Analytical approach to directed sandpile models on the Apollonian network

We investigate a set of directed sandpile models on the Apollonian network, which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659 (1989)) for Euclidian lattices. They are characterized by a single parameter $q$, that restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two and three point correlation functions are amenable to exact treatment, leading to analytical results for the avalanche distributions in the limit of an infinite system, for $q=1,2$. The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems, when larger values of $q$ are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.Comment: 7 pages in two-column format, 10 illustrations, 5 figure

Basin entropy behavior in a cyclic model of the rock-paper-scissors type

We deal with stochastic network simulations in a model with three distinct species that compete under cyclic rules which are similar to the rules of the popular rock-paper-scissors game. We investigate the Hamming distance density and then the basin entropy behavior, running the simulations for some typical values of the parameters mobility, predation and reproduction and for very long time evolutions. The results show that the basin entropy is another interesting tool of current interest to investigate chaotic features of the network simulations that are usually considered to describe aspects of biodiversity in the cyclic three-species model.Comment: 7 pages, 7 figures, 2 tables. To appear in EP

Future dynamics in f(R) theories

The $f(R)$ gravity theories provide an alternative way to explain the current cosmic acceleration without invoking dark energy matter component. However, the freedom in the choice of the functional forms of $f(R)$ gives rise to the problem of how to constrain and break the degeneracy among these gravity theories on theoretical and/or observational grounds. In this paper to proceed further with the investigation on the potentialities, difficulties and limitations of $f(R)$ gravity, we examine the question as to whether the future dynamics can be used to break the degeneracy between $f(R)$ gravity theories by investigating the future dynamics of spatially homogeneous and isotropic dust flat models in two $f(R)$ gravity theories, namely the well known $f(R) = R + \alpha R^{n}$ gravity and another by A. Aviles et al., whose motivation comes from the cosmographic approach to $f(R)$ gravity. To this end we perform a detailed numerical study of the future dynamic of these flat model in these theories taking into account the recent constraints on the cosmological parameters made by the Planck team. We show that besides being powerful for discriminating between $f(R)$ gravity theories, the future dynamics technique can also be used to determine the fate of the Universe in the framework of these $f(R)$ gravity theories. Moreover, there emerges from our numerical analysis that if we do not invoke a dark energy component with equation-of-state parameter $\omega < -1$ one still has dust flat FLRW solution with a big rip, if gravity deviates from general relativity via $f(R) = R + \alpha R^n$. We also show that FLRW dust solutions with $f''<0$ do not necessarily lead to singularity.Comment: 12 pages, 8 figures. V2: Generality and implications of the results are emphasized, connection with the recent literature improved, typos corrected, references adde

Non-nequilibrium model on Apollonian networks

We investigate the Majority-Vote Model with two states ($-1,+1$) and a noise $q$ on Apollonian networks. The main result found here is the presence of the phase transition as a function of the noise parameter $q$. We also studies de effect of redirecting a fraction $p$ of the links of the network. By means of Monte Carlo simulations, we obtained the exponent ratio $\gamma/\nu$, $\beta/\nu$, and $1/\nu$ for several values of rewiring probability $p$. The critical noise was determined $q_{c}$ and $U^{*}$ also was calculated. The effective dimensionality of the system was observed to be independent on $p$, and the value $D_{eff} \approx1.0$ is observed for these networks. Previous results on the Ising model in Apollonian Networks have reported no presence of a phase transition. Therefore, the results present here demonstrate that the Majority-Vote Model belongs to a different universality class as the equilibrium Ising Model on Apollonian Network.Comment: 5 pages, 5 figure

Magnetic models on Apollonian networks

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering an Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices. Numerical iteration of this set of maps leads to exact values for the thermodynamic properties of the model. Different choices for the coupling constants between only nearest neighbors along the lattice are taken into account. For both ferromagnetic and anti-ferromagnetic constants, long range magnetic ordering is obtained. With exception of a size dependent effective critical behavior of the correlation length, no evidence of asymptotic criticality was detected.Comment: 21 pages, 5 figure

Physical regularization for the spin-1/2 Aharonov-Bohm problem in conical space

We examine the bound state and scattering problem of a spin-one-half particle undergone to an Aharonov-Bohm potential in a conical space in the nonrelativistic limit. The crucial problem of the \delta-function singularity coming from the Zeeman spin interaction with the magnetic flux tube is solved through the self-adjoint extension method. Using two different approaches already known in the literature, both based on the self-adjoint extension method, we obtain the self-adjoint extension parameter to the bound state and scattering scenarios in terms of the physics of the problem. It is shown that such a parameter is the same for both situations. The method is general and is suitable for any quantum system with a singular Hamiltonian that has bound and scattering states.Comment: Revtex4, 5 pages, published versio