Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Sandpiles subjected to sinusoidal drive.

This paper considers a sandpile model subjected to a sinusoidal external drive with the period T. We develop a theoretical model for the Green's function in a large T limit, which predicts that the avalanches are anisotropic and elongated in the oscillation direction. We track the problem numerically and show that the system additionally shows a regime where the avalanches are elongated in the perpendicular direction with respect to the oscillations. We find a crossover between these two regimes. The power spectrum of avalanche size and the grains wasted from the parallel and perpendicular directions are studied. These functions show power-law behavior in terms of the frequency with exponents, which run with T.

Edwards-Wilkinson depinning transition in fractional Brownian motion background.

There are various reports about the critical exponents associated with the depinning transition. In this study, we investigate how the disorder strength present in the support can account for this diversity. Specifically, we examine the depinning transition in the quenched Edwards-Wilkinson (QEW) model on a correlated square lattice, where the correlations are modeled using fractional Brownian motion (FBM) with a Hurst exponent of H.We identify a crossover time [Formula: see text] that separates the dynamics into two distinct regimes: for [Formula: see text], we observe the typical behavior of pinned surfaces, while for [Formula: see text], the behavior differs. We introduce a novel three-variable scaling function that governs the depinning transition for all considered H values. The associated critical exponents exhibit a continuous variation with H, displaying distinct behaviors for anti-correlated ([Formula: see text]) and correlated ([Formula: see text]) cases. The critical driving force decreases with increasing H, as the host medium becomes smoother for higher H values, facilitating fluid mobility. This fact causes the asymptotic velocity exponent [Formula: see text] to increase monotonically with H.

Relation between the degree and betweenness centrality distribution in complex networks.

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.

Stochastic and deterministic dynamics in networks with excitable nodes.

Networks of excitable systems provide a flexible and tractable model for various phenomena in biology, social sciences, and physics. A large class of such models undergo a continuous phase transition as the excitability of the nodes is increased. However, models of excitability that result in this continuous phase transition are based implicitly on the assumption that the probability that a node gets excited, its transfer function, is linear for small inputs. In this paper, we consider the effect of cooperative excitations, and more generally the case of a nonlinear transfer function, on the collective dynamics of networks of excitable systems. We find that the introduction of any amount of nonlinearity changes qualitatively the dynamical properties of the system, inducing a discontinuous phase transition and hysteresis. We develop a mean-field theory that allows us to understand the features of the dynamics with a one-dimensional map. We also study theoretically and numerically finite-size effects by examining the fate of initial conditions where only one node is excited in large but finite networks. Our results show that nonlinear transfer functions result in a rich effective phase diagram for finite networks, and that one should be careful when interpreting predictions of models that assume noncooperative excitations.

A self-organized critical model and multifractal analysis for earthquakes in Central Alborz, Iran.

This paper is devoted to a phenomenological study of the earthquakes in central Alborz, Iran. Using three observational quantities, namely the weight function, the quality factor, and the velocity model in this region, we develop a modified dissipative sandpile model which captures the main features of the system, especially the average activity field over the region of study. The model is based on external stimuli, the location of which is chosen (I) randomly, (II) on the faults, (III) on the low active points, (IV) on the moderately active points, and (V) on the highly active points in the region. We uncover some universal behaviors depending slightly on the method of external stimuli. A multi-fractal detrended fluctuation analysis is exploited to extract the spectrum of the Hurst exponent of the time series obtained by each of these schemes. Although the average Hurst exponent depends slightly on the method of stimuli, we numerically show that in all cases it is lower than 0.5, reflecting the anti-correlated nature of the system. The lowest average Hurst exponent is found to be associated with the case (V), in such a way that the more active the stimulated sites are, the lower the average Hurst exponent is obtained, i.e. the large earthquakes are more anticorrelated. Moreover, we find that the activity field achieved in this study provide information about the depth and topography of the basement, and also the area that can potentially be the location of the future large events. We successfully determine a high activity zone on the Mosha Fault, where the mainshock occurred on May 7th, 2020 (M[Formula: see text] 4.9).

Self-similar but not conformally invariant traces obtained by modified Loewner forces.

The two-dimensional Loewner exploration process is generalized to the case where the random force is self-similar with positively correlated increments. We model this random force by a fractional Brownian motion with Hurst exponent H≥1/2≡H_{BM}, where H_{BM} stands for the one-dimensional Brownian motion. By manipulating the deterministic force, we design a scale-invariant equation describing self-similar traces which lack conformal invariance. The model is investigated in terms of the "input diffusivity parameter" κ, which coincides with the one of the ordinary Schramm-Loewner evolution (SLE) at H=H_{BM}. In our numerical investigation, we focus on the scaling properties of the traces generated for κ=2,3, κ=4, and κ=6,8 as the representatives, respectively, of the dilute phase, the transition point, and the dense phase of the ordinary SLE. The resulting traces are shown to be scale invariant. Using two equivalent schemes, we extract the fractal dimension, D_{f}(H), of the traces which decrease monotonically with increasing H, reaching D_{f}=1 at H=1 for all κ values. The left passage probability (LPP) test demonstrates that, for H values not far from the uncorrelated case (small ε_{H}≡H-H_{BM}/H_{BM}), the prediction of the ordinary SLE is applicable with an effective diffusivity parameter κ_{eff}. Not surprisingly, the κ_{eff}'s do not fulfill the prediction of SLE for the relation between D_{f}(H) and the diffusivity parameter.

Self-repelling bipedal exploration process.

A self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters ß_{d} and ß_{h}, so that the former controls the distance (d) between the legs positions, and the latter controls the statistics of self-crossing of the traversed paths. The probability measure for local movements is supposed to be the one for the "true self-avoiding walk" multiplied by a factor exponentially decaying with d. After a transient behavior for short times, a variety of behaviors have been observed for large times depending on the value of ß_{d} and ß_{h}. Our statistical analysis reveals that the system undergoes a crossover between two (small and large ß_{d}) regimes identified in large times (t). In the small ß_{d} regime, the random walkers (identified by the position of the legs of the spider) remain on average in a fixed nonzero distance in the large time limit, whereas in the second regime (large ß_{d}), the absorbing force between the walkers dominates the other stochastic forces. In the latter regime, d decays in a power-law fashion with the logarithm of time. When the system is mapped to a growth process (represented by a height field which is identified by the number of visits for each point), the roughness and the average height show different behaviors in two regimes, i.e., they show a power law with respect to t in the first regime and logt in the second regime. The fractal dimension of the random walker traces and the winding angle are shown to consistently undergo a similar crossover.

Persistent homology of fractional Gaussian noise.

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.

Self-organized criticality in cumulus clouds.

The shape of clouds has proven to be essential for classifying them. Our analysis of images from fair weather cumulus clouds reveals that, in addition to turbulence, they are driven by self-organized criticality. Our observations yield exponents that support the fact the clouds, when projected to two dimensions, exhibit conformal symmetry compatible with c=-2 conformal field theory. By using a combination of the Navier-Stokes equation, diffusion equations, and a coupled map lattice, we successfully simulated cloud formation, and obtained the same exponents.

Role of anaxonic local neurons in the crossover to continuously varying exponents for avalanche activity.

Local anaxonic neurons with graded potential release are important ingredients of nervous systems, present in the olfactory bulb system of mammalians and in the human visual system, as well as in arthropods and nematodes. We develop a neuronal network model including both axonic and anaxonic neurons and monitor the activity tuned by the following parameters: the decay length of the graded potential in local neurons, the fraction of local neurons, the largest eigenvalue of the adjacency matrix, and the range of connections of the local neurons. Tuning the fraction of local neurons, we derive the phase diagram including two transition lines: a critical line separating subcritical and supercritical regions, characterized by power-law distributions of avalanche sizes and durations, and a bifurcation line. We find that the overall behavior of the system is controlled by a parameter tuning the relevance of local neuron transmission with respect to the axonal one. The statistical properties of spontaneous activity are affected by local neurons at large fractions and on the condition that the graded potential transmission dominates the axonal one. In this case the scaling properties of spontaneous activity exhibit continuously varying exponents, rather than the mean-field branching model universality class.

Superstatistical two-temperature Ising model.

The previous approach of the nonequilibrium Ising model was based on the local temperature in which each site or part of the system has its own specific temperature. We introduce an approach of the two-temperature Ising model as a prototype of the superstatistic critical phenomena. The model is described by two temperatures (T_{1},T_{2}) in a zero magnetic field. To predict the phase diagram and numerically estimate the exponents, we develop the Metropolis and Swendsen-Wang Monte Carlo method. We observe that there is a nontrivial critical line, separating ordered and disordered phases. We propose an analytic equation for the critical line in the phase diagram. Our numerical estimation of the critical exponents illustrates that all points on the critical line belong to the ordinary Ising universality class.

Invasion percolation in short-range and long-range disorder background.

In the original invasion percolation model, a random number quantifies the role of necks, or generally the quality of pores, ignoring the structure of pores and impermeable regions (to which the invader cannot enter). In this paper, we investigate invasion percolation (IP), taking into account the impermeable regions, the configuration of which is modeled by ordinary and Ising-correlated site percolation (with short-range interactions, SRI), on top of which the IP dynamics is defined. We model the long-ranged correlations of pores by a random Coulomb potential (RCP). By examining various dynamical observables, we suggest that the critical exponents of Ising-correlated cases change considerably only in the vicinity of the critical point (critical temperature), while for the ordinary percolation case the exponents are robust against the occupancy parameter p. The properties of the model for the long-range interactions [LRI (RCP)] are completely different from the normal IP. In particular, the fractal dimension of the external frontier of the largest hole is nearly 4/3 for SRI far from the critical points, which is compatible with normal IP, while it converges to 1.099±0.04 for RCP. For the latter case, the time dependence of our observables is divided into three parts: the power law (short time), the logarithmic (moderate time), and the linear (long time) regimes. The second crossover time is shown to go to infinity in the thermodynamic limit, whereas the first crossover time is nearly unchanged, signaling the dominance of the logarithmic regime. The average gyration radius of the growing clusters, the length of their external perimeter, and the corresponding roughness are shown to be nearly constant for the long-time regime in the thermodynamic limit.

Edwards-Wilkinson depinning transition in random Coulomb potential background.

The quenched Edwards-Wilkinson growth of the 1+1 interface is considered in the background of the correlated random noise. We use random Coulomb potential as the background long-range correlated noise. A depinning transition is observed in a critical driving force F[over ̃]_{c}≈0.037 (in terms of disorder strength unit) in the vicinity of which the final velocity of the interface varies linearly with time. Our data collapse analysis for the velocity shows a crossover time t^{*} at which the velocity is size independent. Based on a two-variable scaling analysis, we extract the exponents, which are different from all universality classes we are aware of. Especially noting that the dynamic and roughness exponents are z_{w}=1.55±0.05, and α_{w}=1.05±0.05 at the criticality, we conclude that the system is different from both Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) universality classes. Our analysis shows therefore that making the noise long-range correlated, drives the system out of the EW universality class. The simulations on the tilted lattice show that the nonlinearity term (λ term in the KPZ equations) goes to zero in the thermodynamic limit.

Geometry-induced nonequilibrium phase transition in sandpiles.

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treated as the control parameter. By analyzing the three-dimensional avalanches and their two-dimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOC_{T=∞}), along with a tricritical point (at T_{c}, the critical temperature of the host) between them. The transition between these two criticalities is induced by the transition in the support. The SOC_{T=∞} universality class is characterized by the exponent of the avalanche size distribution τ^{T=∞}=1.27±0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.4705]. The tricritical point is characterized by its own critical exponents. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP), and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of ε≡T-T_{c}/T_{c}, defining further critical exponents. Importantly, the finite-size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents ß=0.19±0.02 and ν=0.75±0.05. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against T with the numerical value D_{f}=1.25±0.01.

Elastic backbone phase transition in the Ising model.

The two-dimensional (zero magnetic field) Ising model is known to undergo a second-order paraferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature T_{eb}

Local smoothing in sandpiles: Spanning avalanches, bifurcation, and temporal oscillations.

The manipulation of the self-organized critical systems by repeatedly deliberate local relaxations (local smoothing) is considered. During a local smoothing, the grains diffuse to the neighboring regions, causing a smoothening of the height filed over the system. The local smoothings are controlled by a parameter ζ which is related to the number of local smoothening events in an avalanche. The system shows some new (mass and time) scales, leading to some oscillatory behaviors. A bifurcation occurs at some ζ value, above which some oscillations are observed for the mean number of grains, and also in the autocorrelation functions. These oscillations are associated with spanning avalanches which are due to the accumulation of grains in the smoothed system. The analysis of the rare event waiting time confirms also the appearance of a new time scale.

Interaction-disorder-driven characteristic momentum in graphene, approach of multi-body distribution functions.

Multi-point probability measures along with the dielectric function of Dirac Fermions in mono-layer graphene containing particle-particle and white-noise (out-plane) disorder interactions on an equal footing in the Thomas-Fermi-Dirac approximation is investigated. By calculating the one-body carrier density probability measure of the graphene sheet, we show that the density fluctuation (ζ-1) is related to the disorder strength (ni), the interaction parameter (rs) and the average density ([Formula: see text]) via the relation [Formula: see text] for which [Formula: see text] leads to strong density inhomogeneities, i.e. electron-hole puddles (EHPs), in agreement with the previous works. The general equation governing the two-body distribution probability is obtained and analyzed. We present the analytical solution for some limits which is used for calculating density-density response function. We show that the resulting function shows power-law behaviors in terms of ζ with fractional exponents which are reported. The disorder-averaged polarization operator is shown to be a decreasing function of momentum like ordinary 2D parabolic band systems. It is seen that a disorder-driven momentum qch emerges in the system which controls the behaviors of the screened potential. We show that in small densities an instability occurs in which imaginary part of the dielectric function becomes negative and the screened potential changes sign. Corresponding to this instability, some oscillations in charge density along with a screening-anti-screening transition are observed. These effects become dominant in very low densities, strong disorders and strong interactions, the state in which EHPs appear. The total charge probability measure is another quantity which has been investigated in this paper. The resulting equation is analytically solved for large carrier densities, which admits the calculation of arbitrary-point correlation function.

Scaling properties of monolayer graphene away from the Dirac point.

The statistical properties of the carrier density profile of graphene in the ground state in the presence of particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi et al. [Phys. Rev. E 95, 032112 (2017)2470-004510.1103/PhysRevE.95.032112]. The nonzero chemical potential (µ) in gated graphene has nontrivial effects on electron-hole puddles, since it generates mass in the Dirac action and destroys the scaling behaviors of the effective Thomas-Fermi-Dirac theory. We provide detailed analysis on the resulting spatially inhomogeneous system in the framework of the Thomas-Fermi-Dirac theory for the Gaussian (white noise) disorder potential. We show that the chemical potential in this system as a random surface destroys the self-similarity, and also the charge field is non-Gaussian. We find that the two-body correlation functions are factorized to two terms: a pure function of the chemical potential and a pure function of the distance. The spatial dependence of these correlation functions is double logarithmic, e.g., the two-point density correlation behaves like D_{2}(r,µ)∝µ^{2}exp[-(-a_{D}lnlnr^{ß_{D}})^{α_{D}}] (α_{D}=1.82, ß_{D}=0.263, and a_{D}=0.955). The Fourier power spectrum function also behaves like ln[S(q)]=-ß_{S}^{-a_{S}}(lnq)^{a_{S}}+2lnµ (a_{S}=3.0±0.1 and ß_{S}=2.08±0.03) in contrast to the ordinary Gaussian rough surfaces for which a_{S}=1 and ß_{S}=1/2(1+α)^{-1} (α being the roughness exponent). The geometrical properties are, however, similar to the ungated (µ=0) case, with the exponents that are reported in the text.

Statistical investigation of avalanches of three-dimensional small-world networks and their boundary and bulk cross-sections.

In many situations we are interested in the propagation of energy in some portions of a three-dimensional system with dilute long-range links. In this paper, a sandpile model is defined on the three-dimensional small-world network with real dissipative boundaries and the energy propagation is studied in three dimensions as well as the two-dimensional cross-sections. Two types of cross-sections are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross-section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long-range links (α) increases. This trend is numerically shown to be a power law in a manner described in the paper. Two regimes of α are considered in this work. For sufficiently small αs the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is nontrivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with α just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) m^{2}∼α for the three-dimensional system as well as its two-dimensional cross-sections.