论文标题
分数布朗运动的概率密度和带有吸收壁的分数langevin方程
Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls
论文作者
论文摘要
分数布朗运动和分数Langevin方程是以远程幂律相关为特征的异常扩散过程的模型。我们采用大规模的计算机模拟在两个几何形状中研究这些模型，（i）颗粒在半无限结构域上的扩散，一端具有吸收壁，（ii）（ii）固定状态在有限的间隔内，两端具有吸收性边界和中心的源。我们证明，如果异常扩散指数$α$替换为$ 2-α$，则可以将分数langevin方程的概率密度和其他特性映射到相同噪声的相应量的分数。相反，最近显示出具有反射边界的分数布朗运动的特性和分数朗格文方程在定性上彼此不同。具体而言，我们发现接近吸收墙的概率密度的行为为$ p（x）\ sim x^κ$，距离$ x $的距离长期限制。在分数布朗运动的情况下，$κ$随着以前提出的异常扩散指数$α$ $α$ $α$而变化。我们还将模拟结果与分数布朗运动的扰动分析方法进行了比较。
Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding quantities of fractional Brownian motion driven by the same noise if the anomalous diffusion exponent $α$ is replaced by $2-α$. In contrast, the properties of fractional Brownian motion and the fractional Langevin equation with reflecting boundaries were recently shown to differ from each other qualitatively. Specifically, we find that the probability density close to an absorbing wall behaves as $P(x) \sim x^κ$ with the distance $x$ from the wall in the long-time limit. In the case of fractional Brownian motion, $κ$ varies with the anomalous diffusion exponent $α$ as $κ=2/α-1$, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.
