| An important problem is the construction of theories relating geometry of random structures with flow properties in those structures, because the results have implications in oil recovery, metabolic network research and polymer physics. For this reason, I consider three kinds of random structures and elucidate their flow properties.; Firstly, to improve oil production forecasting, I study traveling-tracer times ttr between sites A and B (injection and extraction) separated by distance r in percolation systems of linear size L through their probability density function (pdf) P(ttr| r, L). I find: (a) a most probable traveling time t*tr satisfying a power law with r, and (b) two power law decay regions for P(ttr| r, L), one for intermediate ttr and another for large ttr with multifractal properties. I explain these results through geometric exponents of percolation.; Secondly, I examine the pdf P(ℓopt| r, L) of optimal path lengths ℓopt between A and B when lattice sites are assigned weights of an exponentially broad distribution. This problem relates to polymer behavior in random potentials. I find a power law decay for P(ℓ opt|r, L), and determine the scaling form of P(ℓopt|r, L). Since optimal paths can jump across so called 'percolation regions', P(ℓ opt|r, L) differs from P(ℓ opt|r, L), the pdf of traveling-tracer lengths ℓ tr of the first model. However, by constraining optimal paths to remain inside such percolation regions, the two problems exhibit similar scaling.; Thirdly, I analyze pdf phiSF(G) of conductance G between two arbitrary nodes of random scale-free networks with degree distribution P(k) ∼ k-lambda or Erdo&huml;s-Renyi networks, where in both networks the links have unit resistance. I predict a power-law distribution phiSF(G) ∼ G 1-2lambda and confirm my predictions by simulations. The power-law tail in phiSF(G) leads to large values of G, improving transport in scale-free networks compared to Erdo&huml;s-Renyi networks. Based on a simple physical picture that I call the transport backbone picture, I show that the conductances are ckAkB/(kA + kB) for any pair of nodes A and B with degrees kA and k B. Thus, a single parameter c characterizes transport on complex networks.; In summary, this work presents new ideas that illustrate the connection between flow and geometry in random structures. |
