Calculation of Spectral Dimension on Inhomogeneous Fractal Lattices

We propose a scheme to derive the spectral dimension of inhomogeneous fractal lattice via renormalization procedure, in which the distribution of masses on the sites of fractals is introduced. The spectral dimension of diamond-type hierarchical lattice and Sierpinski gasket with b = 3 are re-investigated in this way. Moreover, the variants of Sierpinski gaskettype fractals are studied, the results show that the spectral dimension is independent of the details of internal structure of fractal, and thus implies the existence of universality. The source of the universality is also analyzed.     

Random Sierpinski network with scale-free small-world and modular structure

In this paper, we define a stochastic Sierpinski gasket, on the basis of which we construct a network called random Sierpinski network (RSN). We investigate analytically or numerically the statistical characteristics of RSN. The obtained results reveal that the properties of RSN is particularly rich, it is simultaneously scale-free, small-world, uncorrelated, modular, and maximal planar. All obtained analytical predictions are successfully contrasted with extensive numerical simulations. Our network representation method could be applied to study the complexity of some real systems in biological and information fields.     

Comment on "Critical behavior of the chain-generating function of self-avoiding walks on the sierpinski gasket family: the euclidean limit"

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.     

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

     

A physically based connection between fractional calculus and fractal geometry

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.     

Ising models on the lattice Sierpinski gasket

Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.     

Voter model on Sierpinski fractals

We investigate the ordering of voter model on fractal lattices: Sierpinski Carpets and Sierpinski Gasket. We obtain a power-law ordering in all cases, but the dynamics is found to differ significantly for finite and infinite ramification order of investigated fractals.     

Price dynamics from a simple multiplicative random process model

The existence of stylized facts suggests that there might be `universal' mechanism which drives price evolution on financial markets in general. Based on empirical estimates of 10 major indices, we propose a stylized model of endogenous price formation on an aggregate level whose key issue is that price evolution is driven by the `market's' expectations about future growth rates of investment. The model is a multiplicative random process with a stochastic, state-dependent growth rate which establishes a negative feedback component in the price dynamics which admits some far reaching formal analysis. Generated return trails exhibit statistical properties such as 'volatility clustering', multi scaling, and a non-Gaussian distribution which is in quantitative in agreement with stylized facts from empirical asset returns. Additionally non-equilibrium entropies are also considered. These results suggests that the structure of the model mimicks a mechanism which is essential in driving price dynamics of financial markets in general.     

Black-Scholes Formula in Subdiffusive Regime

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.     

Hydrodynamic Limit for a Zero-Range Process in the Sierpinski Gasket

We consider a system of random walks on graph approximations of the Sierpinski gasket, coupled with a zero-range interaction. We prove that the hydrodynamic limit of this system is given by a nonlinear heat equation on the Sierpinski gasket.     

Critical Behavior of Two Interacting Linear Polymer Chains in a Good Solvent

A model of two interacting (chemically different) linear polymer chains is solved exactly using the real-space renormalization group transformation on a family of Sierpinski gasket type fractals and on a truncated 4-simplex lattice. The members of the family of the Sierpinski gasket-type fractals are characterized by an integer scale factorb which runs from 2 to ∞. The Hausdorff dimensiond _F of these fractals tends to 2 from below asb → ∞. We calculate the contact exponenty for the transition from the State of segregation to a State in which the two chains are entangled forb = 2-5. Using arguments based on the finite-size scaling theory, we show that forb→∞, y = 2 - v(b) d _F, wherev is the end-toend distance exponent of a chain. For a truncated 4-simplex lattice it is shown that the system of two chains either remains in a State in which these chains are intermingled in such a way that they cannot be told apart, in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system, or in a State in which they are either zipped or entangled. We show the region of existence of these different phases separated by tricritical lines. The value of the contact exponenty is calculated at the tricritical points.     

Some Two-Point Resistances of the Sierpinski Gasket Network

In this paper, we use the principle of substitution to replace sub-gaskets of the Sierpinski gasket network by an equivalent Y-network which enables the use of only the Delta–Wye transformation and the series and parallel principles to derive some two-point resistances of the Sierpinski gasket network with dimension two.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

Incompatibility networks as models of scale-free small-world graphs

We make a mapping from Sierpinski fractals to a new class of networks, the incompatibility networks, which are scale-free, small-world, disassortative, and maximal planar graphs. Some relevant characteristics of the networks such as degree distribution, clustering coefficient, average path length, and degree correlations are computed analytically and found to be peculiarly rich. The method of network representation can be applied to some real-life systems making it possible to study the complexity of real networked systems within the framework of complex network theory.     

Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system's dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.     

Damage-spreading dynamic scaling for the ising model on the sierpinski gasket fractal

We study the relaxation towards equilibrium of the ferromagnetic Ising model on the Sierpinski gasket, which is a fractal lattice. We do this by performing Monte Carlo simulations, based on the heat-bath dynamics, and investigating the time evolution of the Hamming distance between two different configurations of the model. Starting with an initial damage created in all lattice sites, we calculate the average values of two quantities that characterize the relaxation process: the nonlinear damage relaxation time (tau), and the time for all sites to be undamaged at least once (tau(c)). We find that tau diverges, at low temperatures, with a dynamical exponent z which depends linearly on the inverse of temperature, as predicted by a generalized scaling theory developed by Henley. There is a complete breakdown of scaling for tau(c).     

Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus the Hurst index

Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst indexes H∈(0,1). Special attention has been paid to the impact of the Hurst index on topological properties. It is found that the clustering coefficient C decreases when H increases. We also found that the mean length L of the shortest paths increases exponentially with H for fixed length N of the original time series. In addition, L increases linearly with respect to N when H is close to 1 and in a logarithmic form when H is close to 0. Although the occurrence of different motifs changes with H, the motif rank pattern remains unchanged for different H. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension d_B decreases with H. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with degree, which indicate that the horizontal visibility graphs are assortative. With the increase of H, the Pearson coefficient decreases first and then increases, in which the turning point is around H=0.6. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.     

Are crude oil markets multifractal? Evidence from MF-DFA and MF-SSA perspectives

In this article, we investigated the multifractality and its underlying formation mechanisms in international crude oil markets, namely, Brent and WTI, which are the most important oil pricing benchmarks globally. We attempt to find the answers to the following questions: (1) Are those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? To answer these questions, we applied both multifractal detrended fluctuation analysis (MF-DFA) and multifractal singular spectrum analysis (MF-SSA) based on the partition function, two widely used multifractality detecting methods. We found that both markets exhibit multifractal properties by means of these methods. Furthermore, in order to identify the underlying formation mechanisms of multifractal features, we destroyed the underlying nonlinear temporal correlation by shuffling the original time series; thus, we identified that the causes of the multifractality are influenced mainly by a nonlinear temporal correlation mechanism instead of a non-Gaussian distribution. At last, by tracking the evolution of left- and right-half multifractal spectra, we found that the dynamics of the large price fluctuations is significantly different from that of the small ones. Our main contribution is that we not only provided empirical evidence of the existence of multifractality in the markets, but also the sources of multifractality and plausible explanations to current literature; furthermore, we investigated the different dynamical price behaviors influenced by large and small price fluctuations.     

On the inclusion of collisional correlations in quantum dynamics

We present a formalism to describe collisional correlations responsible for thermalization effects in finite quantum systems. The approach consists in a stochastic extension of time dependent mean field theory. Correlations are treated in time dependent perturbation theory and loss of coherence is assumed at some time intervals allowing a stochastic reduction of the correlated dynamics in terms of a stochastic ensemble of time dependent mean-fields. This theory was formulated long ago in terms of density matrices but never applied in practical cases because of its complexity. We propose here a reformulation of the theory in terms of wave functions and use a simplified 1D model of cluster and molecules allowing to test the theory in a schematic but realistic manner. We illustrate the performance in terms of several observables, in particular global moments of the density matrix and single particle entropy built on occupation numbers. The occupation numbers remain fixed in time dependent mean-field propagation and change when evaluating the correlations, then taking fractional values. They converge asymptotically towards Fermi distributions which is a clear indication of thermalization.