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Cesar Augusto Sangaletti Tercariol

Publications and source records attributed to Cesar Augusto Sangaletti Tercariol.

9 recordsLinked to original sources
arXiv2008

Generalized Exponential Function and some of its Applications to Complex Systems

From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing complex systems can be conveniently written in terms of this generalization of the exponential function. The gamma function is then generalized and we generalize the factorial operation. Also a very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. One obtains, as particular cases, the generalized error function, the Zipf-Mandelbrot probability density function (pdf), the generalized gaussian and Laplace pdf. One can also obtain analytically their cumulative functions and moments.

physics.data-an↗
arXiv2008

The influence of memory in deterministic walks in random media: analytical calculation within a mean field approximation

Consider a random medium consisting of points randomly distributed so that there is no correlation among the distances. This is the random link model, which is the high dimensionality limit (mean field approximation) for the euclidean random point structure. In the random link model, at discrete time steps, the walker moves to the nearest site, which has not been visited in the last $μ$ steps (memory), producing a deterministic partially self avoiding walk (the tourist walk). We have obtained analitically the distribution of the number $n$ of points explored by a walker with memory $μ= 2$, as well as the transient and period joint distribution. This result enables to explain the abrupt change in the exploratory behavior between the cases $μ= 1$ (memoryless, driven by extremal statistics) and $μ= 2$ (with memory, driven by combinatorial statistics). In the $μ= 1$ case, the mean newly visited points in the thermodynamic limit $(N \gg 1)$ is just $ = e = 2.72 ...$ while in the $μ= 2$ case, the mean number $ $ of visited points is proportional to $N^{1/2}$. Also, this result allows us to stabilish an equivalence between the random link model with $μ=2$ and random map (uncorrelated back and forth distances) with $μ=0$ and the drastic change between the cases where the transient time is null compared to non-null transient times.

cond-mat.stat-mech↗
arXiv2008

Continuous growth models in terms of generalized logarithm and exponential functions

Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of non-extensive thermostatistics. We show that these functions are suitable to describe and unify the great majority of continuous growth models, which we briefly review. Physical interpretation to the generalization function parameter is given for the Richards' model, which has an underlying microscopic model to justify it.

q-fin.GN↗
arXiv2007

Exact analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media

Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point which has not been visited in the preceding $μ$ steps. Using open boundary conditions, we have calculated analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited, with $N \gg μ\gg 1$. This approximated expression for $P_N(μ)$ is reasonable even for small $N$ and $μ$ values, as validated by Monte Carlo simulations. We show the existence of a critical memory $μ_1 = \ln N/\ln 2$. For $μ< μ_1 - e/(2\ln2)$, the walker gets trapped in cycles and does not fully explore the system. For $μ> μ_1 + e/(2\ln2)$ the walker explores the whole system. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker needs not to have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order $\log_{2} N$.

cond-mat.dis-nn↗
arXiv2006

Optimum exploration memory and anomalous diffusion in deterministic partially self-avoiding walks in one-dimensional random media

Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding $μ$ steps. We have obtained analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited in this open system, with $N \gg μ\gg 1$. The expression for $P_N(μ)$ evaluated in the mentioned limit is valid even for small $N$ and leads to a transition region centered at $μ_1 = \ln N/\ln 2$ and with width $ε= e/\ln2$. For $μ< μ_1 - ε/2$, the walker gets trapped in cycles and does not fully explore the system. For $μ> μ_1 + ε/2$ the walker explores the whole system. In both cases the walker presents diffusive behavior. Nevertheless, in the intermediate regime $μ\sim μ_1 \pm ε/2$, the walker presents anoumalous diffusion behavior. Since the intermediate region increases as $\ln N$ and its width is constant, a sharp transition is obtained for one-dimensional large systems. The walker does not need to have full memory of its trajectory to explore the whole system, it suffices to have memory of order $μ_1$.

cond-mat.dis-nn↗
arXiv2006

Analytical calculation of neighborhood order probabilities for high dimensional Poissonic processes and mean field models

Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest neighbor (Cox probabilities) plays an important role in spatial statistics. Also, it has been useful in the description of physical processes in disordered media. Here we propose a simpler derivation of Cox probabilities, where we stress the role played by the system dimensionality $d$. In the limit $d \to \infty$, the distances between pair of points become indenpendent (random link model) and closed analytical forms for the neighborhood probabilities are obtained both for the thermodynamic limit and finite-size system. Breaking the distance symmetry constraint drives us to the random map model, for which the Cox probabilities are obtained for two cases: whether a point is its own nearest neighbor or not.

cond-mat.dis-nn↗
arXiv2005

An efficient algorithm to generate large random uncorrelated Euclidean distances: the random link model

A disordered medium is often constructed by $N$ points independently and identically distributed in a $d$-dimensional hyperspace. Characteristics related to the statistics of this system is known as the random point problem. As $d \to \infty$, the distances between two points become independent random variables, leading to its mean field description: the random link model. While the numerical treatment of large random point problems pose no major difficulty, the same is not true for large random link systems due to Euclidean restrictions. Exploring the deterministic nature of the congruential pseudo-random number generators, we present techniques which allow the consideration of models with memory consumption of order O(N), instead of $O(N^2)$ in a naive implementation but with the same time dependence $O(N^2)$.

cond-mat.dis-nn↗
arXiv2005

Analytical Results for the Statistical Distribution Related to Memoryless Deterministic Tourist Walk: Dimensionality Effect and Mean Field Models

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding μsteps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial non-periodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parameterized by the normalized incomplete beta function I_d = I_{1/4}[1/2,(d+1)/2]. The joint distribution S_{μ,d}^{(N)}(t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is: S_{1,d}^{(\infty)}(t,p) = [Γ(1+I_d^{-1}) (t+I_d^{-1})/Γ(t+p+I_d^{-1})] δ_{p,2}, where t=0,1,2,...,\infty, Γ(z) is the gamma function and δ_{i,j} is the Kronecker's delta. The mean field models are random link model, which corresponds to d \to \infty, and random map model which, even for μ= 0, presents non-trivial cycle distribution [S_{0,rm}^{(N)}(p) \propto p^{-1}]: S_{0,rm}^{(N)}(t,p) = Γ(N)/\{Γ[N+1-(t+p)]N^{t+p}\}. The fundamental quantities are the number of explored points n_e=t+p and I_d. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

cond-mat.dis-nn↗