Network Algorithms and Critical Manifolds in Disordered Systems

摘要

We have briefly discussed some of the basic network flow algorithms: Prim's algorithm, Dijkstra's algorithm, the maximum flow algorithms and convex flow algorithms. These algorithms are all efficient and they find the exact solution to nontrivial manifold problems in large systems. These network algorithms provide a new set of tools for the numerical analysis of manifolds of interest in statistical physics. Three notabable successes are the calculation of RBIM interfaces in three dimensions, the inclusion of overhangs in the configuration of a flux line in a disordered medium; and the efficient treatment of muliple flux line pinning problems. Convex nonlinear random resistor networks are a broad class of problems which are accessible to large scale numerical study. We discussed the fact that the combination of nonlinearity and disorder leads to the emergence of critical manifolds in nonlinear resistor networks. Two illustrative examples are: at the critical voltage of varistors current localises on the shortest path; while at the critical current of disordered superconductors, voltage localises on a minimum cut. Finally we showed that the numerical methods used to study models of interest in statistical physics can also be used to analyse problems of interest in material's theory. This was illustrated for the case of polycrystalline materials, where a preliminary result concerning the crossover from intergranular to transgranular character of a critical manifold was presented.