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Register a new userPower-law and log-normal avalanche size statistics in random growth processes
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We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a and variance va. These two control parameters determine if the avalanche size tends to a stationary distribution, (Finite Scale statistics with finite mean and variance or Power-Law tailed statistics with exponent in (1, 3]), or instead to a non-stationary regime with Log-Normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by analytical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions and provide a precise definition of the boundaries between the three regimes.
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We study the time dependence of the grain size distribution N(r,t) during crystallization of a d-dimensional solid. A partial differential equation including a source term for nuclei and a growth law for grains is solved analytically for any dimension d. We discuss solutions obtained for processes described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and growth (RNG). Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates the distribution evolves in time into the logarithmic-normal (lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310, 3135 (2008)]. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.
As a classical state, for instance a digitized image, is transferred through a classical channel, it decays inevitably with the distance due to the surroundings interferences. However, if there are enough number of repeaters, which can both check and recover the states information continuously, the states decay rate will be significantly suppressed, then a classical Zeno effect might occur. Such a physical process is purely classical and without any interferences of living beings, therefore, it manifests that the Zeno effect is no longer a patent of quantum mechanics, but does exist in classical stochastic processes.
We propose a novel algorithm that outputs the final standings of a soccer league, based on a simple dynamics that mimics a soccer tournament. In our model, a team is created with a defined potential(ability) which is updated during the tournament according to the results of previous games. The updated potential modifies a teams future winning/losing probabilities. We show that this evolutionary game is able to reproduce the statistical properties of final standings of actual editions of the Brazilian tournament (Brasileir~{a}o). However, other leagues such as the Italian and the Spanish tournaments have notoriously non-Gaussian traces and cannot be straightforwardly reproduced by this evolutionary non-Markovian model. A complete understanding of these phenomena deserves much more attention, but we suggest a simple explanation based on data collected in Brazil: Here several teams were crowned champion in previous editions corroborating that the champion typically emerges from random fluctuations that partly preserves the gaussian traces during the tournament. On the other hand, in the Italian and Spanish leagues only a few teams in recent history have won their league tournaments. These leagues are based on more robust and hierarchical structures established even before the beginning of the tournament. For the sake of completeness, we also elaborate a totally Gaussian model (which equalizes the winning, drawing, and losing probabilities) and we show that the scores of the Brasileir~{a}o cannot be reproduced. Such aspects stress that evolutionary aspects are not superfluous in our modeling. Finally, we analyse the distortions of our model in situations where a large number of teams is considered, showing the existence of a transition from a single to a double peaked histogram of the final classification scores. An interesting scaling is presented for different sized tournaments.
A bootstrap percolation process on a graph $G$ is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infected and remains so forever. The parameter $rgeq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $beta$, where $2 < beta < 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)gg a_c(n)$, then there is a constant $eps>0$ such that, with high probability, the final set of infected vertices has size at least $eps n$. It turns out that when the maximum degree is $o(n^{1/(beta -1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Theta (n^{1/(beta -1)})$, then $a_c (n)=n^{beta -2 over beta -1}$.
For any branching process, we demonstrate that the typical total number $r_{rm mp}( u tau)$ of events triggered over all generations within any sufficiently large time window $tau$ exhibits, at criticality, a super-linear dependence $r_{rm mp}( u tau) sim ( u tau)^gamma$ (with $gamma >1$) on the total number $ u tau$ of the immigrants arriving at the Poisson rate $ u$. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power law tail with tail exponent $1 < gamma leqslant 2$, the exponent of the super-linear law for $r_{rm mp}( u tau)$ is identical to the exponent $gamma$ of the distribution of fertilities. For $gamma>2$ and for standard branching processes without power law distribution of fertilities, $r_{rm mp}( u tau) sim ( u tau)^2$. This novel scaling law replaces and tames the divergence $ u tau/(1-n)$ of the mean total number ${bar R}_t(tau)$ of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations and an heuristic derivation enlightens its underlying mechanism. We also show that ${bar R}_t(tau)$ is always linear in $ u tau$ even at criticality ($n=1$). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by $r_{rm mp}( u tau)$.
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