Discrete minimisers are close to continuum minimisers for the interaction energy

Abstract

Abstract. Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy Γ-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.

Collective migration and patterning during early development of zebrafish posterior lateral line

“…The remaining set of model coefficients is somehow more technical. In this respect, for the sake of simplicity, we have further assumed that (i) V adh ¼ 2v adh , consistently with [16]; (ii) v t i (t)t j (t) adh =v rep , 5d 3 c =4d 3 a (in particular, we have fixed v adh =v rep ¼ 0:005 and V adh =v rep ¼ 0:01) so that the interactions laws result H-stable, see recent works in [24,29,30] and references therein; and (iii) a sdf 7 ¼ 10a sdf 4 with a sdf 4 [ (0, 0:1] and a sdf 7 [ (0, 1], according to [13]. For the estimate of the remaining parameters, i.e.…”

Modelling chase-and-run migration in heterogeneous populations

“…While explicit analytical explorations are often difficult in agent-based models, we remark on two areas that could benefit from further mathematical exploration. First, we have built on recent investigations on the H-stability properties of adhesive/repulsive pairwise interaction potentials (Cañizo et al 2015;Cañizo and Patacchini 2018;Carrillo et al 2018;Ruelle 1969), that have allowed us to identify parameter regimes under which the model system evolves to a physically realistic configuration. Such analytical studies have in fact offered us welcome constraints for streamlining the tricky process of parametrisation.…”

“…In this respect, it has been shown that the large-time behaviour of particle systems subjected to non-local pairwise interactions is related to the H-stability of the relative kernels and potentials, see Ruelle (1969). In particular, taking advantage of the characterization of H-stable potentials provided in Ruelle (1969), recent works (such as Cañizo et al 2015;Cañizo and Patacchini 2018;Carrillo et al 2018) provide a criterion that determines a subregion of the parameter space of the repulsive-adhesive interacting kernels that results in realistic crystalline cell pattern. Accounting for these analytical results, for any pair αβ ∈ {N N, N P, P N , P P}, the parameters F r and F αβ a must satisfy the relation…”

“…The most dispersed configuration we can obtain under adhesive-repulsive interactions alone arises by setting F P P a = 0 µm s −1 , and is characterised by bounded minimal interparticle distances that do not trespass d r due to the definition of f αβ res in Eq. (4) (see Cañizo et al 2015;Cañizo and Patacchini 2018;Carrillo et al 2018 and references therein for further details). On the other hand, on fixing F P P a we observe that decreasing F r results in a slight decrease of the equilibrium interparticle mean distance, specifically towards a highly packed colony (with a small degree of cell membrane overlap, see Fig.…”

Analysis of Spherical Shell Solutions for the Radially Symmetric Aggregation Equation

“…By the fattening instability condition from [5], one should be able to arrange As seen in [11] for the particle system, as the repulsive power crosses the threshold of stability for a single spherical shell, particles begin to align themselves on multiple spheres. Approaches such as those in [27,17] using Γ-convergence of interaction energy functionals may ultimately bridge the gap.…”