

[21]

J. GarnierG. Papanicolaou and T.-W. Yang, Consensus convergence with stochastic effects, Vietnam Journal of Mathematics, 45 (2017), 51-75. doi: 10.1007/s10013-016-0190-2. Google Scholar

[22]

J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. Google Scholar

[23]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[24]

F. J. Hickernell, A generalized discrepancy and quadrature error bound, Math. Comp., 67 (1998), 299-322. doi: 10.1090/S0025-5718-98-00894-1. Google Scholar

[25]

T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev. E, 83 (2011), 030901. doi: 10.1103/PhysRevE.83.030901. Google Scholar

[26]

S. Mishra, A. Baskaran and M. C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E, 81 (2010), 061916. doi: 10.1103/PhysRevE.81.061916. Google Scholar

[27]

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. Google Scholar

[28]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[29]

O. J. O'Loan and M. R. Evans, Alternating steady state in one-dimensional flocking, Journal of Physics A: Mathematical and General, 32 (1999), L99. doi: 10.1088/0305-4470/32/8/002. Google Scholar

[30]

A. M. ReynoldsG. A. SwordS. J. Simpson and D. R. Reynolds, Predator percolation, insect outbreaks, and phase polyphenism, Current Biology, 19 (2009), 20-24. doi: 10.1016/j.cub.2008.10.070. Google Scholar

[31]

P. RomanczukM. BärW. EbelingB. Lindner and L. Schimansky-Geier, Active brownian particles, The European Physical Journal Special Topics, 202 (2012), 1-162. doi: 10.1140/epjst/e2012-01529-y. Google Scholar

[32]

A. V. Savkin, Coordinated collective motion of groups of autonomous mobile robots: Analysis of Vicsek's model, IEEE Trans. Automat. Control, 49 (2004), 981-983. doi: 10.1109/TAC.2004.829621. Google Scholar

[33]

A. P. Solon and J. Tailleur, Revisiting the flocking transition using active spins, Phys. Rev. Lett., 111 (2013), 078101. doi: 10.1103/PhysRevLett.111.078101. Google Scholar

[34]

A. P. Solon and J. Tailleur, Flocking with discrete symmetry: The 2d active Ising model, Phys. Rev. E, 92 (2015), 042119. doi: 10.1103/PhysRevE.92.042119. Google Scholar

[35]

A. P. Solon, J. -B. Caussin, D. Bartolo, H. Chaté and J. Tailleur, Pattern formation in flocking models: A hydrodynamic description, Phys. Rev. E, 92 (2015), 062111. doi: 10.1103/PhysRevE.92.062111. Google Scholar

[36]

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. Google Scholar

[37]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828. Google Scholar

[38]

C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet, and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput Biol, 8 (2012), e1002642, 11 pp. doi: 10.1371/journal.pcbi.1002642. Google Scholar

[39]

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

[40]

T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. Google Scholar

[41]

C. A. YatesR. ErbanC. EscuderoI. D. CouzinJ. BuhlI. G. KevrekidisP. K. Maini and D. J. T. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106. Google Scholar

Figure 1.  The real (a) and imaginary (b) parts of the complex growth rate of the first mode (the most unstable mode) as a function of $\sigma$ for $h = 5$ (solid black), $h = 6$ (dashed red), $h = 8$ (magenta dotted), and $h = 10$ (blue dot-dashed). Here $\phi$ is given by (46) and $L = 10$. $G$ is given by (47) and it derives from the double-well potential plotted in picture c. We address the linear stability of the order state $\rho_{\xi_e}$. One can see that the threshold value for the noise level $\sigma$ to ensure linear stability is $1.8$ for $h = 5$, $0.85$ for $h = 6$, $1.4$ for $h = 8$, and $2.2$ for $h = 10$. The most stable situation is the one corresponding to $h = 6$.
Figure 2.  The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ at each time step $t_n$ for $h = 2$ (a-b) and $h = 6$ (c-d). The dashed lines in the velocity plots are the solutions of $G(\xi) = \xi$. The dashed lines in the discrepancy plots stand for the value (53) corresponding to a uniform sampling. Here $\Delta t = 0.1$, $N = 500$, and $\sigma = 2$. One can see that the spatial distribution is uniform and the velocity average is 0 when $h = 2$ and $- \xi_e$ when $h = 6$.
Figure 3.  The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ for $\sigma = 0.5$ (a-b), $\sigma = 1$ (c-d), $\sigma = 1.5$ (e-f). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, and $h = 6$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform when $\sigma = 0.5$ while the average velocity is $\xi_e$ and the spatial distribution is uniform when $\sigma = 1$ or $1.5$.
Figure 4.  The empirical position distribution smoothed by kernel density estimation (a and c) and the first $200$ trajectories (b and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 0.5$, and $h = 6$. One can see that the spatial distribution is not uniform and a cluster is moving with an apparent velocity of $3.6$.
Figure 5.  The empirical average velocity $\bar{u}^n$ and the square centered $L^2$-discrepancy $CL_2^2(n)$ for $\sigma = 0.5$ (a-b), $\sigma = 1$ (c-d), $\sigma = 1.5$ (e-f). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean 0 and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, and $h = 6$. One can see that the average velocity is not $\pm \xi_e$ and the spatial distribution is not uniform when $\sigma = 0.5$ or $1$ while the average velocity is $\xi_e$ and the spatial distribution is uniform when $\sigma = 1.5$.
Figure 6.  The empirical average velocity $\bar{u}^n$ (a), the square centered $L^2$-discrepancy $CL_2^2(n)$ (b), and the empirical position distribution smoothed by kernel density estimation (c and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 1$, and $h = 10$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform.
Figure 7.  The empirical average velocity $\bar{u}^n$ (a), the square centered $L^2$-discrepancy $CL_2^2(n)$ (b), and the empirical position distribution smoothed by kernel density estimation (c and d). The initial positions $\{x_i^0\}_{i = 1}^N$ are uniformly sampled over $[0,L]$ and the initial velocities $\{u_i^0\}_{i = 1}^N$ are sampled from the Gaussian distribution with mean $\xi_e$ and variance $\sigma^2/2$. Here $\Delta t = 0.1$, $N = 2000$, $\sigma = 1$, and $h = 5$. One can see that the average velocity is not $\xi_e$ and the spatial distribution is not uniform.
Figure 8.  The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $N = 80$ (a), $N = 100$ (b), $N = 120$ (c), and $N = 140$ (d). Here $\Delta t = 0.1$, $h = 6$ and $\sigma = 5$. The frequencies of the transitions between the two stable order states decay when $N$ increases. The system has less transitions with a higher number of agents.
Figure 9.  The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $\sigma = 4$ (a), $\sigma = 4.5$ (b), $\sigma = 5$ (c), and $\sigma = 5.5$ (d). Here $\Delta t = 0.1$, $N = 100$, and $h = 6$. The frequencies of the transitions between the two stable order states increase when $\sigma$ increases. The system has more transitions with a higher $\sigma$.
Figure 10.  The empirical average velocity $\bar{u}^n$ at each time step $t_n$ for $h = 5$ (a), $h = 5.5$ (b), $h = 6$ (c), and $h = 6.5$ (d). Here $\Delta t = 0.1$, $N = 100$, and $\sigma = 5$. The frequencies of the transitions between the two stable order states decay with $h$.
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