动力学网络介导的粒子相互作用的动力学理论
Kinetic theory of particle interactions mediated by dynamical networks
摘要：
我们提供了一个详细的多尺度分析系统的粒子通过动态网络的链接相互作用。从微观模型开始，通过平均场极限，我们按照[Degond等人，M3AS，2016]。假设网络重构的过程非常快，我们简化描述为一个宏观模型的形式为单一的聚集-扩散方程的颗粒密度。我们定性分析这个方程，解决稳定性的均匀分布的粒子的一般潜力。对于胡克潜在的，我们得到了一个精确的条件，相变，并使用中心流形减少，我们的特点是在不稳定的发病类型的分歧。
英文摘要：
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 [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of 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.
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