Skip to main content
Log in

Anticipation Breeds Alignment

  • Original Article
  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. 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, \(\tau _{ij}^{pq}\in [0,\tau ]\).

  2. 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 \(\mathbf{z}\).

  3. Under a simplifying assumption of a mono-kinetic closure.

  4. 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 \(E_i\).

  5. \(({\langle {r}\rangle }^{2-\beta })'' = -\beta (2-\beta ) r^2{\langle {r}\rangle }^{-2-\beta } + (2-\beta ){\langle {r}\rangle }^{-\beta } = (2-\beta )\big ((1-\beta ) r^2 + 1\big ){\langle {r}\rangle }^{-2-\beta }>0\) for \(\beta \leqslant 1\).

  6. Observe that we do not use the fat tail decay (1.6).

  7. To be pedantic at this point, the time derivative on the left (2.10) exists for almost all t’s by Rademacher theorem, where it coincides with the maximal time derivatives on the left of (2.9)\({}_i\).

  8. Note that a confining potential need not be positive yet \(U\geqslant -aL\) and hence \(1/2N\sum _j |\mathbf{v}_j|^2 \leqslant {\mathcal {E}}(0)+aL\).

  9. 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 \(\frac{1}{N}\sum _i |\mathbf{x}^\tau _i|^2=\frac{1}{2N^2}\sum _{i,j}|\mathbf{x}_i^\tau -\mathbf{x}_j^\tau |^2\).

  10. 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

  1. 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

    Article  MathSciNet  Google Scholar 

  2. Balagué , D., Carrillo , J., Yao , Y.: Confinement for repulsive–attractive kernels. DCDS - B 19(5), 1227–1248, 2014

    Article  MathSciNet  Google Scholar 

  3. Bernoff , A.J., Topaz , C.M.: A primer of swarm equilibria SIAM. J. Appl. Dyn. Syst. 10, 212–250, 2011

    Article  MathSciNet  Google Scholar 

  4. Bertozzi , A.L., Carrillo , J.A., Laurent , T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710, 2009

    Article  MathSciNet  ADS  Google Scholar 

  5. 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

    Article  MathSciNet  Google Scholar 

  6. 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

    Article  MathSciNet  Google Scholar 

  7. 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)

  8. 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)

  9. 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

    Article  MathSciNet  Google Scholar 

  10. 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

    Article  MathSciNet  Google Scholar 

  11. 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

    Article  MathSciNet  Google Scholar 

  12. 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

    Article  MathSciNet  Google Scholar 

  13. Cucker , F., Smale , S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862, 2007

    Article  MathSciNet  Google Scholar 

  14. Cucker , F., Smale , S.: On the mathematics of emergence. Jpn. J. Math. 2(1), 197–227, 2007

    Article  MathSciNet  Google Scholar 

  15. 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

    Article  MathSciNet  Google Scholar 

  16. Dietert, H., Shvydkoy, R.: On Cucker–Smale dynamical systems with degenerate communication. Anal. Appl. (2020)

  17. 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

    Article  MathSciNet  Google Scholar 

  18. 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

    Article  ADS  Google Scholar 

  19. 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

    Article  MathSciNet  Google Scholar 

  20. Gerlee , P., Tunstrøm , K., Lundh , T., Wennberg , B.: Impact of anticipation in dynamical systems. Phys. Rev. E 96, 062413, 2017

    Article  MathSciNet  ADS  Google Scholar 

  21. 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

    Article  Google Scholar 

  22. Guéant, O., Lasry, J.-M.: Pierre–Louis lions mean field games and applications. Paris-Princeton Lectures on Mathematical Finance, pp. 205-266 (2010)

  23. 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

    Article  MathSciNet  Google Scholar 

  24. Ha , S.-Y., Tadmor , E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(3), 415–435, 2008

    Article  MathSciNet  Google Scholar 

  25. He , S., Tadmor , E.: Global regularity of two-dimensional flocking hydrodynamics. Comptes rendus - Mathematique Ser. I(355), 795–805, 2017

    Article  MathSciNet  Google Scholar 

  26. Jabin , P.E.: A review of the mean field limits for Vlasov equations. KRM 7, 661–711, 2014

    Article  MathSciNet  Google Scholar 

  27. Kolokonikov , T., Sun , H., Uminsky , D., Bertozzi , A.: Stability of ring patterns arising from 2d particle interactions. Phys. Rev. E 84, 015203, 2011

    Article  ADS  Google Scholar 

  28. Levine , H., Rappel , W.-J., Cohen , I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101, 2000

    Article  ADS  Google Scholar 

  29. 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)

  30. Morin , A., Caussin , J.-B., Eloy , C., Bartolo , D.: Collective motion with anticipation: flocking, spinning, and swarming. Phys. Rev. E 91, 012134, 2015

    Article  MathSciNet  ADS  Google Scholar 

  31. Motsch , S., Tadmor , E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621, 2014

    Article  MathSciNet  Google Scholar 

  32. 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

    Article  MathSciNet  Google Scholar 

  33. Serfaty, S.: Coulomb gases and Ginzburg–Landau vortices. Zurich Lectures in Advanced Mathematics, 70, Eur. Math. Soc. (2015)

  34. Serfaty, S.: Mean field limit for Coulomb flows. arXiv:1803.08345

  35. Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing. Trans. Math. Appl. 1(1), tnx001 (2017)

  36. 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)

  37. Shvydkoy, R., Tadmor, E.: Topologically-based fractional diffusion and emergent dynamics with short-range interactions. ArXiv:1806:01371v3

  38. Shu , R., Tadmor , E.: Flocking hydrodynamics with external potentials. Arch. Rat. Mech. Anal. 238, 347–381, 2020

    Article  MathSciNet  Google Scholar 

  39. 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)

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eitan Tadmor.

Additional information

Communicated by A. Figalli

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shu, R., Tadmor, E. Anticipation Breeds Alignment. Arch Rational Mech Anal 240, 203–241 (2021). https://doi.org/10.1007/s00205-021-01609-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-021-01609-8

Navigation