Geometric universality and anomalous diffusion in frictional fingers

Abstract

Frictional finger trees are patterns emerging from non-equilibrium processes in particle-fluid systems. Their formation share several properties with growth algorithms for minimum spanning trees (MSTs) in random energy landscapes. We propose that the frictional finger trees are indeed in the same geometric universality class as the MSTs, which is checked using updated numerical simulation algorithms for frictional fingers. We also propose a theoretical model for anomalous diffusion in these patterns, and discuss the role of diffusion as a tool to classify geometry.

Diffusion Entropy and the Path Dimension of Frictional Finger Patterns

“…The path dimension can also be estimated by treating the pattern as a tree structure and using branching statistics, similar to the study of river networks. This method in stead gives d m = 1.20 ± 0.03 [16]. While both of these measurements are consistent with the MST class, they are inconsistent with each other.…”

“…In the case of diffusion in frictional finger patterns we chose the diffusion law associated with the Hänggi-Klimontovich convention, where no drift term associated with diffusivity gradients are present. This choice of diffusion law together with the Einstein relation was recently used to identify the form of the spatially dependent diffusivity for transport in the frictional fingers, where under an isotropy assumption one has D(r) = D 0 r −ξ [16]. As is the case for perfectly hierarchical fractals, the exponent ξ is related to the fractal dimensions of the pattern as…”

“…The deformation takes place where the energy needed to move the boundary is the smallest. A simulated version of the pattern is shown in Figure 1A, based on the algorithms discussed in Olsen et al [16]. It was recently hypothesized by the authors that since the frictional finger patterns are formed through a optimal pathfinding process it may belong to the geometric universality class of minimum spanning trees (MST) [16].…”

“…A simulated version of the pattern is shown in Figure 1A, based on the algorithms discussed in Olsen et al [16]. It was recently hypothesized by the authors that since the frictional finger patterns are formed through a optimal pathfinding process it may belong to the geometric universality class of minimum spanning trees (MST) [16]. These are treestructures constructed on a lattice by assigning a random energy to every lattice link and finding the loopless configuration of links that globally minimizes the total system energy [17].…”

“…In particular, the path dimension can be estimated locally by fixing a sample point in the system and considering the average length ℓ of paths out to a radius of Euclidean length r. To obtain a global "coarse-grained" estimate d m for the path dimension of the system this local path dimension should be averaged over many sample points. This gives a path dimension of d m = 1.25 ± 0.03 [16]. The path dimension can also be estimated by treating the pattern as a tree structure and using branching statistics, similar to the study of river networks.…”