| Authors: |
A. Zoia
and Y. Kantor
and M. Kardar
|
| Editors: |
Luciano Pietronero
and Vittorio Loreto
and Stefano Zapperi
|
| URL: |
http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=258 |
| Tags: |
diffusion
evolution
first
fractures
lengths
loewner
passage
statphys23
stochastic
times
topic-3
|
| Abstract: |
Scale invariant curves, such as coastlines and fracture fronts, abound in nature. Fractal dimension provides the scaling of the length $\ell$ along the curve between two points to their actual separation $R$ in Euclidean space. However, for stochastic shapes, the actual length will fluctuate, and knowledge of its full distribution greatly augments the information about scaling. We numerically investigate such distributions for scale invariant (critical) curves in two dimensions.
While the actual quantity that we measure is the distribution of lengths along the fractal curves, an underlying motivation is to understand the distribution of first passage times for traversing such curves. To provide an example, experiments characterizing the behavior of chemical tracers point out that the flow field of a water-saturated heterogeneous porous sediment or fractured hard rock formation can possibly be regarded as a network of fractal paths. Also, we may wish to track the spread of an inject along a coastline or a crack. In such cases, finding the distribution of first passage times between points of known Euclidean distance requires knowledge of the dynamics of the process as well as the distribution of lengths. If we assume that the motion along the path is independent of its shape (as in a simple damage spreading process, or curvature independent diffusion), we can then obtain the distribution of first passage times from a simple convolution of the distribution of lengths. At the minimum, the results of such an assumption may help to rule out simple hypotheses about the dynamics and/or structures of the fractal shapes in physical models.
Important examples of scale invariant curves in two dimensions are self-avoiding walks (SAWs) and percolation fronts (PFs): additional instances may be obtained from critical systems. These examples suggest that generating and studying stochastic scale-invariant curves is a computationally hard process. Indeed, the past decades have witnessed much effort and progress in efficient numerical algorithms for generating SAWs and critical systems. An important recent development concerns the Stochastic Loewner Evolution (SLE), which provides a way of generating such curves through mappings (in the complex plane) of a simple Markovian random walk. While the main interest in SLE has been as a way of extracting analytic information about critical systems, here we use it as an efficient means to generate curves with different values of the fractal dimension.
One issue with SLE as a numerical tool is that (since it is strictly defined in the continuum limit) the length of the curve is not well defined. This question has also been discussed by Kennedy, and like him we introduce a specific procedure for calculating distances along curves generated by discretized SLE. The mathematical ambiguity makes it essential to verify the correctness of this procedure. We do so by comparing distributions obtained from lattice implementation of SAWs and PFs with those obtained from implementation of SLE with appropriate parameters. Having gained confidence on this procedure, we use SLE to generate different sets of scale invariant curves with fractal dimensions $1<d_f<2$. The scaled distributions depend on the fractal dimension $d_f$, progressing from sharp distributions for $d_f$ close to one to broader forms as $d_f$ approaches two.
Knowledge of the lengths distribution allows to obtain both spatial concentration profile and first passage times for walkers diffusing along these fractal paths.
Despite its simplicity, our model for walkers diffusing along critical curves is able to capture some essential features of experimental evidences and detailed Monte Carlo simulations of chemical injects spreading in fracture networks, concerning both spatial (tracers concentration) and temporal behavior (first passage times distributions). |
@incollection{statphys23_0258,
title = {Distributions of passage times and distances along critical curves},
address = {Genova, Italy},
author = {A. Zoia and Y. Kantor and M. Kardar},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
month = {9-13 July},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=258},
year = {2007},
abstract = {Scale invariant curves, such as coastlines and fracture fronts, abound in nature. Fractal dimension provides the scaling of the length $\ell$ along the curve between two points to their actual separation $R$ in Euclidean space. However, for stochastic shapes, the actual length will fluctuate, and knowledge of its full distribution greatly augments the information about scaling. We numerically investigate such distributions for scale invariant (critical) curves in two dimensions.
While the actual quantity that we measure is the distribution of lengths along the fractal curves, an underlying motivation is to understand the distribution of first passage times for traversing such curves. To provide an example, experiments characterizing the behavior of chemical tracers point out that the flow field of a water-saturated heterogeneous porous sediment or fractured hard rock formation can possibly be regarded as a network of fractal paths. Also, we may wish to track the spread of an inject along a coastline or a crack. In such cases, finding the distribution of first passage times between points of known Euclidean distance requires knowledge of the dynamics of the process as well as the distribution of lengths. If we assume that the motion along the path is independent of its shape (as in a simple damage spreading process, or curvature independent diffusion), we can then obtain the distribution of first passage times from a simple convolution of the distribution of lengths. At the minimum, the results of such an assumption may help to rule out simple hypotheses about the dynamics and/or structures of the fractal shapes in physical models.
Important examples of scale invariant curves in two dimensions are self-avoiding walks (SAWs) and percolation fronts (PFs): additional instances may be obtained from critical systems. These examples suggest that generating and studying stochastic scale-invariant curves is a computationally hard process. Indeed, the past decades have witnessed much effort and progress in efficient numerical algorithms for generating SAWs and critical systems. An important recent development concerns the Stochastic Loewner Evolution (SLE), which provides a way of generating such curves through mappings (in the complex plane) of a simple Markovian random walk. While the main interest in SLE has been as a way of extracting analytic information about critical systems, here we use it as an efficient means to generate curves with different values of the fractal dimension.
One issue with SLE as a numerical tool is that (since it is strictly defined in the continuum limit) the length of the curve is not well defined. This question has also been discussed by Kennedy, and like him we introduce a specific procedure for calculating distances along curves generated by discretized SLE. The mathematical ambiguity makes it essential to verify the correctness of this procedure. We do so by comparing distributions obtained from lattice implementation of SAWs and PFs with those obtained from implementation of SLE with appropriate parameters. Having gained confidence on this procedure, we use SLE to generate different sets of scale invariant curves with fractal dimensions $1<d_f<2$. The scaled distributions depend on the fractal dimension $d_f$, progressing from sharp distributions for $d_f$ close to one to broader forms as $d_f$ approaches two.
Knowledge of the lengths distribution allows to obtain both spatial concentration profile and first passage times for walkers diffusing along these fractal paths.
Despite its simplicity, our model for walkers diffusing along critical curves is able to capture some essential features of experimental evidences and detailed Monte Carlo simulations of chemical injects spreading in fracture networks, concerning both spatial (tracers concentration) and temporal behavior (first passage times distributions).},
keywords = {diffusion evolution first fractures lengths loewner passage statphys23 stochastic times topic-3 }
}
::