Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
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DOI：
10.1140/epjb/e2008-00142-9
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发表时间：
2008-03-01
影响因子：
1.6
通讯作者：
Chavanis, P. H.
中科院分区：
文献类型：
--
作者：
Chavanis, P. H.
研究了一类一般的非线性平均场Fokker-Planck方程与一个有效的广义热力学(E.G.T.)形式主义。这些方程描述了几个物理系统，如细菌种群的趋化性，正则系综中的玻色-爱因斯坦凝聚，多孔介质，广义Cahn-Hilliard方程，Kuramoto模型，BMF模型，Burgers方程，自引力布朗粒子的Smoluchowski-Poisson系统，德拜-Hackel电解质理论，二维湍流...特别地，我们证明了非线性平均场Fokker-Planck方程可以为生物种群的趋化性提供广义的Keller-Segel模型。作为一个例子，我们引入了一个新的趋化性模型，该模型同时考虑了反常扩散和排斥原理(体积填充)的影响。因此，广义热力学的概念可以应用于具体的物理系统。我们还考虑了相空间中的非线性平均场Fokker-Planck方程，并证明了在强摩擦极限下从广义Kramers方程到广义Smoluchowski方程的转换。我们的形式很简单，并用几个明显的例子来说明，这些例子对应于玻尔兹曼、Tsallis、费米-狄拉克和玻色-爱因斯坦等熵。
We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hackel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.