Kinetic and hydrodynamic models of chemotactic aggregation |
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Authors: | Pierre-Henri Chavanis Clément Sire |
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Institution: | Laboratoire de Physique Théorique (IRSAMC, CNRS), Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France |
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Abstract: | We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed. |
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Keywords: | Nonlinear mean-field Fokker-Planck equations Generalized thermodynamics Chemotaxis Gravity Long-range interactions |
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