Flocking With Short-Range Interactions

Abstract

We study the large-time behavior of continuum alignment dynamics based on Cucker–Smale (CS)-type interactions which involve short-range kernels, that is, communication kernels with support much smaller than the diameter of the crowd. We show that if the amplitude of the interactions is larger than a finite threshold, then unconditional hydrodynamic flocking follows. Since we do not impose any regularity nor do we require the kernels to be bounded, the result covers both regular and singular interaction kernels.Moreover, we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average (over balls at small enough scale), but otherwise vacuum is allowed at smaller scales. Consequently, our arguments of hydrodynamic flocking apply, mutatis mutandis, to the agent-based CS model with finitely many Dirac masses. In particular, discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents.

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Notes

  1. 1.

    Here and below we use \(\delta \cdot \) to denote fluctuations, \(\delta W\equiv (\delta W)({{\mathbf {x}}},{{\mathbf {y}}}):=W({{\mathbf {x}}})-W({{\mathbf {y}}})\) with the corresponding weighted norms taken on the product space \({{\mathcal {S}}}\times {{\mathcal {S}}}\), e.g., \(|\delta {{\mathbf {u}}}|^2_2= \int |{{\mathbf {u}}}({{\mathbf {x}}})-{{\mathbf {u}}}({{\mathbf {y}}})|^2 \, \text{ d }\rho ({{\mathbf {x}}})\, \text{ d }\rho ({{\mathbf {y}}})\). Likewise, \((\delta {{\mathbf {v}}})_{ij}={{\mathbf {v}}}_i-{{\mathbf {v}}}_j\) with \(|\delta {{\mathbf {v}}}|_\infty =\max _{i,j}|{{\mathbf {v}}}_i-{{\mathbf {v}}}_j|\) etc.

  2. 2.

    Recall that \(\rho \) is a probability measure

  3. 3.

    \({{\mathbf {x}}}_\pm \) need not be unique — any extreme location will suffice.

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Acknowledgements

Research was supported by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR Grant N00014-1812465. JP was also supported by the Polish MNiSW grant Mobilność Plus no. 1617/MOB/V/2017/0.

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Correspondence to Eitan Tadmor.

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Communicated by Eric Carlen.

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Morales, J., Peszek, J. & Tadmor, E. Flocking With Short-Range Interactions. J Stat Phys 176, 382–397 (2019). https://doi.org/10.1007/s10955-019-02304-5

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Keywords

  • Alignment
  • Cucker–Smale
  • Agent-based system
  • Large-crowd hydrodynamics
  • Interaction kernels
  • Short-range
  • Chain connectivity
  • Flocking

Mathematics Subject Classification

  • 92D25
  • 35Q35
  • 76N10