Using extensive numerical analysis of the fiber bundle model with equal load sharing dynamics we studied the finite-size scaling forms of the relaxation times against the deviations of applied load per fiber from the critical point. Our most crucial result is we have not found any ln(N) dependence of the average relaxation time
We present a study of the fiber bundle model using equal load sharing dynamics where the breaking thresholds of the fibers are drawn randomly from a power law distribution of the form p(b) ∼ b −1 in the range 10 −β to 10 β . Tuning the value of β continuously over a wide range, the critical behavior of the fiber bundle has been studied both analytically as well as numerically. Our results are: (i) The critical load σc(β, N ) for the bundle of size N approaches its asymptotic value σc(β) as σc(β, N ) = σc(β)+AN −1/ν(β) where σc(β) has been obtained analytically as σc(β) = 10 β /(2βe ln 10) for β ≥ βu = 1/(2 ln 10), and for β < βu the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to σc(β) = 10 −β ; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form 1 − 1/(2β ln 10); (iii) the distribution D(∆) of the avalanches of size ∆ follows a power law D(∆) ∼ ∆ −ξ with ξ = 5/2 for ∆ ≫ ∆c(β) and ξ = 3/2 for ∆ ≪ ∆c(β), where the crossover avalanche size ∆c(β) = 2/(1−e10 −2β ) 2 .
The ranges of transmission of the mobiles in a mobile ad hoc network are not uniform in reality. They are affected by the temperature fluctuation in air, obstruction due to the solid objects, even the humidity difference in the environment, etc. How the varying range of transmission of the individual active elements affects the global connectivity in the network may be an important practical question to ask. Here a model of percolation phenomena, with an additional source of disorder, is introduced for a theoretical understanding of this problem. As in ordinary percolation, sites of a square lattice are occupied randomly with probability p. Each occupied site is then assigned a circular disk of random value R for its radius. A bond is defined to be occupied if and only if the radii R_{1} and R_{2} of the disks centered at the ends satisfy a certain predefined condition. In a very general formulation, one divides the R_{1}-R_{2} plane into two regions by an arbitrary closed curve. One defines a point within one region as representing an occupied bond; otherwise it is a vacant bond. The study of three different rules under this general formulation indicates that the percolation threshold always varies continuously. This threshold has two limiting values, one is p_{c}(sq), the percolation threshold for the ordinary site percolation on the square lattice, and the other is unity. The approach of the percolation threshold to its limiting values are characterized by two exponents. In a special case, all lattice sites are occupied by disks of random radii R∈{0,R_{0}} and a percolation transition is observed with R_{0} as the control variable, similar to the site occupation probability.
Desert roses are gypsum crystals that consist of intersecting disks. We determine their geometrical structure using computer assisted tomography. By mapping the geometrical structure onto a graph, the topology of the desert rose is analyzed and compared to a model based on diffusion limited aggregation. By comparing the topology, we find that the model gets a number of the features of the real desert rose right, whereas others do not fit so well.
The random sequential adsorption (RSA) model is a classical model in Statistical Physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously adsorbed one, a sequence of dimer displacements may occur to accommodate the new dimer. The introduction of this simple relaxation dynamics leads to a more dense jamming state than the one obtained with RSA without relaxation. We also consider the anisotropic case, where one dimer orientation is favored over the other, finding a non-monotonic dependence of the jamming coverage on the strength of anisotropy. We find that the density of adsorbed dimers at which percolation occurs is reduced with relaxation, but the value depends on the strength of anisotropy.
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