Minimal path for transport in networks
- 1 May 1993
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review B
- Vol. 47 (18), 11815-11825
- https://doi.org/10.1103/physrevb.47.11815
Abstract
This report examines transport through networks in which transport across each bond in the network requires exceeding a microscopic threshold potential Δ. In particular, we examine the macroscopic gradient ∇ at which transport begins, as a function of the distribution of microscopic thresholds Δ. 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 ∇ for a Bethe- (or Cayley-) tree network. One simple model, based on taking the average of Δ among the percolation-threshold fraction of low-Δ 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 just above the value at the percolation threshold. Evidently the minimal path occasionally passes through these high-Δ bonds in order to access large clusters of low-Δ bonds.
Keywords
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