长记忆构成了与非广延统计力学一致的统一的介观机制
Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
摘要：
我们统一了两个范例介观机制的出现非广泛的统计，即乘性噪声机制导致{ \it线性}福克-普朗克（FP）方程与{ \it非均匀}扩散系数，和非马尔可夫过程导致{ \it非线性} FP方程与{\it均匀}扩散系数。更准确地说，我们考虑方程$\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [ϕ(x,p)p(x,t)]$，其中$D \in {\cal R}$和$F(x)=-\partial V(x) /\partial x$，$V(x)$是扩散发生时的电势。我们的目的是找到是否存在$ϕ(x,p)$，使非齐次线性和齐次非线性FP方程成为统一的方式，（普遍观察到的）$q$指数保持为固定的解决方案。事实证明，这样的解决方案确实存在的广泛的系统，即当$ϕ(x,p)=[A+BV(x)]^θ[p(x,t)]^η$，其中$A$，$B$，$θ$和$η$是（实）常数。我们的主要结果可以概括如下：对于$θ\neq 1$和任意限制势$V(x)$，$p(x,\infty) \propto \lbrace 1-β(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-βV(x)}$，其中$q= 1+ η/(θ-1)$ .本方法统一成一个单一的机制，基本上{ \it长记忆}，目前在文献中讨论和应用的结果。
英文摘要：
We unify two paradigmatic mesoscopic mechanisms for the emergence of nonextensive statistics, namely the multiplicative noise mechanism leading to a {\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion coefficient, and the non-Markovian process leading to the {\it nonlinear} FP equation with {\it homogeneous} diffusion coefficient. More precisely, we consider the equation $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [ϕ(x,p)p(x,t)]$, where $D \in {\cal R}$ and $F(x)=-\partial V(x) /\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim is to find whether $ϕ(x,p)$ exists such that the inhomogeneous linear and the homogeneous nonlinear FP equations become unified in such a way that the (ubiquitously observed) $q$-exponentials remain as stationary solutions. It turns out that such solutions indeed exist for a wide class of systems, namely when $ϕ(x,p)=[A+BV(x)]^θ[p(x,t)]^η$, where $A$, $B$, $θ$ and $η$ are (real) constants. Our main result can be sumarized as follows: For $θ\neq 1$ and arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace 1-β(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-βV(x)}$, where $q= 1+ η/(θ-1)$. The present approach unifies into a single mechanism, essentially {\it long memory}, results currently discussed and applied in the literature.
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