从一个简单的动物生物学的例子，一般的，有点违反直觉的性质的扩散随机游动。结果表明，对于任意（非均匀）纯扩散系统，在任意各向同性均匀入射下，通过系统的平均轨迹长度（即从进入点到第一个离开点的随机游走轨迹的平均长度）与扩散过程的特性无关，因而只取决于系统的几何形状.这种精确的不变性性质可以被看作是对众所周知的平均弦长性质[1]的扩散的推广，从而导致广泛的物理学和生物学应用。让我们首先考虑一个实际的动物生物学例子，它是下文所报告的理论工作的起源。对于不同的物种来说，蚂蚁等昆虫在水平平面上的自发位移可以精确地建模为恒速扩散随机行走[2-4]。我们假设在表面上画了一个半径为R的圆，我们测量蚂蚁进入圆内时在圆内花费的平均时间（见图1）。最后一个假设是，测量是在蚂蚁落在表面上很长一段时间后进行的，因此它们的初始位置的记忆丢失了：这足以确保没有特定的方向是有利的，因此当蚂蚁遇到圆时，它们的入射方向是各向同性分布的。简单的蒙特卡罗模拟（见图2）毫无疑问地表明，对于一个固定的速度，平均相遇时间（从进入圆到第一次离开的时间）只取决于圆的半径。它独立于扩散游走的特性：平均自由程$λ$（两个散射事件之间的平均距离，即两个方向变化之间的平均距离）和单次散射相函数p（us，u i）（入射方向u i的散射方向us的概率密度函数）。此外，该平均时间按R/v缩放，这意味着平均轨迹长度< L >按R缩放。因此，对于任何昆虫物种的不同实验，或者对于任何类型的扩散微粒运动，平均扩散长度都是相同的。有两个原因可以解释为什么这一观察结果最初听起来是违反直觉的。第一个原因是，已知两点之间的扩散轨迹的平均长度近似为d 2 / $λ$，其中d是两点之间的距离（1）。对于较短的平均值（1）这种缩放对于极限d >> $λ$ [5] c EDP Sciences中的无限域是精确的
英文摘要：
Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence, the average length of trajectories through the system (the average length of the random walk trajecto-ries from entry point to first exit point) is independent of the characteristics of the diffusion process and therefore depends only on the geometry of the system. This exact invariance property may be seen as a generalization to diffusion of the well known mean-chord-length property [1], leading to broad physics and biology applications. Let us first consider a practical animal-biology example that was at the origin of the theoretical work reported hereafter. It is well established, for diverse species, that the spontaneous displacement of insects such as ants, on an horizontal planar surface, may be accurately mod-eled as a constant-speed diffusion random-walk [2-4]. We assume that a circle of radius R is drawn on the surface and we measure the average time that ants spend inside the circle when they enter it (see Fig. 1). The last assumption is that measurements are performed a long time after ants were dropped on the surface, so that the memory of their initial position is lost : this is enough to ensure that no specific direction is favored and therefore that when ants encounter the circle, their incident directions are distributed isotropically. Simple Monte Carlo simulations of such experiments (see Fig. 2) indicate without any doubt that the average encounter time (time between entry into the circle and first exit), for a fixed velocity, depends only of the circle radius. It is independent of the characteristics of the diffusion walk : the mean free path $λ$ (average distance between two scattering events, i.e. between two direction-changes), and the single-scattering phase function p(u s , u i) (probability density function of the scattering direction u s for an incident direction u i). Furthermore, this average time scales as R/v, which means that the average trajectory-length < L > scales as R. The average trajectory-length would therefore be the same for different experiments with any insect species-or for any type of diffusive corpuscular motion. There are two reasons why this observation may initially sound counter-intuitive. The first reason is that the average length of diffusion trajectories between two points is known to scale approximately as d 2 /$λ$ where d is the distance between the two points (1). For shorter mean (1)This scaling would be exact for an infinite domain in the limit d >> $λ$ [5] c EDP Sciences
A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables
研究了记忆诱导扩散-超扩散跃迁的系综性质和时均观测量。该模型包括在一个随机步行者的过渡在一个给定的方向上取决于一个加权的线性组合的数量的权利和左以前的过渡。扩散过程是非平稳的，它的概率发展的老化现象。根据特征存储器参数，系综行为可以是正常的，超扩散的，或弹道。相反，时间平均的均方位移等于正常的无驱动随机游动，这使得过程非遍历。此外，与Levy类似，步行[Godec和Metzler，Phys. Rev. Lett. 110，020603（2013）]，对于有限持续时间的轨迹，时间平均位移显然变得随机，具有取决于测量时间并且还取决于记忆特性的特性。这些特征与跃迁概率向其平稳值的非平稳幂律衰减有关。还计算了对偏差的时间平均响应。与Levy Walks [Froemberg and Barkai，Phys.Rev.E87，030104（R）（2013）]相比，响应总是渐近地消失。
