Limited Range Fractality of Randomly Adsorbed Rods

Multiple resolution analysis of two dimensional structures composed of randomly adsorbed penetrable rods, for densities below the percolation threshold, has been carried out using box-counting functions. It is found that at relevant resolutions, for box-sizes, $r$, between cutoffs given by the average rod length and the average inter-rod distance $r_1$, these systems exhibit apparent fractal behavior. It is shown that unlike the case of randomly distributed isotropic objects, the upper cutoff $r_1$ is not only a function of the coverage but also depends on the excluded volume, averaged over the orientational distribution. Moreover, the apparent fractal dimension also depends on the orientational distributions of the rods and decreases as it becomes more anisotropic. For box sizes smaller than the box counting function is determined by the internal structure of the rods, whether simple or itself fractal. Two examples are considered - one of regular rods of one dimensional structure and rods which are trimmed into a Cantor set structure which are fractals themselves. The models examined are relevant to adsorption of linear molecules and fibers, liquid crystals, stress induced fractures and edge imperfections in metal catalysts. We thus obtain a distinction between two ranges of length scales: $r$ where the internal structure of the adsorbed objects is probed, and $< r < r_1$ where their distribution is probed, both of which may exhibit fractal behavior. This distinction is relevant to the large class of systems which exhibit aggregation of a finite density of fractal-like clusters, which includes surface growth in molecular beam epitaxy and diffusion-limited-cluster-cluster-aggregation models.Comment: 10 pages, 7 figures. More info available at http://www.fh.huji.ac.il/~dani/ or http://www.fiz.huji.ac.il/staff/acc/faculty/biham or http://chem.ch.huji.ac.il/employee/avnir/iavnir.htm . Accepted for publication in J. Chem. Phy

Decay Process for Three - Species Reaction - Diffusion System

We propose the deterministic rate equation of three-species in the reaction - diffusion system. For this case, our purpose is to carry out the decay process in our three-species reaction-diffusion model of the form $A+B+C\to D$. The particle density and the global reaction rate are also shown analytically and numerically on a two-dimensional square lattice with the periodic boundary conditions. Especially, the crossover of the global reaction rate is discussed in both early-time and long-time regimes.Comment: 6 pages, 3 figures, Late

On Universality in Human Correspondence Activity

Identifying and modeling patterns of human activity has important ramifications in applications ranging from predicting disease spread to optimizing resource allocation. Because of its relevance and availability, written correspondence provides a powerful proxy for studying human activity. One school of thought is that human correspondence is driven by responses to received correspondence, a view that requires distinct response mechanism to explain e-mail and letter correspondence observations. Here, we demonstrate that, like e-mail correspondence, the letter correspondence patterns of 16 writers, performers, politicians, and scientists are well-described by the circadian cycle, task repetition and changing communication needs. We confirm the universality of these mechanisms by properly rescaling letter and e-mail correspondence statistics to reveal their underlying similarity.Comment: 17 pages, 3 figures, 1 tabl

Exactly Solvable Model of Monomer-Monomer Reactions on a Two-Dimensional Random Catalytic Substrate

We present an \textit{exactly solvable} model of a monomer-monomer $A + B \to \emptyset$ reaction on a 2D inhomogeneous, catalytic substrate and study the equilibrium properties of the two-species adsorbate. The substrate contains randomly placed catalytic bonds of mean density $q$ which connect neighboring adsorption sites. The interacting $A$ and $B$ (monomer) species undergo continuous exchanges with corresponding adjacent gaseous reservoirs. A reaction $A + B \to \emptyset$ takes place instantaneously if $A$ and $B$ particles occupy adsorption sites connected by a catalytic bond. We find that for the case of \textit{annealed} disorder in the placement of the catalytic bonds the reaction model under study can be mapped onto the general spin $S = 1$ (GS1) model. Here we concentrate on a particular case in which the model reduces to an exactly solvable Blume-Emery-Griffiths (BEG) model (T. Horiguchi, Phys. Lett. A {\bf 113}, 425 (1986); F.Y. Wu, Phys. Lett. A, {\bf 116}, 245 (1986)) and derive an exact expression for the disorder-averaged equilibrium pressure of the two-species adsorbate. We show that at equal partial vapor pressures of the $A$ and $B$ species this system exhibits a second-order phase transition which reflects a spontaneous symmetry breaking with large fluctuations and progressive coverage of the entire substrate by either one of the species.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

Exactly solvable model of A + A \to 0 reactions on a heterogeneous catalytic chain

We present an exact solution describing equilibrium properties of the catalytically-activated A + A \to 0 reaction taking place on a one-dimensional lattice, where some of the sites possess special "catalytic" properties. The A particles undergo continuous exchanges with the vapor phase; two neighboring adsorbed As react when at least one of them resides on a catalytic site (CS). We consider three situations for the CS distribution: regular, annealed random and quenched random. For all three CS distribution types, we derive exact results for the disorder-averaged pressure and present exact asymptotic expressions for the particles' mean density. The model studied here furnishes another example of a 1D Ising-type system with random multi-site interactions which admits an exact solution.Comment: 7 pages, 3 Figures, appearing in Europhysics Letter

Possible Experimental Test of Continuous Medium Model for Fractal Media

We use the fractional integrals to describe fractal media. We consider the fractal media as special ("fractional") continuous media. We discuss the possible experimental testing of the continuous medium model for fractal media that is suggested in Phys. Lett. A. 336 (2005) 167-174. This test is connected with measure of period of the Maxwell pendulum with fractal medium cylinder.Comment: 9 page

Binary Reactive Adsorbate on a Random Catalytic Substrate

We study the equilibrium properties of a model for a binary mixture of catalytically-reactive monomers adsorbed on a two-dimensional substrate decorated by randomly placed catalytic bonds. The interacting $A$ and $B$ monomer species undergo continuous exchanges with particle reservoirs and react ($A + B \to \emptyset$) as soon as a pair of unlike particles appears on sites connected by a catalytic bond. For the case of annealed disorder in the placement of the catalytic bonds this model can be mapped onto a classical spin model with spin values $S = -1,0,+1$, with effective couplings dependent on the temperature and on the mean density $q$ of catalytic bonds. This allows us to exploit the mean-field theory developed for the latter to determine the phase diagram as a function of $q$ in the (symmetric) case in which the chemical potentials of the particle reservoirs, as well as the $A-A$ and $B-B$ interactions are equal.Comment: 12 pages, 4 figure