[统计数据] 爱因斯坦主方程的非线性扩散 [推广有奖]

摘要翻译：
我们推广了Einstein随机游动过程的主方程，认为在$r$位置的粒子跳跃长度$j$晶格位的概率$p_j(r)$是粒子分布函数$f(r，t)$的泛函。通过多尺度展开，我们得到了一个广义的平流扩散方程。从广义方程允许标度解($f(r；t)=t^{-\gamma}\phi(r/t^{\gamma})$)的要求出发，我们证明了幂律$P_j(r)\propto f(r)^{\alpha-1}$($\alpha>1$)。所得解具有$q$-指数形式，与Monte-Carlo模拟结果一致，为验证非线性扩散方程提供了微观基础。虽然其流体力学极限与唯象多孔介质方程等价，但如蒙特卡罗计算所证明的那样，在一般情况下，有一些额外的项是不能忽略的
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英文标题：
《Nonlinear diffusion from Einstein's master equation》
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作者：
J.P. Boon and J.F. Lutsko
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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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 $P_j(r) \propto f(r)^{\alpha - 1}$ (with $\alpha > 1$) follows from the requirement that the generalized equation admits of scaling solutions ($ f(r;t) = t^{-\gamma}\phi (r/t^{\gamma}) $). 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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PDF链接：
https://arxiv.org/pdf/709.1194