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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. [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. [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. [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. [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. [6] A. Budhiraja, P. 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. [7] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142. [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. [9] H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the vicsek model, The European Physical Journal B, 64 (2008), 451-456. [10] I. D. Couzin, J. Krause, N. 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. [11] A. Czirók, A.L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Phys. Rev. Lett., 82 (1999), 209-212. [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. [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. [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. [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. [16] M. H. DeGroot and M. J. Schervish, Probability and Statistics, Addison-Wesley, Boston, 2012. [17] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-5320-4. [18] L. Edelstein-Keshet, J. 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. [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. [20] K.-T. Fang, C.-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. [21] J. Garnier, G. 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. [22] J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. [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. [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. [25] T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev. E, 83 (2011), 030901. doi: 10.1103/PhysRevE.83.030901. [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. [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. [28] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. [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. [30] A. M. Reynolds, G. A. Sword, S. 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. [31] P. Romanczuk, M. Bär, W. Ebeling, B. 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. [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. [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. [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. [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. [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. [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. [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. [39] 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. [40] T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. [41] C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. 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.

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. [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. [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. [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. [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. [6] A. Budhiraja, P. 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. [7] J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142. [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. [9] H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the vicsek model, The European Physical Journal B, 64 (2008), 451-456. [10] I. D. Couzin, J. Krause, N. 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. [11] A. Czirók, A.L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Phys. Rev. Lett., 82 (1999), 209-212. [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. [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. [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. [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. [16] M. H. DeGroot and M. J. Schervish, Probability and Statistics, Addison-Wesley, Boston, 2012. [17] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, Berlin, 1998. doi: 10.1007/978-1-4612-5320-4. [18] L. Edelstein-Keshet, J. 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. [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. [20] K.-T. Fang, C.-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. [21] J. Garnier, G. 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. [22] J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. [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. [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. [25] T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev. E, 83 (2011), 030901. doi: 10.1103/PhysRevE.83.030901. [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. [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. [28] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. [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. [30] A. M. Reynolds, G. A. Sword, S. 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. [31] P. Romanczuk, M. Bär, W. Ebeling, B. 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. [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. [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. [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. [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. [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. [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. [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. [39] 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. [40] T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004. [41] C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. 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.
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 $h = 6$.
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 $h = 6$.
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 $1.5$.
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 $3.6$.
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 $\sigma = 1.5$.
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.
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.
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.
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 $\sigma$.
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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