Formation of clumps and patches in self-aggregation of finite-size particles

Abstract

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Hückel [1] and Keller-Segel [2] models are: a) the mobility µ of particles depends on the locally-averaged particle density and b) linear diffusion acts on that locally-averaged particle density. The cases both with and without diffusion are considered here. Surprisingly, these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states. When 1 µ remains positive, the evolution of collapsed states in our model reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerging from smooth initial conditions, even in one spatial dimension. If µ vanishes for some averaged density, the evolution leads to spontaneous formation of jammed patches (weak solution with density having compact support). Simulations confirm that a combination of these patches forms the final state for the system.

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

“…The purely gradient flow case has been studied for self-interacting individuals via pairwise potentials arising in the modelling of animal collective behavior: flocks, schools or swarms formed by insects, fishes and birds. The simplest models based on ODEs systems [15,24,29,43,44] led to continuum descriptions [19,18,14,37,42,47,48] for the evolution of densities of individuals. It is this class of models that we focus on here, although we will draw parallels to well-known problems and results from the incompressible flow literature.…”

Emergent singular solutions of nonlocal density-magnetization equations in one dimension

“…In a series of papers, Holm, Putkaradze and Tronci [6,7,8,9,10,11] have focused on the derivation of aggregation equations that possess emergent singular solutions. Continuum aggregation equations have been used to model gravitational collapse and the subsequent emergence of stars [12], the localization of biological populations [13,14,15], and the self-assembly of nanoparticles [16].…”

Swarming on Random Graphs II

“…If np (1 + ✏)c 0 () log n then lemma 3.3 su ces to guarantee that each diagonal entry d ii of the diagonal component (20) satisfies d ii  c 1 Np with probability at least 1 c 0 n (1+✏/2) . As a consequence, the union bound implies that there exists c 1 > 0 so that…”

“…Classical examples from physics and chemistry include the distribution of electrons in the Thomson problem [44,34,2,50,8,9,25] and VSEPR theory. More modern applications in these areas include protein folding [33,42], colloid stability [46,47,22] and the self-assembly of nanoparticles into supramolecular structures [17,20,16,53]. In biology, similar mathematical models help explain the complex phenomena observed in flocking [29,10,48], viral capsids [18,52], locust swarms [12,5] and colonies of bacteria [45,11] or ants [4].…”