Mass Transport Models on Networks

对一维和二维系统中物质输运过程的研究导致了对统计系统失衡的理解的相当大的进展，对于这些统计系统，通常缺少指导原则和分类方案。通常这些研究集中在简单的规则拓扑。另一方面，自然系统中也存在着各种各样的失平衡过程，这些失平衡过程目前被概括为复杂网络。对复杂网络动力学过程的分析还处于起步阶段。一类重要的过程是质量运输（质量在一般意义上的粒子流，能量，信息或交通），我们将实现无处不在的噪音，固有的任何运输，通过随机描述。我们将专注于非平凡拓扑上的零程过程的推广，这可以作为重正化群精神中更大网络的构建块，但仍然是分析上易于处理的。对于网络动力学领域，这应该提供一个分析的理解，以及嵌入到一个更大的背景下，并在网络模拟中观察到的静态分类的步骤。从统计物理学的角度出发，我们的项目旨在推广到更现实的几何形状，真实的过程，如细胞内的蛋白质运输，和更现实的（例如开放）边界条件，非零范围的过程与短期和长期的相互作用和反馈，波动定理的扩展和瞬态分析的方法的稳态。我们两个小组在质量输运和数值模拟方面的专业知识（W。Janke）和网络动力学和分析计算（H。Meyer-Ortmanns）将相互补充，并在合并时带来富有成效的合作。

The study of mass transport processes in one-and two dimensional systems led to considerable progress in the understanding of statistical systems out-of-equilibrium, for which guiding principles and classification schemes are missing in general. Usually these studies focus on simple regular topologies. On the other hand, a variety of out-of-equilibrium processes is going on in natural systems, which are nowadays summarized under complex networks. The analysis of dynamical processes on complex networks is just in their infancy. An important class of processes is mass transport ( mass in a generic sense of flow of particles, energy, information or traffic) for which we shall implement the ubiquitous noise, inherent in any transport, via a stochastic description. We shall focus on generalizations of the zero-range process on non-trivial topologies, which may serve as building blocks for larger networks in the spirit of the renormalization group, but remain analytically tractable. For the field of network dynamics this should provide an analytic understanding as well as steps towards an embedding into a larger context and a classification of stationary states observed in network simulations. From the perspective of statistical physics out-of-equilibrium, our project aims at generalizations towards more realistic geometries, real processes such as protein traffic inside cells, and more realistic (e.g. open) boundary conditions, non-zero range processes with short- and long-range interactions and feedbacks, extensions of fluctuation theorems and analysis of transients in the approach of stationary states. The expertise of our two groups in mass transport and numerical simulations (W. Janke) and network dynamics and analytical calculations (H. Meyer-Ortmanns) will supplement each other and lead to a productive collaboration when it is combined.