Convergent Calculation of the Asymptotic Dimension of Diffusion Limited Aggregates: Scaling and Renormalization of Small Clusters

Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003

Diffusion Limited Aggregation with Power-Law Pinning

Using stochastic conformal mapping techniques we study the patterns emerging from Laplacian growth with a power-law decaying threshold for growth $R_N^{-\gamma}$ (where $R_N$ is the radius of the $N-$ particle cluster). For $\gamma > 1$ the growth pattern is in the same universality class as diffusion limited aggregation (DLA) growth, while for $\gamma < 1$ the resulting patterns have a lower fractal dimension $D(\gamma)$ than a DLA cluster due to the enhancement of growth at the hot tips of the developing pattern. Our results indicate that a pinning transition occurs at $\gamma = 1/2$, significantly smaller than might be expected from the lower bound $\alpha_{min} \simeq 0.67$ of multifractal spectrum of DLA. This limiting case shows that the most singular tips in the pruned cluster now correspond to those expected for a purely one-dimensional line. Using multifractal analysis, analytic expressions are established for $D(\gamma)$ both close to the breakdown of DLA universality class, i.e., $\gamma \lesssim 1$, and close to the pinning transition, i.e., $\gamma \gtrsim 1/2$.Comment: 5 pages, e figures, submitted to Phys. Rev.

Scaling exponent of the maximum growth probability in diffusion-limited aggregation

An early (and influential) scaling relation in the multifractal theory of Diffusion Limited Aggregation(DLA) is the Turkevich-Scher conjecture that relates the exponent \alpha_{min} that characterizes the ``hottest'' region of the harmonic measure and the fractal dimension D of the cluster, i.e. D=1+\alpha_{min}. Due to lack of accurate direct measurements of both D and \alpha_{min} this conjecture could never be put to serious test. Using the method of iterated conformal maps D was recently determined as D=1.713+-0.003. In this Letter we determine \alpha_{min} accurately, with the result \alpha_{min}=0.665+-0.004. We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.Comment: 4 pages, 5 figure

The use of small angle neutron scattering with contrast matching and variable adsorbate partial pressures in the study of porosity in activated carbons

The porosity of a typical activated carbon is investigated with small angle neutron scattering (SANS), using the contrast matching technique, by changing the hydrogen/deuterium content of the absorbed liquid (toluene) to extract the carbon density at different scattering vector (Q) values and by measuring the p/p0 dependence of the SANS, using fully deuterated toluene. The contrast matching data shows that the apparent density is Q-dependent, either because of pores opening near the carbon surface during the activation processor or changes in D-toluene density in nanoscale pores. For each p/p0 value, evaluation of the Porod Invariant yields the fraction of empty pores. Hence, comparison with the adsorption isotherm shows that the fully dry powder undergoes densification when liquid is added. An algebraic function is developed to fit the SANS signal at each p/p0 value hence yielding the effective Kelvin radii of the liquid surfaces as a function of p/p0. These values, when compared with the Kelvin Equation, show that the resultant surface tension value is accurate for the larger pores but tends to increase for small (nanoscale) pores. The resultant pore size distribution is less model-dependent than for the traditional methods of analyzing the adsorption isotherms

Laplacian growth with separately controlled noise and anisotropy

Conformal mapping models are used to study competition of noise and anisotropy in Laplacian growth. For that, a new family of models is introduced with the noise level and directional anisotropy controlled independently. Fractalization is observed in both anisotropic growth and the growth with varying noise. Fractal dimension is determined from cluster size scaling with its area. For isotropic growth we find d = 1.7, both at high and low noise. For anisotropic growth with reduced noise the dimension can be as low as d = 1.5 and apparently is not universal. Also, we study fluctuations of particle areas and observe, in agreement with previous studies, that exceptionally large particles may appear during the growth, leading to pathologically irregular clusters. This difficulty is circumvented by using an acceptance window for particle areas.Comment: 13 pages, 15 figure

Convergence of the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity

For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.Comment: 8 pages and 8 fig

Multiple tier fuel cycle studies for waste transmutation.

As part of the U.S. Department of Energy Advanced Accelerator Applications Program, a systems study was conducted to evaluate the transmutation performance of advanced fuel cycle strategies. Three primary fuel cycle strategies were evaluated: dual-tier systems with plutonium separation, dual-tier systems without plutonium separation, and single-tier systems without plutonium separation. For each case, the system mass flow and TRU consumption were evaluated in detail. Furthermore, the loss of materials in fuel processing was tracked including the generation of new waste streams. Based on these results, the system performance was evaluated with respect to several key transmutation parameters including TRU inventory reduction, radiotoxicity, and support ratio. The importance of clean fuel processing ({approx}0.1% losses) and inclusion of a final tier fast spectrum system are demonstrated. With these two features, all scenarios capably reduce the TRU and plutonium waste content, significantly reducing the radiotoxicity; however, a significant infrastructure (at least 1/10 the total nuclear capacity) is required for the dedicated transmutation system

Scaling Analysis of Fluctuating Strength Function

We propose a new method to analyze fluctuations in the strength function phenomena in highly excited nuclei. Extending the method of multifractal analysis to the cases where the strength fluctuations do not obey power scaling laws, we introduce a new measure of fluctuation, called the local scaling dimension, which characterizes scaling behavior of the strength fluctuation as a function of energy bin width subdividing the strength function. We discuss properties of the new measure by applying it to a model system which simulates the doorway damping mechanism of giant resonances. It is found that the local scaling dimension characterizes well fluctuations and their energy scales of fine structures in the strength function associated with the damped collective motions.Comment: 22 pages with 9 figures; submitted to Phys. Rev.

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure