Abstract
We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, \(\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)\), with anticipation increment \(\tau >0\). As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected to emerge into flocking due to alignment of velocities, and spatial concentration due to confining potentials. We treat both the discrete dynamics and large crowd hydrodynamics, proving the decisive role of anticipation in driving such systems with attractive potentials into velocity alignment and spatial concentration. We also study the concentration effect near equilibrium for anticipated-based dynamics of pair of agents governed by attractive–repulsive potentials.
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Notes
Expanding (AT) in \(\tau \) we obtain (\(\Phi \)U) with matrices \(\displaystyle \Phi _{ij}= \overline{D^2U}_{ij}:=\int _{0}^1 D^2U(|(\mathbf{x}_i-\mathbf{x}_j)+\tau s(\mathbf{v}_i-\mathbf{v}_j)|)\,\mathrm {d}{s}\), depending on states \((\mathbf{x}_i,\mathbf{v}_i)\) and \((\mathbf{x}_j,\mathbf{v}_j)\). Their (pq) entries are given by \((\overline{D^2U}_{ij})_{pq}=(D^2U)_{pq}(|\mathbf{x}_i(t;{\tau _{ij}^{pq}})-\mathbf{x}_j(t;{\tau _{ij}^{pq}})|)\), evaluated in anticipated positions, \(\mathbf{x}(t;\tau _{ij}^{pq})=\mathbf{x}+\tau _{ij}^{pq}\mathbf{v}\), at some intermediate times, \(\tau _{ij}^{pq}\in [0,\tau ]\).
Throughout this paper, we use the notation \({\langle {r}\rangle }^s := (1+r^2)^{s/2}\) for scalar r and \({\langle {\mathbf{z}}\rangle }= {\langle {|\mathbf{z}|}\rangle }\) for vectors \(\mathbf{z}\).
Under a simplifying assumption of a mono-kinetic closure.
In fact \(E_i\) is not a proper particle energy, since \(\sum _i E_i \ne N E\) (the pairwise potential is counted twice). However, it is the ratio of the kinetic energy and potential energy in (2.5) which is essential, as one would like to eliminate all the positive terms with indices i in (2.5), in order to avoid exponential growth of \(E_i\).
\(({\langle {r}\rangle }^{2-\beta })'' = -\beta (2-\beta ) r^2{\langle {r}\rangle }^{-2-\beta } + (2-\beta ){\langle {r}\rangle }^{-\beta } = (2-\beta )\big ((1-\beta ) r^2 + 1\big ){\langle {r}\rangle }^{-2-\beta }>0\) for \(\beta \leqslant 1\).
Observe that we do not use the fat tail decay (1.6).
Note that a confining potential need not be positive yet \(U\geqslant -aL\) and hence \(1/2N\sum _j |\mathbf{v}_j|^2 \leqslant {\mathcal {E}}(0)+aL\).
We may assume without loss of generality, that the two time invariant moments vanish, \(\sum \mathbf{x}_i =\sum \mathbf{v}_i=0\), and hence \(\frac{1}{N}\sum _i |\mathbf{x}^\tau _i|^2=\frac{1}{2N^2}\sum _{i,j}|\mathbf{x}_i^\tau -\mathbf{x}_j^\tau |^2\).
Without loss of generality we use the normalization \(\int \rho _0(\mathbf{x})\,\mathrm {d}{\mathbf{x}}=\int \rho (t,\mathbf{x})\,\mathrm {d}{\mathbf{x}}=1\).
References
Balagué , D., Carrillo , T.J.A., Laurent , R.G.: Dimensionality of local minimizers of the interaction energy. Arch. Rat. Mech. Anal. 209, 1055–1088, 2013
Balagué , D., Carrillo , J., Yao , Y.: Confinement for repulsive–attractive kernels. DCDS - B 19(5), 1227–1248, 2014
Bernoff , A.J., Topaz , C.M.: A primer of swarm equilibria SIAM. J. Appl. Dyn. Syst. 10, 212–250, 2011
Bertozzi , A.L., Carrillo , J.A., Laurent , T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710, 2009
Bertozzi , A.L., Kolokolnikov , T., Sun , H., Uminsky , D., von Brecht , J.: Ring patterns and their bifurcations in a nonlocal model of biological swarms. Commun. Math. Sci. 13, 955–985, 2015
Bertozzi , A.L., Laurent , T., Léger , F.: Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions. Math. Models Methods Appl. Sci. 22(supp01), 1140005, 2012
Carrillo, J.A., Choi, Y.-P., Hauray, M.: The derivation of swarming models: mean-field limit and Wasserstein distances. In: Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation, Series: CISM Inter. Centre for Mech. Sci. Springer, vol. 533, pp. 1–45 (2014)
Carrillo, J.A., Choi, Y.-P., Perez, S.: A review on attractive–repulsive hydrodynamics for consensus in collective behavior Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology Bellomo, N., Degond, P., Tadmor, E. (eds.). Birkhäuser (2017)
Carrillo , J.A., Choi , Y.-P., Tadmor , E., Tan , C.: Critical thresholds in 1D Euler equations with non-local forces. Math. Models Methods Appl. Sci. 26(1), 185–206, 2016
Carrillo , J.A., D’Orsogna , M.R., Panferov , V.: Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2, 363–378, 2009
Carrillo , J., Fornasier , M., Rosado , J., Toscani , G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42(218), 218–236, 2010
Carrillo , J.A., Huang , Y., Martin , S.: Nonlinear stability of flock solutions in second-order swarming models. Nonlinear Anal. Real World Appl. 17, 332–343, 2014
Cucker , F., Smale , S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862, 2007
Cucker , F., Smale , S.: On the mathematics of emergence. Jpn. J. Math. 2(1), 197–227, 2007
Danchin , R., Mucha , P.B., Peszek , J., Wróblewski , B.: Regular solutions to the fractional Euler alignment system in the Besov spaces framework. Math. Models Methods Appl. Sci. 29(1), 89–119, 2019
Dietert, H., Shvydkoy, R.: On Cucker–Smale dynamical systems with degenerate communication. Anal. Appl. (2020)
Do , T., Kiselev , A., Ryzhik , L., Tan , C.: Global regularity for the fractional Euler alignment system. Arch. Ration. Mech. Anal. 228(1), 1–37, 2018
D’Orsogna, M.R., Chuang, Y.L., Bertozzi, A.L., Chayes, L.: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302, 2006
Figalli, A., Kang, M.-J.: A rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignment. Anal. PDE 1293, 843–866, 2019
Gerlee , P., Tunstrøm , K., Lundh , T., Wennberg , B.: Impact of anticipation in dynamical systems. Phys. Rev. E 96, 062413, 2017
Golse , F.: On the dynamics of large particle systems in the mean field limit. In macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity. Lect. Notes Appl. Math. Mech. 3, 1–144, 2016
Guéant, O., Lasry, J.-M.: Pierre–Louis lions mean field games and applications. Paris-Princeton Lectures on Mathematical Finance, pp. 205-266 (2010)
Ha , S.-Y., Liu , J.-G.: A simple proof of the Cucker–Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7(2), 297–325, 2009
Ha , S.-Y., Tadmor , E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(3), 415–435, 2008
He , S., Tadmor , E.: Global regularity of two-dimensional flocking hydrodynamics. Comptes rendus - Mathematique Ser. I(355), 795–805, 2017
Jabin , P.E.: A review of the mean field limits for Vlasov equations. KRM 7, 661–711, 2014
Kolokonikov , T., Sun , H., Uminsky , D., Bertozzi , A.: Stability of ring patterns arising from 2d particle interactions. Phys. Rev. E 84, 015203, 2011
Levine , H., Rappel , W.-J., Cohen , I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101, 2000
Minakowski, P., Mucha, P.B., Peszek, J., Zatorska, E.: Singular Cucker–Smale dynamics. In: Bellomo, N., Degond, P., Tadmor, E., (eds.) Active Particles—Volume 2—Theory, Models, Applications. Birkhäuser-Springer, Boston, USA (2019)
Morin , A., Caussin , J.-B., Eloy , C., Bartolo , D.: Collective motion with anticipation: flocking, spinning, and swarming. Phys. Rev. E 91, 012134, 2015
Motsch , S., Tadmor , E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621, 2014
Poyato , D., Soler , J.: Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker–Smale models. Math. Models Methods Appl. Sci. 27(6), 1089–1152, 2017
Serfaty, S.: Coulomb gases and Ginzburg–Landau vortices. Zurich Lectures in Advanced Mathematics, 70, Eur. Math. Soc. (2015)
Serfaty, S.: Mean field limit for Coulomb flows. arXiv:1803.08345
Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing. Trans. Math. Appl. 1(1), tnx001 (2017)
Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing III: fractional diffusion of order \(0\le \alpha \le 1\). Physica D 376–377, 131–137 (2018)
Shvydkoy, R., Tadmor, E.: Topologically-based fractional diffusion and emergent dynamics with short-range interactions. ArXiv:1806:01371v3
Shu , R., Tadmor , E.: Flocking hydrodynamics with external potentials. Arch. Rat. Mech. Anal. 238, 347–381, 2020
Tadmor, E., Tan, C.: Critical thresholds in flocking hydrodynamics with non-local alignment. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2028), 20130401 (2014)
Acknowledgements
Research was supported by NSF Grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR Grant N00014-1812465. ET thanks the hospitality of the Institut of Mittag-Leffler during fall 2018 visit which initiated this work, and of the Laboratoire Jacques-Louis Lions in Sorbonne University during spring 2019, with support through ERC Grant 740623 under the EU Horizon 2020, while concluding this work. We thank the anonymous reviewers for careful reading and insightful comments which help improving an earlier draft of the paper.
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Shu, R., Tadmor, E. Anticipation Breeds Alignment. Arch Rational Mech Anal 240, 203–241 (2021). https://doi.org/10.1007/s00205-021-01609-8
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DOI: https://doi.org/10.1007/s00205-021-01609-8