Morphology of Laplacian random walks

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Morphology of Laplacian random walks

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November 2010 EPL, 92 (2010) 36004 doi: 10.1209/0295-5075/92/36004

www.epljournal.org

Morphology of Laplacian random walks J. Ø. H. Bakke1,2 , P. Ray3 and A. Hansen1(a) 1

Department of Physics, Norwegian University of Science and Technology - N-7496 Trondheim, Norway Schlumberger Stavanger Research - N-4068 Stavanger, Norway 3 Institute of Mathematical Sciences - Chennai 600 113, India 2

received 12 April 2010; accepted in final form 22 October 2010 published online 26 November 2010 PACS PACS PACS

62.20.M- – Structural failure of materials 46.50.+a – Fracture mechanics, fatigue and cracks 05.40.Fb – Random walks and Levy flights

Abstract – Roughness of random walks in the presence of a Laplacian field is studied in two dimensions for various strengths of the field parametrized by η. We find an ηc ∼ 4.5 ± 0.3 at which a transition occurs from a tortuous fractal structure to a one-dimensional profile of the walk. At ηc , the walks are self-affine with a roughness exponent ζ = 0.80 ± 0.05. For increasing η-values, the roughness exponent increases. c EPLA, 2010 Copyright 

Growth processes in presence of disorder and a competing Laplacian field have received much attention [1]. Examples of such processes are diffusion limited aggregation, dielectric breakdown, deposition and fracture. In these processes, the Laplacian field favours the growth to take place along the direction of maximum field gradient whereas disorder, generally, tends to change the direction of growth in random fashion. The morphology of the emerging structures, as a result, depends crucially on the strength of the field. A typical example of a Laplacian growth is the dielectric breakdown model (DBM) where the growth probability at a point is proportional to the local electric field to the power κ [1]. The cluster generated in DBM is a fractal with fractal dimension varying continuously with κ [1]. It is suggested [2] that the structure becomes one-dimensional and non-fractal at a finite value of κ = κc . Numerical simulations indicate κc = 4 [3,4] in two dimensions, though this has been questioned recently [5]. In this letter, we study the roughness morphology of Laplacian random walks (LRW) [6]. LRW is defined by growing a walk on a lattice, in which at every step a Laplacian field V is computed over the entire lattice with boundary conditions that the field vanishes on the walk and grows logarithmically with the distance from the walk. The probability pij for selecting a site j near the tip i for growth is (∇V )ηij (1) pij =  η , k (∇V )ik (a) E-mail:

[email protected]

where the sum in the denominator is over all the possible growth sites k and η  0. η determines the strength of the Laplacian field in controlling the growth probability. In LRW, growth occurs only at the tip of the walk and this distinguishes it from DBM. We demonstrate below that the LRW is self-affine and hence asymptotically flat —i.e., its width grows slower than its length. This implies that the influence of the roughness of a particular section of the LRW on the growth probability Pij decreases with the distance of the section to the tip of the LRW where the growth occurs. Hence, Pij must be statistically stationary. The fractal dimension of LRW varies continuously with η from 1.75 at η = 0 (also called indefinitely growing selfavoiding walk or IGSAW) to 1 at large η [6]. For η = 1, the configurations of LRW can be mapped to that of looperased self-avoiding walks for which the fractal dimension is known to be 1.25 exactly [7]. We study LRW for different values of η. We measure, particularly, the roughness and the correlations in the transverse fluctuations of the walks. We find a transition in the morphology of the walks for η = ηc ∼ 4.5 ± 0.3. For η < ηc , the walk morphology is determined predominantly by the backbends which under SOS (solid-on-solid) approximation give rise to large jumps —overhangs— in walk profiles. When the walk is fractal, the jump distribution follows a power law with an exponent α = D, where D is the fractal dimension of the walk [8]. These jumps also give rise to multi-affine behaviour of the profiles at small length scales [9]. This means that moments of the height difference ∆h(δ) = h(x + δ) − h(x), |∆h(δ)|n 1/n scales a

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J. Ø. H. Bakke et al.

Fig. 1: (Colour on-line) Samples profiles for LRW with anisotropic boundary conditions are shown. Three different values for the field range parameter are used: η = 0.5 (top), η = 5.0 (center) and η = 10.00 (bottom). The full walk for η = 0.5 is shown as the dashed line.

δ ζn , where ζn is a function of n. This is in contrast to the mono-affine case when ζn = ζ. For η > ηc , the walk tends to follow its previous step and the size of a typical growth step becomes much larger than shift in the walk away from the growth direction. The walks become one dimensional and the ovehangs loose their statistical significance. At ηc , the walks show clear evidence of self-affinity: the distribution of the height difference ∆h(δ) follows a Gaussian distribution with a scaling of the standard deviation σ proportional to δ ζ with a unique roughness exponent ζ = 0.80 ± 0.05. Higher moments scale as the standard deviation, and, hence, the walk is mono-affine. We generate walks on a square lattice of size L × L tilted by 45◦ and placed between two parallel bus-bars having voltages V = 1 and 0. We apply open boundary condition at the two free sides of the lattice rather than periodic boundary conditions as with the latter, as the latter would make it possible for the LRW to run into itself. A walk originates from a site with V = 0 and every step of the walk involves the following processes: i) voltages are determined at every lattice sites by solving the Laplace’s equation ∇2 V = 0, ii) a new step is added to the last added step of the walk in accordance to eq. (1), iii) the voltage to the newly added site is set to zero. The processes are repeated till the walk reaches the other side of the network. The walk profile h(x) is extracted from the walk S(x, y) by taking the minimum value of the possibly multi-valued surface at position x: h(x) = miny S(x, y) (solid-on-solid or SOS approximation). Figure 1 shows the profiles of the walks for different η-values. Results are obtained for various system sizes L ∈ {16, 32, 64, 128, 256, 512} and for large number of η-values from 0.5 to 10.0. Here, we will discuss mostly the results for η = 0.5, 4.5 and 7.0. We find two distinct regions: (1) low-η region (η < 4.5), where the walks have backbends at every scale. The overhangs or the jumps ∆ from the SOS approximation plays crucial role in determining the morphology of the walks as they signify that the walk is fractal or not. We

Fig. 2: (Colour on-line) The jump distribution p(∆) ∝ |∆|−1−α for η = 0.5 (), η = 4.5 (•) and η = 7.0 (×). The straight lines are power law fits to the different data sets. The apparent α-values are α0.5 = 1.3, α4.5 = 2.1 and α7.0 = 3.0.

compute the jump distribution p(|∆|) (where ∆ = h(y + 1) − h(y) is the height difference between two consecutive points) which is found to follow a power law: p(|∆|) ∝ |∆|−1−α (fig. 2). The exponent α depends on η. It varies from ≈ 1.3 for η = 0.5 to give an apparent value of ≈ 3.5 for η = 7. For large η-values the exponent is very questionable as p(|∆|) falls off rapidly with |∆| and gives few data points. The exponent may be related to the fractal dimension of the walks as already pointed out [8]. We do not determine ηc from the behavior of the jump distribution directly. This is too inaccurate. Rather, we determine the transition point indirectly by comparing different methods for measuring the roughness exponent of the walks, that are sensitive and insensitive to the prsence of jumps. The transition occurs where the different methods go from giving matching results to giving different results [11]. This will be discussed in detail below. Region (2) is for large η-values (η > 4.5) where randomness in walk directions is low and the parts of the walk where consecutive steps are in the same direction become large (see fig. 1). The walks are dominated by few very large segments. To quantify the number of long straight segments we determine the distribution p(s) of straight segments of length s in the walks. In fig. 3 we present p(s) for three different disorders. For low η = 0.5, p(s) falls off exponentially with s suggesting that at low η there is hardly any straight region in the walks. As η increases there emerges a power law region in p(s) vs. s and the exponential cut-off moves to higher s-values. The exponent for the power law is −1.5, which implies that the path is totally dominated by the largest straight segment in the thermodynamic limit. This is seen in fig. 1 where the profile for η = 10.0 has two large segments making up almost half of the length. When η becomes larger than approximately 7, the profiles becomes so straight that it is difficult to consider them as rough for the system sizes we have considered. By using the profiles from the SOS approximations we have measured the self-affine local roughness exponent ζ for the different η-values. ζ can be measured by many different methods [10,11]. One method is to measure

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Fig. 3: (Colour on-line) The distribution p(s) of the number s of consecutive steps in the same direction. The distributions for three different disorder parameters are shown. η = 0.5 (), η = 4.5 (•) and η = 7.0 (×). The straight line is p(s) ∝ s−1.5 .

Fig. 4: (Colour on-line) The roughness exponent ζ, measured by the power spectrum density method, is plotted against η. Measurements are done on the original profiles h(x) (), the profiles with only the sign change information h0 (x) (•) and randomly reshuffled profiles h∆ (x) (×). Results on h(x) and h0 (x) are almost identical, while the errors are less than ±0.03 for all the three profiles.

the second-order correlation function C2 (l) = |h(x + l) − h(x)|2  ∝ l2ζ or its Fourier transform the power spectrum density P (k) ∝ k −(2ζ+1) , where k is the spatial frequency [12]. One can also use the average wavelet coefficients analysis W [h](a) ∝ a1/2+ζ where W [h](a) is the averaged wavelet coefficients at scale a [13]. In the following, we present the results obtained using the power spectrum density and the averaged wavelet coefficient analysis. The choice of these two methods was based on the evaluation of the methods done in [11]. Figures 4 and 5 show the results for ζ obtained by the two above-mentioned methods. By comparing the ζ-values measured on the original profiles h(x) (), one notices, in the two figures, a large discrepancy between the two measurement methods for η < 4.5. The values from the averaged wavelet coefficients analysis are higher than the values from the power spectrum density analysis. For η  4.5 the measured values are equal inside the error bars. Local roughness exponent measurements by the characteristic width methods gave the same results as those we got from the averaged wavelet analysis. The second-order correlation function gave the same results as the power spectrum density analysis.

Fig. 5: (Colour on-line) The roughness exponent ζ, measured by the averaged wavelet coefficients method, is plotted against η. Measurements are done on the original profiles h(x) (), the profiles with only the sign change information h0 (x) (•) and the randomly reshuffled profiles h∆ (x) (×). The effect of the power law distributed jumps are observed for small η in h(x) and h∆ (x). The errors are less than ±0.02 for all three profiles.

We have also measured the roughness of two modified sets of the original walk profiles. The motivation behind this is to understand where the difference in the values of ζ measured by different methods comes from [11,14]. The first modification was to construct profiles where we kept only the information of the signs in the height differences in the profiles. This was done by constructing the following modified profile: h0 (i) =

i 

sgn(∆(j))|∆(j)|0 ,

(2)

j=1

where ∆(j) = h(j + 1) − h(j) and sgn(x) returns the sign of x as ±1. The roughness measured on these modified profiles will then only measure the contribution from the long range correlations in the sign as all amplitude information is removed. The ζ-values of these modified profiles are also shown in figs. 4 and 5 as ζ0 (•). For the profile h0 (x), the ζ-values measured by the averaged wavelet coefficients method and power spectrum density analysis are in accordance for all η-values. Moreover, the results agree with that obtained by the power spectrum density analysis on the original h(x) profiles. This shows that the power spectrum density method measures the contribution of the long range correlations in the sign changes to the roughness exponent. To measure the possible contribution(s) to the roughness factors other than the correlations in the sign changes, we construct another modified set from the original profiles as h∆ (i) =

i 

∆r (j),

(3)

j=1

where ∆r (j) is a randomly chosen jump from a position j in the profile. In these profiles any effect of the correlations in the sign of the jumps to the roughness is destroyed. The results for the roughness exponent measurements on these randomly reshuffled profiles are shown in figs. 4

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J. Ø. H. Bakke et al. and 5 as ζ∆ (×). From the figures one finds that above η = 4.5, ζ remains close to one half as one would expect for an uncorrelated profile. Below η = 4.5, ζ-values measured with averaged wavelet coefficients and power spectrum density differ. For the power spectrum density measurements in fig. 4, ζ remains close to one half for all η, while the averaged wavelet coefficients measurements in fig. 5 give a roughness exponent higher than 0.5 when η < 4.5. The roughness exponent for the original profiles appears to be the combination of the roughness exponents from h0 and h∆ , ζ = (ζ0 − 0.5) + (ζ∆ − 0.5) + 0.5. A profile with an uncorrelated power law jump distribution and 0 < α < 2 is a L´evy flight. The local roughness exponent for such a flight is ζL = 1/α [15]. When the roughness exponent is measured using h∆ , the result is not consistent with the that calculated from the measurements of α. This suggests that the profiles for small η-values are not purely L´evy flights. Recent studies of the height fluctuation distributions p(∆h, δ), for both experimental and numerical studies of fracture profiles have shown that the distribution is Gaussian on larges scales [16,17]. In this LRW model one observes in fig. 6 that the distribution is Gaussian for η close to 4.5. The standard deviation of the Gaussian σ ∝ δ ζ with ζ = 0.80 ± 0.05. For smaller η-values it is wider than a Gaussian and for larger η the it is narrower. This shows that only for η-values close to 4.5 will the profiles from the LRW be a fractional Brownian motion. We note that when δ is of the order of one step, the distribution P (∆h, δ) becomes equal to the jumps size distribution, p(|∆h|). To summarise we have shown that the anisotropic Laplacian Random walk is self-affine and have a Gaussian height fluctuation distribution at ηc = 4.5 ± 0.3. We therefore propose that the walk at ηc is a fractional Brownian walk with ζ = 0.80 ± 0.05. For lower η-values and corresponding higher degrees of disorder we find the profiles to be a combination of a fractional Brownian motion and a L´evy flight. The large number of big jumps for η < 4.5 can be attributed to two effects. The first effect is the random fluctuations in the growth. The second effect is the restriction of the growth to the tip of the walk, and not to the place on the existing walk surface with the highest voltage gradient. For η > 4.5 the fluctuation in the walk direction decreases and the growth of the walk is guided by the lattice. Above η ≈ 7.0 the profiles can no longer be considered rough. In summary, we find that Laplacian processes produce highly rough profiles. At η < ηc ≈ 4.5, the surfaces are multi-affine due to the presence of jumps, which in turn are caused by backbends. At η = ηc , the profile is selfaffine characterized by a roughness exponent which is approximately 0.8. At larger η-values, the roughness exponent increases. The roughness exponent is much larger than those for most surfaces one finds in nature or from theoretical models [18]. We believe these walks belong to a distinctly different universality class at η = ηc .

Fig. 6: (Colour √ on-line) Rescaled height fluctuation distribution √ p(∆h, δ) 2πσ 2 plotted vs. the rescaled height difference ∆h/ 2σ 2 for a) η = 3.0, b) η = 4.5 and c) η = 6.0. The length scales shown here are δ = 64 (), δ = 128 •) and δ = 256 (). The solid line is a least-square fit to a Gaussian done on the δ = 128 data.

This may provide explanation of the anomalousy high roughness exponent that one finds in situations like fracture surfaces [16] and invading interfaces [19]. Although the LRW growth rules are different from that of DBM it is interesting to observe that the transition from a Gaussian height-height distribution to a wider distribution is observed for η ∈ [4.0, 5.0], close to the proposed phase transition at κc = 4 for DBM. The LRW profiles can also possibly be grown with the use of iterated conformal mapping [4,5,20–22]. A natural extension would be to use this method with the boundary conditions used

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Morphology of Laplacian random walks here to get access to large system sizes. One would also get off-lattice results which would be very interesting especially for the large η-values where the lattice is evidently influencing the cluster growth. ∗∗∗ JØHB thanks the IMSC, Chennai, for hospitality during parts of this work. The numerical work was done with support of the Norwegian High-Performance Computing Consortium NOTUR, and the research program Computational Science and Visualization at NTNU. REFERENCES [1] Meakin P., Fractals, Scaling and Growth far from Equilibrium (Cambride University Press, Cambridge) 1998. [2] Derrida B. and Hakim V., Phys. Rev. A, 45 (1992) 8759. ´nchez A., Guinea F., Sander L. M., Hakim V. and [3] Sa Louis E., Phys. Rev. E, 48 (1993) 1296. [4] Hastings M. B., Phys. Rev. Lett., 87 (2001) 175502. [5] Mathiesen J., Jensen M. H. and Bakke J. Ø. H., Phys. Rev. E, 77 (2008) 066203. [6] Lyklema J. W., Evertsz E. and Pietronero L., Europhys. Lett., 2 (1986) 77. [7] Majumdar S. N., Phys. Rev. Lett., 68 (1992) 2329.

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