arXiv 0709.11942007-09-08cond-mat.stat-mech

爱因斯坦主方程的非线性扩散

Nonlinear diffusion from Einstein's master equation

J. P. Boon, J. F. Lutsko

备注：: 7 pages incl. 3 figs
DOI：: 10.1209/0295-5075/80/60006

摘要：

我们推广了Einstein的无规行走过程主方程，考虑到粒子在位置$r$上跳跃长度为$j$格点的概率，$P_j(r)$是粒子分布函数$f(r,t)$的泛函.通过多尺度展开，得到了一个广义对流扩散方程。我们表明，幂律$P_j(r) \propto f(r)^{α- 1}$（与$α> 1$）遵循的要求，广义方程的缩放解决方案（$ f(r;t) = t^{-γ}ϕ(r/t^γ) $）。解具有$q$ -指数形式，并与蒙特卡罗模拟的结果一致，从而为验证非线性扩散方程提供了微观基础.虽然它的流体动力学极限等价于唯象多孔介质方程，但有一些额外的项，一般来说，不可忽略，如Monte-Carlo计算所证明的。

英文摘要：

We generalize Einstein's master equation for random walk processes by considering that the probability for a particle at position $r$ to make a jump of length $j$ lattice sites, $P_j(r)$ is a functional of the particle distribution function $f(r,t)$. By multiscale expansion, we obtain a generalized advection-diffusion equation. We show that the power law $P_j(r) \propto f(r)^{α- 1}$ (with $α> 1$) follows from the requirement that the generalized equation admits of scaling solutions ($ f(r;t) = t^{-γ}ϕ(r/t^γ) $). The solutions have a $q$-exponential form and are found to be in agreement with the results of Monte-Carlo simulations, so providing a microscopic basis validating the nonlinear diffusion equation. Although its hydrodynamic limit is equivalent to the phenomenological porous media equation, there are extra terms which, in general, cannot be neglected as evidenced by the Monte-Carlo computations.}

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