Kinetic Theory of Particle Interactions Mediated by Dynamical Networks

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.

[1]  Maria E. Schonbek,et al.  Decay rates for a class of diffusive-dominated interaction equations , 2011, 1106.5880.

[2]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Michael E. Fisher,et al.  THE STABILITY OF MANY-PARTICLE SYSTEMS , 1966 .

[4]  J. Carrillo,et al.  Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems , 2014, 1406.4040.

[5]  Andrew J. Bernoff,et al.  A Primer of Swarm Equilibria , 2010, SIAM J. Appl. Dyn. Syst..

[6]  John W. Barrett,et al.  Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers , 2014, 1407.3763.

[7]  D. Slepčev,et al.  Existence of Ground States of Nonlocal-Interaction Energies , 2014, 1405.5146.

[8]  J. A. Smith,et al.  Comparison of Hard-Core and Soft-Core Potentials for Modelling Flocking in Free Space , 2009, 0905.2260.

[9]  Pierre Degond,et al.  Evolution of wealth in a non-conservative economy driven by local Nash equilibria , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  David A Weitz,et al.  Cross-link-governed dynamics of biopolymer networks. , 2010, Physical review letters.

[11]  P. Degond,et al.  Simple mechanical cues could explain adipose tissue morphology. , 2017, Journal of theoretical biology.

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Pierre Degond,et al.  Continuum model for linked fibers with alignment interactions , 2015, 1505.05027.

[14]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .

[15]  D. Weitz,et al.  Elastic Behavior of Cross-Linked and Bundled Actin Networks , 2004, Science.

[16]  M. Haragus,et al.  Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems , 2010 .

[17]  A. Bertozzi,et al.  State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System , 2006, nlin/0606031.

[18]  Lorenzo Pareschi,et al.  Opinion dynamics over complex networks: kinetic modeling and numerical methods , 2016, ArXiv.

[19]  Nigel Clarke,et al.  "Bending to stretching" transition in disordered networks. , 2007, Physical review letters.

[20]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[21]  Andrea L. Bertozzi,et al.  Blow-up in multidimensional aggregation equations with mildly singular interaction kernels , 2009 .

[22]  J. Carrillo,et al.  A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure , 2014, 1402.4252.

[23]  J. Carrillo,et al.  Double milling in self-propelled swarms from kinetic theory , 2009 .

[24]  J. Carrillo,et al.  Existence of Compactly Supported Global Minimisers for the Interaction Energy , 2014, 1405.5428.

[25]  A. Bertozzi,et al.  Self-propelled particles with soft-core interactions: patterns, stability, and collapse. , 2006, Physical review letters.

[26]  D. Ruelle.  Statistical Mechanics: Rigorous Results , 1999 .

[27]  Lincoln Chayes,et al.  The McKean–Vlasov Equation in Finite Volume , 2009, 0910.4615.

[28]  R. Fetecau,et al.  Emergent behaviour in multi-particle systems with non-local interactions , 2013 .