Submitted to: Physica B
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 7/20/2003
Publication Date: 10/20/2003
Citation: Skaggs, T.H. 2003. Effects of finite system-size and finite inhomogeneity on the conductivity of broadly distributed resistor networks. Physica B. 338:266-269.
Interpretive Summary: Although scientific progress over the last century has produced a reasonably complete theoretical explanation of the way water and chemicals move in soils and other porous media, certain aspects of our basic understanding are lacking. For example, chemicals often move faster and farther in soil than is predicted by conventional theory. This paper investigates a theory known as "critical path analysis" that may help explain the rapid movement of water and chemicals. Whereas conventional theory supposes that all pores are active in transmitting fluids and chemicals, critical path analysis hypothesizes that connected chains of larger pores create discrete pathways of low resistance and that fluid flow and chemical transport occur primarily on those low resistance pathways. This paper looks at one particular aspect of the critical path theory, namely the number of low resistance pathways that are expected exist over a given distance. Using numerical simulation, we find that the number of pathways is greater than the number predicted by previous theoretical arguments. This work will benefit scientists seeking to understand the basic physics of fluid flow and chemical transport in soils and other porous media, and could ultimately lead to better predictive tools for managing agricultural lands.
Technical Abstract: Monte Carlo simulation is used to investigate the critical path calculation of the conductivity of a random resistor network that has a logarithmically broad distribution of bond conductances. It has been argued that in three dimensions the conductivity prefactor exponent y is equal to the percolation correlation length exponent n, but past numerical computations have always found y<n. Finite-size effects are usually blamed but have never been documented. Our analysis of Monte Carlo data also finds y<n, but we show that the result is not due to finite-size effects. Instead, the observed y<n is due to the effects of finite inhomogeneity. The conductivity is controlled by critical conductors, but the distance between current carrying pathways is less than presumed in the theoretical arguments that lead to y=n. The shorter separation distance results in y<n.