网络拓扑上传输过程的位移
Displacement of transport processes on networked topologies
摘要：
考虑一个粒子，它的位置沿着网络的边缘演化。粒子位移的一个定义是粒子当前位置和初始位置之间网络上最短路径的长度。这样的定义未能包含粒子穿过的实际路径的信息。在这项工作中，我们考虑另一种定义的网络拓扑上的粒子的位移。使用这个定义，我们称之为缠绕距离，我们证明，布朗粒子，限制到网络可以诱导过渡的均方位移从扩散到弹道行为，$\langle x^2(t) \rangle \propto t^2$很长一段时间。一个多尺度的方法是用来推导出一个宏观的粒子的位移演化方程，并揭示了拓扑条件是否会发生这种过渡的均方位移。此外，对于满足这种拓扑条件的网络，我们确定了位移过渡到长时间行为的时间尺度的预测。最后，我们将网络上位移的研究扩展到一类具有扩散性质的输运过程，发现均方位移在长时间内受到网络拓扑结构和输运过程性质的影响。
英文摘要：
Consider a particle whose position evolves along the edges of a network. One definition for the displacement of a particle is the length of the shortest path on the network between the current and initial positions of the particle. Such a definition fails to incorporate information of the actual path the particle traversed. In this work we consider another definition for the displacement of a particle on networked topologies. Using this definition, which we term the winding distance, we demonstrate that for Brownian particles, confinement to a network can induce a transition in the mean squared displacement from diffusive to ballistic behaviour, $\langle x^2(t) \rangle \propto t^2$ for long times. A multiple scales approach is used to derive a macroscopic evolution equation for the displacement of a particle and uncover a topological condition for whether this transition in the mean squared displacement will occur. Furthermore, for networks satisfying this topological condition, we identify a prediction of the timescale upon which the displacement transitions to long-time behaviour. Finally, we extend the investigation of displacement on networks to a class of anomalously diffusive transport processes, where we find that the mean squared displacement at long times is affected by both network topology and the character of the transport process.
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