Existence of Compactly Supported Global Minimisers for the Interaction Energy

Cañizo, José A.; Carrillo, José A.; Patacchini, Francesco S.

doi:10.1007/s00205-015-0852-3

Cited by 81 publications

(192 citation statements)

References 47 publications

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“…Here we also would like to note that during the review process of this manuscript we noticed that the radial symmetry and uniform boundedness assumption have been relaxed in the recent preprints [23,24] in certain regimes of p and q-values of power-law potentials whereas the results in [25,26] extend our existence results to more general potentials.…”

Section: Introductionmentioning

confidence: 82%

“…Now, first we note that since M m ω N , the function ρ is in the admissible class A, where ω N = π N/2 Γ (N/2+1) denotes the volume of the unit ball in R N . Next, we check that ρ is a critical point of the functional E, i.e., that ρ satisfies (25). As noted in Remark 2.5, the attractive term of the energy simplifies when q = 2.…”

Section: Ground State For Quadratic Attraction and Newtonian Repulsionmentioning

confidence: 98%

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On minimizers of interaction functionals with competing attractive and repulsive potentials

Choksi

Fetecau

Topaloglu

2015

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider a family of interaction functionals consisting of power-law potentials with attractive and repulsive parts and use the concentration compactness principle to establish the existence of global minimizers. We consider various minimization classes, depending on the signs of the repulsive and attractive power exponents of the potential. In the special case of quadratic attraction and Newtonian repulsion we characterize in detail the ground state.

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Section: Introductionmentioning

confidence: 82%

Section: Ground State For Quadratic Attraction and Newtonian Repulsionmentioning

confidence: 98%

On minimizers of interaction functionals with competing attractive and repulsive potentials

Choksi

Fetecau

Topaloglu

2015

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…We denote by t 0 the unit of time and by x 0 the unit of space. A straightforward scaling argument allows us to interpret (8) and (15) obtained in the previous section as scaled equations. Upon choosing time and space units, we can also interpret the coefficients as scaled coefficients in these units.…”

Section: Preliminariesmentioning

confidence: 99%

“…Such bifurcations were previously studied in [14] from a "thermodynamical" point of view, i.e., by looking at the minimizers of the free energy functional; we present here a dynamical point of view and make the connection with the thermodynamical approach. In the case without diffusion, this free energy functional reduces to the interaction energy, whose minimizers have been studied in [8,28,12]; for numerical studies in this direction we refer to [11]. In particular, global minimizers exist provided the associated potential is H-unstable, a classical notion in statistical mechanics linked to the phase transitions in the system [19,27].…”

mentioning

confidence: 99%

Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Barré¹,

Degond²,

Zatorska³

2017

Multiscale Model. Simul.

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Abstract. We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [P. Degond, F. Delebecque, and D. Peurichard, Math. Models Methods Appl. Sci., 26 (2016), pp. 269-318]. Assuming that the process of remodeling the network is very fast, we simplify the description to a macroscopic model taking the form of a single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset.

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“…These clusters are then deposited at regular intervals along the horizontal myoseptum and will mature into full active neuromasts. The rest of the primordium (still constituted by pseudo-mesenchymal cells) eventually continues its migration ( Dambly-Chaudière et al, 2007;Durdu et al, 2014;Nechiporuk and Raible, 2008 ).…”

Section: Early Migration Of the Zebrafish Posterior Lateral Line (Pllmentioning

confidence: 99%

Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles

Carrillo

Colombi

Scianna

2018

Journal of Theoretical Biology

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a b s t r a c tThe description of the cell spatial pattern and characteristic distances is fundamental in a wide range of physio-pathological biological phenomena, from morphogenesis to cancer growth. Discrete particle models are widely used in this field, since they are focused on the cell-level of abstraction and are able to preserve the identity of single individuals reproducing their behavior. In particular, a fundamental role in determining the usefulness and the realism of a particle mathematical approach is played by the choice of the intercellular pairwise interaction kernel and by the estimate of its parameters. The aim of the paper is to demonstrate how the concept of H-stability, deriving from statistical mechanics, can have important implications in this respect. For any given interaction kernel, it in fact allows to a priori predict the regions of the free parameter space that result in stable configurations of the system characterized by a finite and strictly positive minimal interparticle distance, which is fundamental when dealing with biological phenomena. The proposed analytical arguments are indeed able to restrict the range of possible variations of selected model coefficients, whose exact estimate however requires further investigations (e.g., fitting with empirical data), as illustrated in this paper by series of representative simulations dealing with cell colony reorganization, sorting phenomena and zebrafish embryonic development.

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Existence of Compactly Supported Global Minimisers for the Interaction Energy

Abstract: 13.07.15.KB. Ok to add published version to spiral, OA pape

Cited by 81 publications

References 47 publications

On minimizers of interaction functionals with competing attractive and repulsive potentials

On minimizers of interaction functionals with competing attractive and repulsive potentials

Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles

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