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

New model equations are derived for the dynamics of self-aggregation of finite-size particles. Differences from standard Debye–Hückel [P. Debye, E. Hückel, Zur Theorie der Elektrolyte: (2): Das Grenzgesetz für die Elektrische Leiftfahrigkeit (On the theory of electrolytes 2: limiting law of electrical conductivity), Phys. Z. 24 (1923) 305–325] and Keller–Segel [E.F. Keller, L.A. Segel, J. Theoret. Biol. 26 (1970) 399–415; E.F. Keller, L.A. Segel, J. Theoret. Biol. 30 (1971) 225–234] 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 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 the spontaneous formation of jammed patches (a weak solution with the density having compact support). Simulations confirm that a combination of these patches forms the final state for the system.