Angstmann C N, Donnelly I C, Henry B I, Langlands T A M
School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022811. doi: 10.1103/PhysRevE.88.022811. Epub 2013 Aug 20.
We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.
我们研究了在网络顶点上作随机游走的粒子的输运情况，这些粒子受到与顶点和时间相关的驱动力作用。我们使用连续时间随机游走推导了这种输运的广义主方程，该随机游走以跳跃和等待时间密度为特征，作为基础随机过程。驱动力通过控制随机游走粒子的跳跃密度中与顶点和时间相关的偏差来纳入。作为一个特殊情况，我们考虑粒子驱动力与相邻顶点上粒子的浓度成正比，类似于空间连续体中的自趋化吸引。我们对该系统的代数和数值研究揭示了一种有趣的对聚集模式形成，其中稳态由少量孤立相邻顶点对处的高浓度粒子组成。如果在顶点上的输运是随机的，即没有驱动力，稳态就不会表现出这种对聚集。因此，输运网络上对聚集的表现可能是类似自趋化驱动力的一个特征。
