Exact scaling in surface growth with power-law noise

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ-stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Characteristic Lyapunov vectors in chaotic time-delayed systems

We compute Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in delay-differential equations with large time delay. We find that characteristic LVs, and backward (Gram-Schmidt) LVs, exhibit long-range correlations, identical to those already observed in dissipative extended systems. In addition we give numerical and theoretical support to the hypothesis that the main LV belongs, under a suitable transformation, to the universality class of the Kardar-Parisi-Zhang equation. These facts indicate that in the large delay limit (an important class of) delayed equations behave exactly as dissipative systems with spatiotemporal chaos.

Statistical characterization of thermally evaporated rough CaF2 films

Thermal deposition of CaF2 onto a glass substrate creates a nanoscale rough surface. A series of samples with differing nominal CaF2 film thicknesses have been fabricated, and the topography has been investigated using atomic force microscopy. Measured values for the statistical characterization of the samples are presented including the exponents describing the scaling behavior of the surfaces. We find that the roughness exponent alpha=0.88+/-0.03 , the growth exponent beta=0.75+/-0.03 , and the dynamical exponent z=alpha/beta=1.17+/-0.06 . We also measure the multifractal spectra and nearest neighbor height difference probability distribution. The results are consistent with noise dominated by a power-law distribution with exponent mu+1 approximately equal to 4.6. Profilometer measurements were used to determine the porosity phi of the deposited films, which we find to be constant for all film thicknesses with phi approximately 0.46 .

Evidence for power-law dominated noise in vacuum deposited CaF2

We have studied the surface roughness of CaF2 vacuum deposited on glass using atomic force microscopy for film coverages spanning an order of magnitude. We find the roughness exponent alpha=0.88+/-0.03, the growth exponent beta=0.75+/-0.03, and the dynamic exponent z=alpha/beta=1.17+/-0.06. Multifractality is also present, along with power-law behavior in the nearest neighbor height difference probability distribution. The results indicate noise dominated by a power-law distribution with exponent micro+1 approximately 4.6.

Directed polymers and interfaces in random media: free-energy optimization via confinement in a wandering tube

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0<d<2 involves at the same time (i) a confinement in a favorable tube of radius R(S) approximately L (nu(S) ) with nu(S) =1/(4-d)<1/2 (ii) a superdiffusive behavior R approximately Lnu with nu=(3-d)/(4-d)>1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0<D<2. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ( alpha<0 ), whereas it is not for growing correlations ( alpha>0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.