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Edited by: Giancarlo Ruocco, Center for Life NanoScience (IIT), Italy

Reviewed by: Matteo Colangeli, University of L'Aquila, Italy; Constantinos Simserides, National and Kapodistrian University of Athens, Greece

This article was submitted to Condensed Matter Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The authors investigate, using both analytical and numerical methods, the entropy associated with a diffusion process inside frictional finger patterns. The entropy obtained from simulations of diffusion inside the pattern is compared to analytical predictions based on an effective continuum description. The analytical result predicts that the entropy depends in a particular way on the path dimension of the system, which governs the scaling of simple paths in the system. The findings indicates that there is a close analogy between the frictional fingers in the continuum and minimum spaning trees on the lattice, as the path dimension is found, through studies of the entropy, to be close to the defining value for the minimum spanning tree universality class.

Patterns with complex geometry and topology are ubiquitous in Nature. When transport processes take place inside such patterns, their dynamical properties are typically anomalous due to the non-trivial geometry. The case of anomalous diffusive transport, where the mean-square displacement scales non-linearly with time, has been studied in some detail since the 1980s and remains a popular topic to this day [

Fricitonal patterns are space-filling bifurcating two-dimensional geometries that arise due to instabilities in frictional fluids. Experimentally, the frictional finger patterns are produced by preparing a mixture of glass beads and liquid in a Hele-Shaw cell before pumping out of liquid from the center of the cell. When the cell has open ends, this forces air into the glass bead / liquid mixture resulting in a deformation of the boundary of the mixture [

Figure showing a frictional finger pattern

It was recently hypothesized by the authors that since the frictional finger patterns are formed through a optimal path-finding process it may belong to the geometric universality class of minimum spanning trees (MST)[_{f} or minimum path dimension _{m}, the latter being the scaling exponent connecting Euclidean and intrinsic distance measures. Since on loopless structures there is an unique path connecting any two points we will simply refer to _{m} as the path dimension. The 2D MST class has _{f} = 2 and _{m} = 1.22 ± 0.01 [

The rest of this paper is outlined as follows. Section 2 discusses the diffusion entropy associated with the effective continuum description, which is based on a simple power-law scaling of the diffusivity. The entropy associated with the corresponding Fokker-Planck equation is calculated analytically and compared to results obtained using fractional diffusion equations. Section 3 discusses a new numerical implementation of random walkers in frictional finger patterns that is expected to increase both efficiency and accuracy. The entropy is calculated, and compared to analytical predictions. Finding the best fit of the analytical prediction as system parameters are varied gives a value for the path dimension. Concluding remarks are offered in section 4.

To study the diffusion process in the frictional finger patterns on a large length scale we use state-dependent diffusion equations where the diffusivity can depend of the particle position. Microscopically, Brownian particles move throughout the pattern with a constant diffusivity. However, the collisions with the walls of the confining geometry affects the macroscopic transport properties. We imagine that after a sort of coarse-graining or homogenization procedure the diffusion process can be described by an overdamped Langevin equation of the form

where

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 _{f} − 2 + _{m} = _{m}, where we used the space-filling property of the finger pattern [

The corresponding density evolution equation in the Hänggi-Klimontovich interpretation takes the following form

This generalized Ficks equation has a well-known solution for radial power-law diffusivity, taking the form [

where _{S} = 4/(2 + ξ) is the spectral dimension and the normalization used is ∫ _{θ}ρ(_{m}, while it is known that in noisy real systems this dimension can vary locally. Again we expect that on large space and time scales the inhomogeneities average out, producing a single global path dimension

Given the above solution Equation (3) the entropy of the diffusion process can be calculated analytically. What type of entropy we consider is not of great importance here, as long as it is the same entropy that is calculated later in section 3 in the numerical methods. This is because at the end of the day, we are interested in using the numerical measurements of the entropy as an indirect measurement of the path dimension for the frictional fingers. From an information theoretic perspective there are dozens of entropies that could be considered, most of which can be thought of as an analytical continuation of the Shannon-Gibbs entropy which is recovered as some entropic parameter is tuned correctly [

The Shannon-Gibbs entropy for the particle density takes the form [

We will write Equation (3) in the form ρ(^{2+ξ}/

Since our distribution is a simple shifted Gaussian a change of variables easily allows us to find this moment. Using the integral

With μ = 3 + ξ and ν = 2 + ξ, the entropy can be calculated as

where we used

In Equation (7) we see that the global path dimension

where the diffusion exponent is given by

To calculate the numerical entropy we construct a simplified discrete random walk-based model for the diffusion process. To make these simulations more efficient, we make some simplifying assumptions. The biggest simplification comes from applying a topological contraction on the pattern so that the finger widths are set zero, effectively turning the problem into a one dimensional one. The resulting skeletonized version of the pattern, showed in

When performing the numerical simulations the one-dimensional skeletonized pattern is discretized before a discrete random walk process is released. In the resulting discrete "morphological graph" of the pattern there are no additional inhomogeneities associated with transition probabilities over links as all the inhomogeneity we are interested in stems from the pattern itself. In practice, the discretization is obtained by pixelating the skeletonized pattern and treating the pixels as sites for the random walker. A cartoon of the pixels are shown in

To calculate the entropy numerically we use the Gibbs-Shannon formula for the discrete random walk

where ρ_{i}(

The system is initialized with all particles released at the same position, as the analytical solution assumes a Dirac delta-like initial condition. _{0} is allowed to vary throughout the system. This may indicate that a more realistic diffusivity is _{0} is a slowly varying function taking into account small inhomogeneities in the pattern not captured by the simplified power-law model.

The entropy associated with three randomly chosen initial positions inside the finger pattern. The point were chosen inside a circle of ~50% pattern radius in order to avoid the outer boundaries of the pattern. The dotted line is a reference line 0.62 log_{10}(

Averaged entropy

In this paper we have studied the entropy of diffusion in frictional finger patterns. In addition to being a measure of how fast the non-equilibrium process is evolving, the entropy is also considered as a tool for studying the systems coarser geometry as the diffusing particles explore the large-scale structure at late times. Our results show that the (coarse) path dimension takes a value close to

The datasets generated for this study are available on request to the corresponding author.

KO performed analytical calculations and wrote the paper. JC developed numerical code crucial for the paper, analyzed the pattern, and aided in the writing process.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Eirik G. Flekkøy, Luiza Angheluta, and Signe Kjelstrup for insightful input and discussions during this work.