Minimal path for transport in networks

This report examines transport through networks in which transport across each bond in the network requires exceeding a microscopic threshold potential {Δ}Vᵢᵐⁱⁿ. In particular, we examine the macroscopic gradient {∇}Vᵐⁱⁿ at which transport begins, as a function of the distribution of microscopic thresholds {Δ}Vᵢᵐⁱⁿ. 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 {∇}Vᵐⁱⁿ for a Bethe- (or Cayley-) tree network. One simple model, based on taking the average of {Δ}Vᵢᵐⁱⁿ among the percolation-threshold fraction of low-{Δ}Vᵢᵐⁱⁿ 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}Vᵢᵐⁱⁿ just above the value at the percolation threshold. Evidently the minimal path occasionally passes through these high-{Δ}Vᵢᵐⁱⁿ bonds in order to access large clusters of low-{Δ}Vᵢᵐⁱⁿ bonds.