Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Barré, Julien; Degond, Pierre; Zatorska, Ewelina

doi:10.1137/16m1085310

Cited by 28 publications

(57 citation statements)

References 26 publications

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“…Our strategy is based on an argument by contradiction using the nonlinear integral equation satisfied by the steady states obtained from the Euler-Lagrange conditions for the critical points; see Section 3. We conjecture that this is one of the reasons behind the metastability observed in numerical simulations [6,7,35,37] in this context. Note that our result asserts that no stationary state of (1.1) exists for ε > 0 in the whole space R d ; however, for bounded domains with no-flux boundary conditions, ground states exist by compactness and lower semicontinuity of the energy.…”

Section: Introductionmentioning

confidence: 79%

“…We refer the reader to [19,48] for further considerations on H-stability and on the existence/nonexistence of global minimizers without noise. Typical examples include Morse and repulsive-attractive powerlaw potentials [28], as well as compactly supported repulsive-attractive potentials [7,37]. Our second main result (Theorem 3.1) gives a negative answer for repulsive-attractive interaction potentials which are smooth enough showing that no critical points or stationary states of the energy (1.2) exist as soon as the linear diffusion is triggered with ε > 0, no matter how small the noise strength ε is.…”

Section: Introductionmentioning

confidence: 83%

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Existence of ground states for aggregation-diffusion equations

2019

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We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 83%

Existence of ground states for aggregation-diffusion equations

2019

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“…The Barré-Degond-Zatorska model for interacting dynamical networks. The Barré-Degond-Zatorska system [BDZ17] models particles that interact through a dynamical network of links. Each particle interacts with its closest neighbours through cross-links modelled by springs which are randomly created and destroyed.…”

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confidence: 99%

Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

Carrillo

Gvalani

Pavliotis

et al. 2019

Arch Rational Mech Anal

100

127

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We study the McKean-Vlasov equationwith periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller-Segel model for bacterial chemotaxis, and the noisy Hegselmann-Krausse model for opinion dynamics. 13 3.1. Trend to equilibrium in relative entropy 13 3.2. Linear stability analysis 14 4. Bifurcation theory 15 5. Phase transitions for the McKean-Vlasov equation 20 5.1. Discontinuous transition points 22 5.2. Continuous transition points 24 6. Applications 31 6.1. The generalised Kuramoto model 31 6.2. The noisy Hegselmann-Krause model for opinion dynamics 34 6.3. The Onsager model for liquid crystals 34 6.4. The Barré-Degond-Zatorska model for interacting dynamical networks 36 6.5. The Keller-Segel model for bacterial chemotaxis 36 Appendix A. Results from bifurcation theory 39 References 41

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“…They describe the behavior of each agent and its interaction with the surrounding agents over time, offering a description of the system at the microscopic scale (see e.g. [3,6,17]). However, these models are computationally expensive and are not suited for the study of large systems.…”

Section: Introductionmentioning

confidence: 99%

“…

We provide a numerical study of the macroscopic model of [3] derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodelling process is very fast, the macroscopic model takes the form of a single aggregation diffusion equation for the density of particles.

…”

mentioning

confidence: 99%

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions

et al. 2017

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We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation–diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the one-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale.

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Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

Cited by 28 publications

References 26 publications

Existence of ground states for aggregation-diffusion equations

Existence of ground states for aggregation-diffusion equations

Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions

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