Existence of Ground States of Nonlocal-Interaction Energies

Simione, Robert; Slepčev, Dejan; Topaloglu, Ihsan

doi:10.1007/s10955-015-1215-z

Cited by 59 publications

(100 citation statements)

References 42 publications

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“…Note that from (28) it is clear that F α,β (0) = 0 for all values of α, β. On the other hand, the picture above suggests that changing the values of parameters α, β may cause that F α,β will achieve negative values.…”

Section: 2mentioning

confidence: 99%

“…and −J 2 (z) that are included in the definition of F α,β (z) from (28). Note that from (28) it is clear that F α,β (0) = 0 for all values of α, β.…”

Section: 2mentioning

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%

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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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Section: 2mentioning

confidence: 99%

“…and −J 2 (z) that are included in the definition of F α,β (z) from (28). Note that from (28) it is clear that F α,β (0) = 0 for all values of α, β.…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

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Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Barré¹,

Degond²,

Zatorska³

2017

Multiscale Model. Simul.

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“…It follows that the evolution is a gradient flow in a weighted metric [42,67,8,9,30,61]. The time derivative of the energy is…”

Section: 2mentioning

confidence: 99%

“…which can be used to analyze the existence and stability of equilibria [8,9,39,30,61]. Despite its popularity in the literature, one shortfall of this model is that it does not produce so-called well-spaced groups.…”

mentioning

confidence: 99%

Biological Aggregation Driven by Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn--Hilliard Approximation

Bernoff¹,

Topaz²

2016

SIAM J. Appl. Dyn. Syst.

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Abstract. Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. 1. Introduction. This paper has three primary aims. First, beginning with a partial integrodifferential equation model of biological aggregation, we show that a long-wave approximation yields a degenerate Cahn-Hilliard equation akin to models describing phase separation in materials science and evolution of thin fluid films. Second, we study solutions of the degenerate Cahn-Hillard model and demonstrate that they compare well with those of the original model. Using the Cahn-Hilliard model is advantageous because it eliminates many of the analytical and computational challenges of nonlocal equations, particularly in higher dimensions. Our methodology is to study the energy minimizers of this local equation, rather than studying the time evolution of the original nonlocal equation. Finally, we use the Cahn-Hilliard model as a testbed to assess whether imposing an external potential modeling the environment, e.g., food sources, can be used to control peak density in aggregations, which is of interest for locust swarms.Biological aggregations such as bird flocks, fish schools, and insect swarms are driven by social interactions between group members. Typical social forces include attraction, repulsion,

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The Equilibrium Measure for a Nonlocal Dislocation Energy

Mora

Rondi

Scardia³

2018

Comm Pure Appl Math

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In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels. © 2018 Wiley Periodicals, Inc.

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Existence of Ground States of Nonlocal-Interaction Energies

Cited by 59 publications

References 42 publications

Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Biological Aggregation Driven by Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn--Hilliard Approximation

The Equilibrium Measure for a Nonlocal Dislocation Energy

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