Minimal path for transport in networks

Abstract

This report examines transport through networks in which transport across each bond in the network requires exceeding a microscopic threshold potential \ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$. In particular, we examine the macroscopic gradient \ensuremath{\nabla}${\mathit{V}}^{\mathrm{min}}$ at which transport begins, as a function of the distribution of microscopic thresholds \ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$. Applications of this ``minimal path'' or ``breakdown'' problem include electrical conduction through networks of diodes and the flow of Bingham plastics through porous media. Two simple models are examined, including a solution for \ensuremath{\nabla}${\mathit{V}}^{\mathrm{min}}$ for a Bethe- (or Cayley-) tree network. One simple model, based on taking the average of \ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$ among the percolation-threshold fraction of low-\ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$ bonds, agrees remarkably well with both the Bethe-tree results and with the Monte Carlo studies for square and cubic networks. However, the Bethe-tree model shows that the minimal path samples the low end of this fraction most heavily. Doing so, it is aided by the existence of bonds with \ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$ just above the value at the percolation threshold. Evidently the minimal path occasionally passes through these high-\ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$ bonds in order to access large clusters of low-\ensuremath{\Delta}${\mathit{V}}_{\mathit{i}}^{\mathrm{min}}$ bonds.