Random Walks with Long-Range Self-Repulsion on Proper Time

We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. Analytic results on the exponent $\nu$ are obtained. They are in good agreement with Monte Carlo simulations in two dimensions. A numerical study of the scaling functions and of the efficiency of the algorithm is also presented.Comment: 25 pages latex, 4 postscript figures, uses epsf.sty (all included) IFUP-Th 13/92 and SNS 14/9

Probability Distribution of the Shortest Path on the Percolation Cluster, its Backbone and Skeleton

We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and Phi(S)(r|l), giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path of length l are separated by an Euclidean distance r. Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents g1=df+dmin-d and g1B=g1S-3dmin-d, which determine the scaling behavior of the distribution functions in the limit x=r/l^(nu) much less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l) proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g1, g1B, and g1S are in very good agreement with our numerical results.Comment: 10 pages, 3 figure

Structural Information in Two-Dimensional Patterns: Entropy Convergence and Excess Entropy

We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies. We show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension. We compare and contrast entropy-convergence with mutual-information and structure-factor techniques for quantifying and detecting spatial structure.Comment: 11 pages, 5 figures, http://www.santafe.edu/projects/CompMech/papers/2dnnn.htm

Dynamical Scaling from Multi-Scale Measurements

We present a new measure of the Dynamical Critical behavior: the "Multi-scale Dynamical Exponent (MDE)"Comment: 9 pages,Latex, Request figures from [email protected]

Epidemic processes with immunization

We study a model of directed percolation (DP) with immunization, i.e. with different probabilities for the first infection and subsequent infections. The immunization effect leads to an additional non-Markovian term in the corresponding field theoretical action. We consider immunization as a small perturbation around the DP fixed point in d<6, where the non-Markovian term is relevant. The immunization causes the system to be driven away from the neighbourhood of the DP critical point. In order to investigate the dynamical critical behaviour of the model, we consider the limits of low and high first infection rate, while the second infection rate remains constant at the DP critical value. Scaling arguments are applied to obtain an expression for the survival probability in both limits. The corresponding exponents are written in terms of the critical exponents for ordinary DP and DP with a wall. We find that the survival probability does not obey a power law behaviour, decaying instead as a stretched exponential in the low first infection probability limit and to a constant in the high first infection probability limit. The theoretical predictions are confirmed by optimized numerical simulations in 1+1 dimensions.Comment: 12 pages, 11 figures. v.2: minor correction

Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps

We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, that corresponds to the limit $N\to 0$ of an $N$-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions concerning the critical crossover functions, finding a good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal lscrossover behavior of our data for any finite range.Comment: 43 pages, revte

Monte Carlo computation of correlation times of independent relaxation modes at criticality

We investigate aspects of universality of Glauber critical dynamics in two dimensions. We compute the critical exponent $z$ and numerically corroborate its universality for three different models in the static Ising universality class and for five independent relaxation modes. We also present evidence for universality of amplitude ratios, which shows that, as far as dynamic behavior is concerned, each model in a given universality class is characterized by a single non-universal metric factor which determines the overall time scale. This paper also discusses in detail the variational and projection methods that are used to compute relaxation times with high accuracy

Nonequilibrium relaxation of the two-dimensional Ising model: Series-expansion and Monte Carlo studies

We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time t and by Monte Carlo simulation. Both the magnetization (m) and energy series are obtained up to 12-th order. An accurate estimate from series analysis for the dynamical critical exponent z is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, z_eff(t) = - {1/8} d ln t /d ln m, directly from a ratio of three-spin correlation to m. Extrapolation to t = infinity leads to an estimate z = 2.169 +/- 0.003.Comment: 9 pages including 2 figure

Geodiversity assessment of Paraná state (Brazil): an innovative approach

Geodiversity is considered as the natural range of geological, geomorphological, and soil features, including their assemblages, relationships, properties, interpretations, and systems. A method developed for the quantitative assessment of geodiversity was applied to Parana ́ , a Brazilian state with an area of about 200,000 km2. The method is based on the overlay of a grid over different maps at scales ranging from 1/500,000 to 1/650,000, with the final Geodiversity Index the sum of five partial indexes calculated on a 25 9 25 km grid. The partial indexes represent the main components of geodi- versity, including geology (stratigraphy and lithology), geomorphology, paleontology, and soils. The fifth partial index covers mineral occurrences of geodiversity, such precious stones and metals, energy and industrial minerals, mineral waters, and springs. The Geodiversity Index takes the form of an isoline map that can be used as a tool in land-use planning, particularly in identifying priority areas for conservation, management, and use of natural resources at the state level.The Portuguese authors express their gratitude for the financial support given by the Fundacao para a Ciencia e a Tecnologia to the Centro de Geologia da Universidade do Porto, which partially supports this research. The Brazilian author expresses his gratitude for the financial support given by the CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) (Process Number 200074/2011-3)