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 Δ${\mathit{V}}_{\mathit{i}}^{\min }$. In particular, we examine the macroscopic gradient ∇${\mathit{V}}^{\min }$ at which transport begins, as a function of the distribution of microscopic thresholds Δ${\mathit{V}}_{\mathit{i}}^{\min }$. 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 ∇${\mathit{V}}^{\min }$ for a Bethe- (or Cayley-) tree network. One simple model, based on taking the average of Δ${\mathit{V}}_{\mathit{i}}^{\min }$ among the percolation-threshold fraction of low-Δ${\mathit{V}}_{\mathit{i}}^{\min }$ 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 Δ${\mathit{V}}_{\mathit{i}}^{\min }$ just above the value at the percolation threshold. Evidently the minimal path occasionally passes through these high-Δ${\mathit{V}}_{\mathit{i}}^{\min }$ bonds in order to access large clusters of low-Δ${\mathit{V}}_{\mathit{i}}^{\min }$ bonds.