Narrow Results

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first-order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly nonlocal interaction, we derive diffusive corrections to the first-order system which lead to the combination of a heat flow of the harmonic map and Landau–Lifschitz–Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau–Lifschitz–Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.

Myxobacteria are social bacteria, that can glide in two dimensions and form counter-propagating, interacting waves. Here, we present a novel age-structured, continuous macroscopic model for the movement of myxobacteria. The derivation is based on microscopic interaction rules that can be formulated as a particle-based model and set within the Self-Organized Hydrodynamics (SOH) framework. The strength of this combined approach is that microscopic knowledge or data can be incorporated easily into the particle model, whilst the continuous model allows for easy numerical analysis of the different effects. However, we found that the derived macroscopic model lacks a diffusion term in the density equations, which is necessary to control the number of waves, indicating that a higher order approximation during the derivation is crucial. Upon ad hoc addition of the diffusion term, we found very good agreement between the age-structured model and the biology. In particular, we analyzed the influence of a refractory (insensitivity) period following a reversal of movement. Our analysis reveals that the refractory period is not necessary for wave formation, but essential to wave synchronization, indicating separate molecular mechanisms.