Avalanche spatial structure and multivariable scaling functions: Sizes, heights, widths, and views through windows

Yan-Jiun Chen, Stefanos Papanikolaou, James P. Sethna, Stefano Zapperi, and Gianfranco Durin

Phys. Rev. E 84, 061103 – Published 1 December 2011

Abstract

We introduce a systematic method for extracting multivariable universal scaling functions and critical exponents from data. We exemplify our insights by analyzing simulations of avalanches in an interface using simulations from a driven quenched Kardar-Parisi-Zhang (qKPZ) equation. We fully characterize the spatial structure of these avalanches—we report universal scaling functions for size, height, and width distributions, and also local front heights. Furthermore, we resolve a problem that arises in many imaging experiments of crackling noise and avalanche dynamics, where the observed distributions are strongly distorted by a limited field of view. Through artificially windowed data, we show these distributions and their multivariable scaling functions may be written in terms of two control parameters: the window size and the characteristic length scale of the dynamics. For the entire system and the windowed distributions we develop accurate parametrizations for the universal scaling functions, including corrections to scaling and systematic error bars, facilitated by a novel software environment SloppyScaling.

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Received 22 July 2011

DOI:https://doi.org/10.1103/PhysRevE.84.061103

Authors & Affiliations

Yan-Jiun Chen, Stefanos Papanikolaou, and James P. Sethna

Laboratory of Atomic and Solid State Physics (LASSP), Clark Hall, Cornell University, Ithaca, New York 14853-2501, USA

Stefano Zapperi

IENI-CNR,Via R. Cozzi 53, I-20125 Milano, Italy and ISI Foundation, viale S. Severio 65, Torino, Italy

Gianfranco Durin

Istituto Nazionale di Ricerca Metrologica, strada delle Cacce 91, I-10135 Torino, Italy and ISI Foundation, viale S. Severio 65, Torino, Italy

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Vol. 84, Iss. 6 — December 2011

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Images

Figure 1
Windowed distributions. (a) The full system of avalanches. This figure shows a qKPZ simulation with avalanches. (b) A limited field of view: we can only see part of the system. The avalanches inside the artificial viewing window are brightly colored, and those outside are washed out. Notice that the avalanches within the window are cut off at top and bottom, and (more importantly for short wide avalanches with roughness exponent $ζ < 1$ ) on the two sides. (c) The size distributions for the different types of avalanches, (d) internal 00 avalanches, (e) split 10 avalanches, (f) split 01 avalanches, and (g) spanning 11 avalanches.Reuse & Permissions
Figure 2
Scale invariance in crackling noise. Three simulations of crackling noise, with different “demagnetization fields” $k$ . (a) $k = 10^{- 5}$ , (b) $k = 10^{- 6}$ , (c) $k = 10^{- 7}$ ( $k$ is what controls the typical avalanche sizes in a given system, giving a characteristic width $L_{k}$ ). Larger demagnetization fields stop avalanches more strongly, hence large $k$ corresponds to smaller avalanches. The colored regions represent avalanches. The fronts are moving from bottom to top. Notice that the two simulations are statistically similar to one another apart from a rescaling of heights and widths. Note that most of the area is covered by the largest avalanches.Reuse & Permissions
Figure 3
$A (S)$ vs $P (S)$ . Here you can see the difference between the more traditional $P (S)$ and the area-weighted $A (S)$ . (a) $P (S)$ : most of the area under the curve is from small avalanches, where nonuniversal lattice effects are important. (b) $A (S)$ : the normalization is dominated by large avalanches, avoiding the lattice effects, so that we can focus instead on the dependence on the large scale cutoffs— $L_{k}$ and, in later sections, the window size. The data here are from qKPZ simulations of different $k$ with different simulation size $L$ . The lines in (b) are a result of a joint fit with the maximum height and width distributions. The fitting function and fitting parameters are shown in Table .Reuse & Permissions
Figure 4
Area-weighted avalanche distributions of (a) maximum heights, and (b) maximum widths for qKPZ simulations at different $k$ , and simulation sizes $L$ (dots are binned data). The critical exponents were jointly fit with the size distributions $A (S | L_{k})$ of Fig. 3, using the scaling forms of Eqs. (11, 12, 13). The best fit values for the critical exponents, parameters for universal scaling functions, and non-universal corrections are given in Table .Reuse & Permissions
Figure 5
Size distribution $A (S | L_{k})$ scaling collapse. We collapse the size distribution data using the universal exponents of $τ = 1.24$ , $ν_{k} = 0.45$ , and $ζ = 0.62$ , the best fit values of the joint fit between $A (S | L_{k})$ , $A (h | L_{k})$ , and $A (w | L_{k})$ .Reuse & Permissions
Figure 6
Height distribution $A (h | L_{k})$ scaling collapse. We collapse the simulation data using the universal exponents of $τ = 1.24$ , $ν_{k} = 0.45$ , and $ζ = 0.62$ , the best fit values of the joint fit between $A (S | L_{k})$ , $A (h | L_{k})$ , and $A (w | L_{k})$ .Reuse & Permissions
Figure 7
Width distribution $A (w | L_{k})$ scaling collapse. We collapse the width distribution data using the universal exponents of $τ = 1.24$ , $ν_{k} = 0.45$ , and $ζ = 0.62$ , the best fit values of the joint fit between $A (S)$ , $A (h)$ , and $A (w)$ .Reuse & Permissions
Figure 8
Here are the area-weighted local height distributions, the fraction of area taken up by a cross sectional height $h_{x}$ . The fits shown in the figure were with the form of Eq. (15), where $A_{h_{x} k}$ is the scaling function of the fit to 11 spanning avalanches Eq. (32), which cross both window boundaries, taking $W = 0$ . Details for this function are explained in Sec. 5.Reuse & Permissions
Figure 9
Internal avalanches data and fit. Shown here are the area-weighted size distributions for internal (00) avalanches. The lines are the joint best fit of $A_{00}$ , $A_{10}$ , and $A_{11}$ to the functional forms of Eqs. (23), (26), and (32), whereas the shaded areas are the fluctuations in theory corresponding to the systematic covariant errors on our exponents and parameters (individual parameter best fit values and errors are quoted in Table ).Reuse & Permissions
Figure 10
Split avalanches data and fit. Shown here are the area-weighted size distributions for split (10) avalanches with different $k$ and window size $W$ . The lines are the joint best fit of $A_{00}$ , $A_{10}$ , and $A_{11}$ to Eqs. (23), (26), and (32), whereas the shaded areas are the fluctuations in theory corresponding to the covariant systematic errors on our exponents and parameters (individual parameter best fit values and errors, Table ).Reuse & Permissions
Figure 11
Spanning avalanches data and fit. Shown here are the area-weighted size distributions for spanning (11) avalanches with different $k$ and window size $W$ . The lines are the joint best fit value using the functional forms using Eqs. (23), (26), and (32), whereas the shaded areas are the fluctuations in theory corresponding to the systematic errors on our exponents and parameters (individual parameter best fit values and errors are quoted in Table ).Reuse & Permissions
Figure 12
An avalanche cut by windows in the extreme limits. Drawn here are two fronts separated by an avalanche event. Here we are depicting cases where this avalanche (or its segment) is the maximum avalanche size for the 00, 10, and 11 at different window widths. Boxes of different widths and colors are used to show the cases in which this may happen. The main avalanche may count as a 00 avalanche for a wide window, while part of it would count as a 11 avalanche for a smaller window; it could also count as a 10 avalanche if it happens to cross the window boundary. This figure illustrates our arguments for the shape of the cutoff (the exponent $n_{z z}$ ) given a window size $W$ for the 00, 11, 10 cases. Another small avalanche is drawn for the 11 case to show that the minimum size to cross the 11 window also introduces a separate cutoff.Reuse & Permissions
Figure 13
Internal avalanches scaling function. (a) Scaling collapse showing $A_{00} (s_{k}, W_{k})$ as a function of $s_{k}$ . The separate curves show the effects of the scaled window size $W_{k}$ . (b) Logarithmic contour plot of best fit scaling function against both scaling variables $s_{k}$ and $W_{k}$ . Each contour reflects a drop of a factor of $e$ in the scaling function. The black dots are at locations of the simulated data points used in the fit indicating where the fit should be a reliable prediction. The red solid line is $\log_{10} W_{k} = n_{00} \log_{10} s_{k}$ , which is the slope at the large avalanche cutoff, with $n_{00} = 1.62$ , the best fit value ( $n_{00} = 1 + ζ = 1.63$ is the expected value from our heuristic arguments).Reuse & Permissions
Figure 14
Split avalanches scaling function. (a) Scaling collapse showing $A_{10} (s_{k}, W_{k})$ as a function of $s_{k}$ . The separation between curves shows the dependence on the scaled window size $W_{k}$ . (b) Logarithmic contour plot of best fit scaling function against both scaling variables $s_{k}$ and $W_{k}$ . Each contour reflects a drop of a factor of $e$ in the value of the scaling function. The black dots are at locations of the simulated data points used in the fit. The red solid line is $\log_{10} W_{k} = n_{10} \log_{10} s_{k}$ , which is the slope at the large avalanche cutoff, with $n_{10} = 1.62$ , the best fit value.Reuse & Permissions
Figure 15
Spanning avalanches scaling function. (a) Scaling collapse showing $A_{10} (s_{k}, W_{k})$ as a function of $s_{k}$ . The curves move leftward and become more sharply rounded with increasing $W_{k}$ . (b) Logarithmic contour plot of best fit scaling function. Each contour reflects a drop of a factor of $e$ in the scaling function. The black dots are at locations of the simulated data points used in the fit. The upper orange solid line is the slope of the contour plot at the small avalanche cutoff, and has $\log_{10} W_{k} = 1 / 1.65 \log_{10} s_{k}$ where $1.65$ is the best fit $n_{11}$ value. The orange dashed has $\log W_{k} = 1 / 1.63 \log_{10} s_{k}$ where $1.63$ is the $1 + ζ$ value. The lower red solid line is $\log_{10} W_{k} = \log_{10} s_{k}$ , which is the slope at the large avalanche cutoff.Reuse & Permissions
Figure 16
Here we collapse the size distributions with exponents $τ = 1.25$ , $ν_{k} = 0.44$ , and $ζ = 0.68$ . Notice that the collapses are similar to the ones shown in Fig. 5. Here only the combination of $ν_{k} (1 + ζ)$ affect the scaling collapse; the large shifts in $ν_{k}$ and $ζ$ mostly cancel in the product, yielding similar collapses.Reuse & Permissions
Figure 17
Height distribution collapse. Here we collapse the size distributions with exponents $τ = 1.25$ , $ν_{k} = 0.44$ , and $ζ = 0.68$ . Here only the combination of $ν_{k} ζ$ affect the scaling collapse; the large shifts in $ν_{k}$ and $ζ$ mostly cancel in the product, yielding similar collapses. Comparing this with the collapse shown in Fig. 6, we see that the two collapses are comparable in quality, where in Fig. 6 the large avalanche cutoff is collapsed nicely, and here the smaller avalanches are collapsed better.Reuse & Permissions
Figure 18
Width distribution collapse. Here we collapse the size distributions with exponents $τ = 1.25$ , $ν_{k} = 0.44$ , and $ζ = 0.68$ . Notice that the collapses are similar to the ones shown in Fig. 7. Here only the combination of $ν_{k}$ affects the scaling collapse and $(τ - 2) (1 + ζ)$ affects the shape of the scaling collapse. Since $ν_{k}$ does not change significantly, the quality of the collapses are similar.Reuse & Permissions
Figure 19
Height-height correlations for the qKPZ simulations. Shown here are the roughness exponents for various simulation sizes $L$ and $k$ . We measure the height-height correlation function $C (r) \sim 〈{[h (x + r) - h (r)]}^{2}〉$ . A power law fit shows $ζ$ falls between 0.63 and 0.68. The lower red-dashed line shows $ζ = 0.63$ and upper black-dashed line shows $ζ = 0.68$ . The lines were shifted to show each individual power law.Reuse & Permissions
Figure 20
Roughness exponents for the qKPZ simulations. Shown here are the measurements of the local-log slope $\ln [C (r) / C (r / 2)] / \ln (2)$ of the height-height correlation function; this is a measure for the roughness exponents for various simulation sizes $L$ and $k$ . The results of the local-log slope $\ln [C (r) / C (r / 2)] / \ln (2)$ is consistent with what is seen in Fig. 19. The lower red-dashed line shows $ζ = 0.63$ , corresponding to directed-percolation depinning (DPD) and suggested by literature to be the correct value for our model. Whereas the upper black-dashed line shows $ζ = 0.68$ , which is the result of our fits of windowed avalanche distributions. In the blowup of the region of $r / 2 = 10 - 100$ , we can see that there is a trend of larger $ζ$ corresponding to larger $k$ simulations.Reuse & Permissions

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