By Adrian Muntean, Federico Toschi

Multiscale types in social purposes mix mean-field and kinetic equations with both microscopic or macroscopic point descriptions. during this publication the reader will locate not just a large spectrum of multiscale research effects (like convergence proofs), but in addition essentially very important info similar to derivations of mean-field equations, ways to deal with difficult contacts numerically, to version team habit, to quantitative estimate microscopic/macroscopic segregation of competing species, to quantitative comprehend the boundaries of validity of mass-action kinetics for easy reactions.

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P. Choi and M. Hauray up to time T > 0. Assume that ρ0 has compact support, that the initial 0 positions X N,0 := {Xi0 }N i=1 are iid with law ρ , and that (1 + α)p < p−1 d, 2p − 1 with α ≥ 0. Then the propagation of chaos holds in the sense that P sup d1 (μN (t), ρ(t)) ≥ t∈[0,T ] C N γ/d → 0, as N → +∞, where γ is a positive constant satisfying p (2p − 1)(1 + α) < γ < 1. 2. The condition on α gets more and more restrictive as p gets smaller and smaller. In d = 2, even for p = ∞ the condition is empty for α ≥ 0.

Franks, and S. A. Levin, Eﬀective leadership and decision making in animal groups on the move, N ature 433:513–516, 2005. F. Cucker and S. Smale, Emergent behavior in ﬂocks, I EEE Trans. Automat. Control 52:852-862, 2007. C. De Lellis and L. Sz´ekelyhidi, The Euler equations as a diﬀerential inclusion, Ann. of Math. 170:1417–1436, 2009. R. Dobrushin, Vlasov equations, F unct. Anal. Appl. 13:115–123, 1979. A. P. Choi and M. Hauray M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. Chayes, Selfpropelled particles with soft-core interactions: patterns, stability, and collapse, P hys.

The projection (9) can also be formulated in a dual manner: since qn+1 = PK (qn + τ U(qn )) there exists a collection of Lagrange multipliers (λij )(i,j)∈Λ such that qn + τ U(qn ) = qn+1 − λij Gij (qn+1 ), i,j which can be written qn+1 − qn + B λ = U(qn ). τ Yet, in spite of its formal simplicity, this formulation does not directly lead to a tractable numerical scheme, since the matrix B (together with the set Λ of active couples) depends on the unknown qn+1 . It reﬂects the implicit (and highly nonlinear) character of the scheme.