A Review on Attractive–Repulsive Hydrodynamics for Consensus in Collective Behavior

Abstract

This survey summarizes and illustrates the main qualitative properties of hydrodynamics models for collective behavior. These models include a velocity consensus term together with attractive-repulsive potentials leading to non-trivial flock profiles. The connection between the underlying particle systems to the swarming hydrodynamic equations is performed through kinetic theory modelling arguments. We focus on Lagrangian schemes for the hydrodynamic systems showing the different qualitative behavior of the systems and its capability of keeping properties of the original particle models. We illustrate known results concerning large time profiles and blow-up in finite time of the hydrodynamic systems to validate the numerical scheme. We finally explore unknown situations making use of the numerical scheme showcasing a number of conjectures based on the numerical results.

Nonparametric inference of interaction laws in systems of agents from trajectory data

“…This suggests the following coercivity condition: Definition 7.1 (Coercivity condition). We say that the dynamical system defined in (14) together with the probability measure µ 0 on R dN , satisfies the coercivity condition on H with a constant c L > 0, if…”

“…Systems of interacting agents arise in a wide variety of disciplines, including Physics, Biology, Ecology, Neurobiology, Social Sciences, and Economics (see e.g. [14,45,78,67] and references therein). Agents may represent particles, atoms, cells, animals, neurons, people, rational agents, opinions, etc...…”

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

“…In particular, substituting in (12) the interaction function I given in (3) yields 1 We use the notation a b to mean that there exists a constant C > 0 such that a ≤ Cb. Figure 3: The asymptotic speed distribution (16) for various traffic densities ρ ∈ [0, 1] with V ∞ (ρ) given in (7), P (ρ) like in (4) with µ = 2, λ = 1 and a(ρ) = ρ(1−ρ), which fulfils the constraint (17).…”

“…The derivation of hydrodynamic equations in the non-classical setting of multi-agent systems is instead a currently underexplored topic due to the general lack of information about the asymptotic statistical behaviour of the system. Some quite recent works in this direction are [16,23,32]. In this work we focus on the case of vehicular traffic, in which only the total mass of vehicles is conserved by the microscopic interactions.…”

Emergent behavior of Cucker–Smale flocking particles with heterogeneous time delays

“…We also emphasize that the strategy used in [9] requires the strictly positive lower bound assumption for the weight function ψ, which is not needed in our framework. We refer to [3,2] and references therein for recent surveys on Cucker-Smale type flocking models. We now introduce the main assumptions and state the main result.…”