论文标题
伞状梳子中的异常扩散
Anomalous diffusion in umbrella comb
论文作者
论文摘要
考虑在圆形梳子中进行异常运输。循环运动发生在固定半径上，而半径则沿圆连续分布。研究了与反射和周期性边界条件有关的异常运输的两种情况。圆形扩散的反射边界条件的第一种情况对应于圆形梳子上2D梳子fokker-Planck方程的共形映射。这种拓扑结构的运动称为伞形梳梳模型。在这种情况下，反射边界条件施加在圆形（旋转器）运动上，而径向运动对应于几何布朗尼运动，而无限上的零边界条件消失到零边界条件。径向扩散通过对数正态分布描述，该分布与$ e^t $的均方一平方位移（MSD）相对应。第二种情况对应于周期性边界条件的圆形扩散，而在无穷大处的零边界条件下，向外径向扩散。在这种情况下，径向运动对应于正常的扩散。在两种情况下，循环运动都是余弦函数的叠加，可导致固定的bernoulli多项式用于概率分布。
Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of $e^t$. The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.
