基于Fisher信息论的异常扩散的简单数学模型
A simple mathematical model for anomalous diffusion via Fisher's information theory
摘要：
从基于先前提出的熵函数$S_q[p]=\int dx p(x)(-\ln p(x))^q$的相对熵开始，我们找到了相应的Fisher信息测度。函数重新定义后，我们最大化的Fisher信息测量的新功能，并获得一个微分算子，减少到空间坐标二阶导数的$q\to 1$限制。然后，我们提出了一个简单的微分方程的反常扩散，并表明其解决方案是一个推广的功能在巴伦布拉特-帕特尔解决方案。我们发现，均方位移，直到$q$依赖常数，具有根据$<x^2>\sim K^{1/q}t^{1/q}$的时间依赖性，其中参数$q$取值$q=\frac{2n-1}{2n+1}$（超扩散）和$q=\frac{2n+1}{2n-1}$（亚扩散），$\forall n\geq 1$。
英文摘要：
Starting with the relative entropy based on a previously proposed entropy function $S_q[p]=\int dx p(x)(-\ln p(x))^q$, we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the $q\to 1$ limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization of the functions in the Barenblatt-Pattle solution. We find that the mean squared displacement, up to a $q$-dependent constant, has a time dependence according to $<x^2>\sim K^{1/q}t^{1/q}$, where the parameter $q$ takes values $q=\frac{2n-1}{2n+1}$ (superdiffusion) and $q=\frac{2n+1}{2n-1}$ (subdiffusion), $\forall n\geq 1$.
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