Equilibria for an Aggregation Model with Two Species

Evers, Joep H. M.; Fetecau, Razvan C.; Kolokolnikov, Théodore

doi:10.1137/16m1109527

Cited by 16 publications

(18 citation statements)

References 39 publications

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“…In this paper we study a type of particle system that is less intensively studied: a system with two species, called positive and negative particles [GLP10, CXZ16, DFF13, BBP17, EFK17,vM18]. In our setting, we also consider singular interactions.…”

Section: Introductionmentioning

confidence: 99%

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals.Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale δn > 0, which converges to zero as the number of particles n tends to infinity.We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on both the theory of λ-convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a δn-regularised version of the Groma-Balogh equations, and a priori estimates for the Groma-Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n → ∞ we require δn to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if δn converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations.These results show how the validity of the Groma-Balogh equations as the limit of manyparticle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

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Section: Introductionmentioning

confidence: 99%

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Garroni

Meurs

Peletier³

et al. 2019

Arch Rational Mech Anal

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“…The simplified model for the interacting dislocations which we consider in this paper is a onedimensional interacting particle system with two species (see §1.1). Interacting particle systems with multiple species are of rapidly increasing interest; see, e.g., [CXZ16,DFF13,BBP16,EFK16] and the references therein for applications to dislocation networks, cellular aggregation, granular media, pedestrian movement, opinion formation and predator-pray models. A common challenge in these particle systems is the passage to the many-particle limit.…”

Section: Introductionmentioning

confidence: 99%

“…Discussion and conclusion. Motivated by missing rigorous micro-to-macro connections for dislocation density models, we consider a class of interacting particle system (given by the energy E n (2)) consisting of two species, which is also related to other applications [DFF13, BBP16,EFK16]. For the physically interesting scaling regimes of the parameters α n and γ n , we prove that E n Γ-converges to E (Theorem 1.1).…”

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confidence: 99%

Many-particle limits and non-convergence of dislocation wall pile-ups

Meurs

2017

Nonlinearity

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Abstract. The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repelling force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models.Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale n of the decay of the interaction potential. Our main results are the Γ-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for n sufficiently large, and the non-convergence of the gradient flows for n sufficiently small.

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“…with generic cross-interaction kernels K ρ and K η is investigated in [22], whereas studies on the shape of stationary states can be found in [17,27]. Concerning the predator-prey modelling and patterns formation, in [15,46] a minimal version of (1) has been considered with only one predator and arbitrarily many prey subject to (different) singular potentials.…”

Section: 3mentioning

confidence: 99%

“…The goal of this paper is to show that the model above catches one of the main features that occur in nature, namely the formation of spatial patterns where the predators are surrounded of empty zones and the prey aggregates around, that is usually observed in fish schools or in flock of sheeps, see [32,38]. In the fully aggregative case, namely system (2) with d 1 = d 2 = 0, the formation of these types of patterns has been studied in several papers, see [15,27,21,46] and references therein.…”

mentioning

confidence: 98%

Multiple patterns formation for an aggregation/diffusion predator-prey system

Fagioli¹,

Jaafra²

2021

NHM

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We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant −α, with α > 0. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant α, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

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Equilibria for an Aggregation Model with Two Species

Cited by 16 publications

References 39 publications

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

Many-particle limits and non-convergence of dislocation wall pile-ups

Multiple patterns formation for an aggregation/diffusion predator-prey system

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