Speaker
Description
I will present the result of a collaboration with Pierre Degond and Sara Merino-Aceituno. We study the emergence of band patterns in the Vicsek model, a minimal agent-based model of alignment dynamics with noise. Agent-based simulations on periodic domains display coexisting ordered (high-density, aligned) bands and disordered (low-density, non-aligned) regions, a phenomenon not explained by classical parameter-driven phase transitions. We review prior kinetic and macroscopic results that identify an spatial phase transition~\cite{degond_phase_2015}~: depending on the local density $\rho$ relative to a critical threshold $\rho_c$, different PDEs govern different spatial regions—Self-Organized Hydrodynamics (SOH, \cite{degond_continuum_2008}) in the ordered regime ($\rho > \rho_c$) and a degenerate diffusive correction (at order $\varepsilon$) in the disordered regime ($\rho < \rho_c$). Building on this framework, we propose a model that couples the ordered and disordered macroscopic equations to simulate the continuum dynamics~\cite{motsch_numerical_2011}, with the goals of reproducing band formation at the macroscopic level, exploring pattern formation, and connecting the linear stability properties of the coupled model with those of SOH.
Bibliography
@article{degond_continuum_2008,
title = {{Continuum} {Limit} {Of} {Self}-{driven} {Particles} {With} {Orientation} {Interaction}},
author = {Degond, Pierre and Motsch, Sébastien},
year = {2008},
month = aug,
journal = {Mathematical Models and Methods in Applied Sciences},
volume = {18},
number = {supp01},
pages = {1193--1215},
doi = {10.1142/s0218202508003005},
issn = {0218-2025, 1793-6314}
}
@article{degond_phase_2015,
title = {Phase {Transitions}, {Hysteresis}, and {Hyperbolicity} for {Self}-{Organized} {Alignment} {Dynamics}},
author = {Degond, Pierre and Frouvelle, Amic and Liu, Jian-Guo},
year = {2015},
month = apr,
journal = {Archive for Rational Mechanics and Analysis},
volume = {216},
number = {1},
pages = {63--115},
doi = {10.1007/s00205-014-0800-7},
issn = {0003-9527, 1432-0673}
}
@article{motsch_numerical_2011,
title = {Numerical {Simulations} of a {Nonconservative} {Hyperbolic} {System} with {Geometric} {Constraints} {Describing} {Swarming} {Behavior}},
author = {Motsch, Sebastien and Navoret, Laurent},
year = {2011},
month = jul,
journal = {Multiscale Modeling \& Simulation},
volume = {9},
number = {3},
pages = {1253--1275},
doi = {10.1137/100794067},
issn = {1540-3459, 1540-3467}
}
