Lutsko James F, Boon Jean Pierre
Physics Department, Code Postal 231, Université Libre de Bruxelles, 1050 Bruxelles, Belgium.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022108. doi: 10.1103/PhysRevE.88.022108. Epub 2013 Aug 7.
We present a master equation formulation based on a Markovian random walk model that exhibits subdiffusion, classical diffusion, and superdiffusion as a function of a single parameter. The nonclassical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean squared displacement [<r(2)(t)>~t(γ) with 0<γ≤1.5] but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and superdiffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the master equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.
我们提出了一种基于马尔可夫随机游走模型的主方程公式，该模型根据单个参数展现出亚扩散、经典扩散和超扩散现象。非经典扩散行为是通过考虑一群游走者之间的相互作用而产生的。在宏观层面，这会产生一个非线性福克 - 普朗克方程。扩散行为不仅体现在均方位移[<r(2)(t)>~t(γ)，其中0<γ≤1.5]中，还体现在福克 - 普朗克方程自相似标度解的存在上。我们根据扩散粒子之间的吸引和排斥相互作用对亚扩散和超扩散进行了物理解释，并分析讨论了指数γ的极限值。基于主方程的模拟结果在所有三种扩散区域都与非线性福克 - 普朗克方程的解析解一致。
