论文标题
布朗尼运动的卑鄙平方绕组角度在坚不可摧的圆柱体周围
Mean square winding angle of Brownian motion around an impenetrable cylinder
论文作者
论文摘要
对于均值的布朗运动（即扩散），在时间t之后的均方根缠绕角度得出了一个精确的公式，它是无限长的半径A的无限长圆柱，从轴上从半径r（> a）开始。令人惊讶的是，对于a = 0的更简单的问题，围绕直线的均方根绕组角度却是立即无限的，但是起点位置很远。数学布朗运动的分裂小，快速，随机步行步骤允许在直线的零厚度周围无绑的绕组。如下所述，如果需要的话，是一种补救措施，是为了符合该线非零厚度，即难以穿透的圆柱体。该问题立即减少到平面中的圆盘上的2D，因为3D Brownian运动的轴向分量与其他运动无关。得出确切的均方根绕组角后，在狭窄的气缸a << r的极限中评估积分，突出了以前近似处理解决的短和长扩散时间的极限。
An exact formula is derived, as an integral, for the mean square winding angle of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius a, having started at radius R(>a) from the axis. Strikingly, for the simpler problem with a=0, the mean square winding angle around a straight line, is long known to be instantly infinite however far away the starting point lies. the fractally small, fast, random walk steps of mathematical Brownian motion allow unbounded windings around the zero thickness of the straight line. A remedy if it is required, is to accord the line non-zero thickness, an impenetrable cylinder, as analysed here. The problem straight away reduces to a 2D one of winding around a disc in a plane since the axial component of the 3D Brownian motion is independent of the others. After deriving the exact mean square winding angle, the integral is evaluated in the limit of a narrow cylinder a<<R, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.
