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In this work, we study the effects of geometric confinement on the point statistics in a quasi-low-dimensional system. Specifically, we focus on the nearest-neighbor statistics. Accordingly, we have performed comprehensive numerical simulations of binomial point process on quasi-one-dimensional rectangle strips for different values of the confinement ratio defined as the ratio of the strip width to the mean nearest-neighbor distance. We found that the nearest-neighbor distance distributions (NNDDs) conform to an extreme value Weibull distribution with the shape parameter depending on the confinement ratio, while the process intensity remains constant. This finding reveals the reduction of effective spatial degrees of freedom in a quasi-low-dimensional system under the geometric confinement. The scale dependence of the number of effective spatial degrees of freedom is found to obey the crossover ansatz. We stress that the functional form of the crossover ansatz is determined by the nature of the studied point process. Accordingly, different physical processes in the quasi-low-dimensional system obey different crossover ansatzes. The relevance of these results for quasi-low-dimensional systems is briefly highlighted. [ABSTRACT FROM AUTHOR]

The main goal of this work is to develop a robust framework for an exhaustive description of essential properties of a fractal object. For this purpose, the inherent features of fractal sets are scrutinized. The topological, metrological, morphological, and topographical attributes of fractal systems are delineated. The criteria of the fractal connectedness are established. The characteristics of the fractal connectivity and ramification are ascertained. The index of the fractal loopiness is introduced. The quantifications of the fractal heterogeneity, lacunarity, and anisotropy are briefly sketched out. A set of key attributes which enable a proper characterization of fractal system are suggested. [ABSTRACT FROM AUTHOR]

In this work, we study the effects of geometric confinement on random walks and diffusion processes in systems of reduced dimensionality. Extensive Monte Carlo simulations of Gaussian random walks were performed on rectangular strips of infinite length. A special emphasis is made on the crossover from two- to one-dimensional diffusion in the Fickian regime. We found that the crossover behavior is controlled by the ratio of the strip width to the standard deviation of the walker step length distribution. Specifically, the characteristic time of crossover behavior scales quadratically with this ratio. Furthermore, the time dependence of the number of effective spatial degrees of freedom of the random walker on the strip is found to obey an ansatz characterized by the universal power-law exponent. This allows us to formulate the diffusion equation with the time dependent number of effective spatial degrees of freedom in the quasi-one-dimensional system. [ABSTRACT FROM AUTHOR]

This article explores the opinion dynamics of a double coalition opinion against a third opinion under majority rule updates on odd fixed size connected groups. For this purpose, coalition benefit criteria and three opinion formation models which extend the 2-state majority rule model on lattices are introduced. The proposed models focus on the coalition profit of its constituent coalition opinions and cover the possible final scenarios from coalition alliance perspective: either minor opinion or major opinion is favored, or dynamics do not favor to any coalition opinion. Opinion exchanges take place on a torus embedded lattice network of a 3-state system having in consideration tie configurations and two rules to break them: either by random choice or leaving ties unaltered. Models were analyzed in the statistical mechanics spirit through Monte Carlo simulations without node replacement. Estimations for coalition benefits, the growth of coalition ties, and consensus probabilities are reported. The loss of coalition strengths due to coalition ties and its indecision is indicated. In particular, the logistic decay of consensus probability is due to the logistic adaptive growth of coalition ties. Scaling behaviors for consensus time and coalition ties in terms of network size are suggested. The results of numerical simulations are discussed in the context of social influence and social dynamics. [ABSTRACT FROM AUTHOR]

Abstract This Letter is focused on the impact of network topology on the site percolation. Specifically, we study how the site percolation threshold depends on the network dimensions (topological d and fractal D), degree of connectivity (quantified by the mean coordination number Z ∞), and arrangement of bonds (characterized by the connectivity index Q also called the ramification exponent). Using the Fisher's containment principle, we established exact inequalities between percolation thresholds on fractal networks contained in the square lattice. The values of site percolation thresholds on some fractal lattices were found by numerical simulations. Our findings suggest that the most relevant parameters to describe properly the values of site percolation thresholds on fractal networks contained in square lattice (Sierpiński carpets and Cantor tartans) and based on the square lattice (weighted planar stochastic fractal and Cantor lattices) are the mean coordination number and ramification exponent, but not the fractal dimension. Accordingly, we propose an empirical formula providing a good approximation for the site percolation thresholds on these networks. We also put forward an empirical formula for the site percolation thresholds on d -dimensional simple hypercubic lattices. Highlights • Effects of network features on the site percolation is studied. • Percolation threshold strongly depends on connectivity degree and ramification. • Percolation threshold is independent of network fractal dimension. • Exact inequalities for percolation thresholds are established. • Formulas for estimate the site percolation thresholds are suggested. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Water discharges from Euclidean and fractal reservoirs are studied. • Inverse Menger sponge is used to model the fractal pore network. • Conditions for Torricelli and Darcy regimes of discharge are established. • Darcy's discharge from fractal reservoir is slower than from Euclidean one. • Torricelli's discharge from fractal pore network is faster than from periodic one. Abstract We perform a comparative study of water discharge from Euclidean and fractal reservoirs. The Euclidean reservoir is modeled by a filter with a translationally invariant cubic network of straight square channels. As a model of fractal reservoir with the scale invariant pore network we use a pre-fractal inverse Menger sponge. The gravity-driven discharge experiments were carried out with two different setups. In the first setup, a porous filter is fixed in container above a lateral exit duct. In the second setup, the exit duct is connected to one of lateral exits of the pore network. In both cases, we found that the squared outflow velocity is a linear function of water table elevation above exit orifice. Hence, the water discharge is governed by the Bernoulli's principle rather than by the Darcy's law. Furthermore, we found that filters placed above the exit duct have no effect on the discharge rate, because absolute permeabilities of both filters are large enough to supply water into the volume between filter's bottom and exit orifice. Conversely, when the exit duct is connected to the lateral exit of the pore network, the water discharge from the inverse Menger sponge is faster than from the Euclidean reservoir. Experimental data fittings with a modified Torricelli's equation suggest that a loss of mechanical energy of flowing water in the fractal pore network is less than in the translationally invariant network of channels. This finding reveals that an effective wetted area along a flow path of least resistance in the self-similar pore network is smaller less than in the periodic network of straight channels. So, the fractal features of the pore network can have a strong impact on hydrological processes in natural reservoirs and karst aquifers which commonly possess statistical scale invariance. [ABSTRACT FROM AUTHOR]

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan d s is equal to its fractal (self-similarity) dimension D . Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Lévy stable if D > 1.5 . Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Lévy walk leads to ballistic diffusion. The Lévy walk with rests leads to super-diffusion, if D > 3 , or sub-diffusion, if 1.5 < D < 3 . [ABSTRACT FROM AUTHOR]

There is no way to build a nontrivial Markov process having continuous trajectories on a totally disconnected fractal embedded in the Euclidean space. Accordingly, in order to delineate the diffusion process on the totally disconnected fractal, one needs to relax the continuum requirement. Consequently, a diffusion process depends on how the continuum requirement is handled. This explains the emergence of different types of anomalous diffusion on the same totally disconnected set. In this regard, we argue that the number of effective spatial degrees of freedom of a random walker on the totally disconnected Cantor set is equal to n s p = [ D ] + 1 , where [ D ] is the integer part of the Hausdorff dimension of the Cantor set. Conversely, the number of effective dynamical degrees of freedom ( d s ) depends on the definition of a Markov process on the totally disconnected Cantor set embedded in the Euclidean space E n ( n ≥ n s p ) . This allows us to deduce the equation of diffusion by employing the local differential operators on the F α -support. The exact solutions of this equation are obtained on the middle- ϵ Cantor sets for different kinds of the Markovian random processes. The relation of our findings to physical phenomena observed in complex systems is highlighted. [ABSTRACT FROM AUTHOR]

In this Letter, we emphasize that methods of fractal homogenization should take into account a loop structure of the fractal, as well as its connectivity and geodesic metric. The fractal attributes can be quantified by a set of dimension numbers. Accordingly, physical problems on fractals can be mapped onto the boundary values problems in the fractional-dimensional space with metric induced by the fractal topology. The solutions of these problems represent analytical envelopes of non-analytical functions defined on the fractal. Some examples are briefly discussed. The interplay between effects of fractal connectivity, loop structure, and mass distributions on electromagnetic fields in fractal media is highlighted. The effects of fractal connectivity, geodesic metric, and loop structure are outlined. [ABSTRACT FROM AUTHOR]

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff ( D H ), topological Hausdorff ( D t H ), and spectral ( d s ) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers ( D H , D t H , d s ), rather than solely by the spatial dimension ( d ). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension D t H , whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension D H . The effect of the network connectivity on the site percolation threshold is revealed. [ABSTRACT FROM AUTHOR]

In this Letter, the analytical expression of topological Hausdorff dimension D t H is derived for some kinds of infinitely ramified Sierpiński carpets. Furthermore, we deduce that the Hausdorff dimension of the union of all self-avoiding paths admitted on the infinitely ramified Sierpiński carpet has the Hausdorff dimension D H s a = D t H . We also put forward a phenomenological relation for the fractal dimension of the random walk on the infinitely ramified Sierpiński carpet. The effects of fractal attributes on the transport properties are highlighted. Possible applications of the developed tools are briefly outlined. [ABSTRACT FROM AUTHOR]

In this work, we prove that the topological Hausdorff dimension of critical percolation cluster (CPC) in two dimensions is equal to D t H = D r b + 1 = 7 / 4 , where D r b is the Hausdorff dimension of the set of red bonds. Hence, the CPC is infinitely ramified. We also argue that the mapping from the Euclidean metric to the geodesic metric on the CPC is governed by the Hausdorff dimension of the cluster skeleton D s c = D H / d ℓ > d min , where D H , d ℓ , and d min are the Hausdorff and the connectivity (chemical) dimensions of the CPC and the fractal dimension of the minimum path, respectively. Then we introduce the notion of the topological connectivity dimension d t ℓ . This allows us to establish the exact upper and lower bounds for the connectivity dimension d ℓ of the CPC in d = 2 . The upper and lower bounds for some other dimension numbers were established using the relations between dimension numbers. Narrow ranges defined by these bounds are much smaller than the error bars of numerical estimates reported in literature. Accordingly, the exact values of some dimension numbers are conjectured. [ABSTRACT FROM AUTHOR]

We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived. [ABSTRACT FROM AUTHOR]

In this Letter, we introduce a three-state majority vote model in which each voter adopts a state of a majority of its active neighbors, if exist, but the voter becomes uncommitted if its active neighbors are in a tie, or all neighbors are the uncommitted. Numerical simulations were performed on square lattices of different linear size with periodic boundary conditions. Starting from a random distribution of active voters, the model leads to a stable non-consensus state in which three opinions coexist. We found that the “magnetization” of the non-consensus state and the concentration of uncommitted voters in it are governed by an initial composition of system and are independent of the lattice size. Furthermore, we found that a configuration of the stable non-consensus state undergoes a second order percolation transition at a critical concentration of voters holding the same opinion. Numerical simulations suggest that this transition belongs to the same universality class as the Ising percolation. These findings highlight the effect of an updating rule for a tie between voter neighbors on the critical behavior of models obeying the majority vote rule whenever a strict majority exists. [ABSTRACT FROM AUTHOR]

The construction of the ternary Cantor set is generalized into the context of hyperbolic numbers. The partial order structure of hyperbolic numbers is revealed and the notion of hyperbolic interval is defined. This allows us to define a general framework of the fractal geometry on the hyperbolic plane. Three types of the hyperbolic analogues of the real Cantor set are identified. The complementary nature of the real Cantor dust and the real Sierpinski carpet on the hyperbolic plane are outlined. The relevance of these findings in the context of modern physics are briefly discussed. [ABSTRACT FROM AUTHOR]

In this Letter, we report experimental and theoretical studies of Newtonian fluid flow through permeable media with fractal porosity. Darcy flow experiments were performed on samples with a deterministic pre-fractal pore network. We found that the seepage velocity is linearly proportional to the pressure drop, but the apparent absolute permeability increases with the increase of sample length in the flow direction L . We claim that a violation of the Hagen–Poiseuille law is due to an anomalous diffusion of the fluid momentum. In this regard we argue that the momentum diffusion is governed by the flow metric induced by the fractal topology of the pore network. The Darcy-like equation for laminar flow in a fractal pore network is derived. This equation reveals that the apparent absolute permeability is independent of L , only if the number of effective spatial degrees of freedom in the pore-network ν is equal to the network fractal (self-similarity) dimension D , e.g. it is in the case of fractal tree-like networks. Otherwise, the apparent absolute permeability either decreases with L , if ν < D , e.g. in media with self-avoiding fractal channels, or increases with L , if ν > D , as this is in the case of the inverse Menger sponge. [ABSTRACT FROM AUTHOR]

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil–Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals. [ABSTRACT FROM AUTHOR]

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space Fν equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in Fν is inferred. The Laplacian operator in Fν is constructed. This allows us to map physical problems on fractals into the corresponding problems in Fν. In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted. [ABSTRACT FROM AUTHOR]

Using an agent-model, we conducted an analysis of a two-party system, in order to identify a pattern of behavior in the influence exercised by the number core votes on the proportion of the number of annulled votes, which are cast during an election. In a 2d toroidal grid network, where each node is connected with its four closest neighbors, we introduce the concept of an opinion network with uniformly distributed nodes that can have one of three different states, also known as “spin”: +1, −1 and 0. Initially, the states +1 and −1 correspond respectively to “active votes” for candidate A and candidate B, with the specific feature that these are not able to change their state “core vote” (i.e., that do not change their political preferences). The rest is established as having a state of 0, known as “undecided voters”, who will try to become “active votes” in favor of one of the two contending candidates, during the process of interaction with their four closest neighbors (because of their influence, in real life these would be interpreted as the voters’ immediate surroundings). The model dynamics tend to a state of equilibrium, in which there are no more voters changing their opinion upon interaction with his four closest neighbors. Finally, active voters who decide become part of the valid vote, while the undecided become part of the annulled vote. [ABSTRACT FROM AUTHOR]

Crumpled metal foils are a new type of lightweight cellular materials. The engineering applications of crumpled foils require a deep understanding of their response to mechanical loads. In order to establish a comprehensive understanding of the mechanical behavior of crumpled foils, in the present study, the uniaxial compression experiments were performed on cylindrical samples of different packing densities manufactured by die compaction of randomly crumpled aluminum foils. This has allowed us to deduce the constitutive stress-strain relationship and quantify the relaxation behavior of crumpled foils. Consequently, we determine the mechanical properties (apparent Young modulus, yield stress, and longitudinal stiffness modulus) that govern the deformation behavior of crumpled samples in the whole range of relative deformation. The power-law dependence of mechanical properties on the initial packing density is revealed. The stress and strain relaxation behavior of crumpled foils is also elucidated. These findings provide further insight into the deformation behavior and relaxation mechanisms of crumpled aluminum foils. The knowledge of the mechanical and relaxation characteristics of crumpled aluminum foils is useful for their engineering application. [ABSTRACT FROM AUTHOR]

Spin dynamics on networks allows us to understand how a global consensus emerges out of individual opinions. Here, we are interested in the effect of heterogeneity in the initial geographic distribution of a competing opinion on the competitiveness of its own opinion. Accordingly, in this work, we studied the effect of spatial heterogeneity on the majority rule dynamics using a three-state spin model, in which one state is neutral. Monte Carlo simulations were performed on square lattices divided into square blocks (cells). Accordingly, one competing opinion was distributed uniformly among cells, whereas the spatial distribution of the rival opinion was varied from the uniform to heterogeneous, with the median-to-mean ratio in the range from 1 to 0. When the size of discussion group is odd, the uncommitted agents disappear completely after Ttr = 3.30 ± 0.05 update cycles, and then the system evolves in a two-state regime with complementary spatial distributions of two competing opinions. Even so, the initial heterogeneity in the spatial distribution of one of the competing opinions causes a decrease of this opinion competitiveness. That is, the opinion with initially heterogeneous spatial distribution has less probability to win, than the opinion with the initially uniform spatial distribution, even when the initial concentrations of both opinions are equal. We found that although the time to consensus TC >> Ttr, the opinion's recession rate is determined during the first 3.3 update cycles. On the other hand, we found that the initial heterogeneity of the opinion spatial distribution assists the formation of quasi-stable regions, in which this opinion is dominant. The results of Monte Carlo simulations are discussed with regard to the electoral competition of political parties. [ABSTRACT FROM AUTHOR]

This paper is devoted to the crumpling of thin matter. The Edwards-like statistical mechanics of crumpling networks in a crushed self-avoiding sheet with finite bending rigidity is developed. The statistical distribution of crease lengths is derived. The relationship between sheet packing density and hydrostatic pressure is established. The entropie contribution to the crumpling network rigidity is outlined. The effects of plastic deformations and sheet self-contacts on crumpling mechanics are discussed. Theoretical predictions are in good agreement with available experimental data and results of numerical simulations. Thus, the findings of this work provide further insight into the physics of crumpling and mechanical properties of crumpled soft matter. [ABSTRACT FROM AUTHOR]

Abstract: The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed. [Copyright &y& Elsevier]

The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged. [ABSTRACT FROM AUTHOR]

Abstract: We construct three-dimensional space with the scale-dependent metric and the corresponding Minkowski space–time with the scale-dependent fractal ( ) and spectral ( ) dimensions. The local derivatives based on scale-dependent metrics are defined and differential vector calculus in is developed. We state that provides a unified phenomenological framework for dimensional flow observed in quite different models of quantum gravity. Nevertheless, the main attention is focused on the special case of flat space–time with the scale-dependent Cantor-dust-like distribution of admissible states, such that increases from on the scale to in the infrared limit , where is the characteristic length (e.g. the Planck length, or characteristic size of multi-fractal features in heterogeneous medium), whereas in all scales. Possible applications of approach based on the scale-dependent metric to systems of different nature are briefly discussed. [Copyright &y& Elsevier]

We study stress relaxation in hand folded aluminum foils subjected to the uniaxial compression force F{X). We found that once the compression ratio is fixed (X -- const) the compression force decreases in time as F oc F0P(J), where P{t) is the survival probability time distribution belonging to the domain of attraction of max-stable distribution of the Fréchet type. This finding provides a general physical picture of energy dissipation in the crumpling network of a crushed elastoplastic foil. The difference between energy dissipation statistics in crushed viscoelastic papers and elastoplastic foils is outlined. Specifically, we argue that the dissipation of elastic energy in crushed aluminum foils is ruled by a multiplicative Poisson process governed by the maximum waiting time distribution. The mapping of this process into the problem of transient random walk on a fractal crumpling network is suggested. [ABSTRACT FROM AUTHOR]

We study the static and dynamic properties of networks of crumpled creases formed in hand crushed sheets of paper. The fractal dimensionalities of crumpling networks in the unfolded (flat) and folded configurations are determined. Some other noteworthy features of crumpling networks are established. The physical implications of these findings are discussed. Specifically, we state that self-avoiding interactions introduce a characteristic length scale of sheet crumpling. A framework to model the crumpling phenomena is suggested. Mechanics of sheet crushing under external confinement is developed. The effect of compaction geometry on the crushing mechanics is revealed. [ABSTRACT FROM AUTHOR]

Abstract: Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed. [Copyright &y& Elsevier]

We study the effects of ambient air humidity on the dynamics of imbibition in a paper. We observed that a quick increase of ambient air humidity leads to depinning and non-Washburn motion of wetting fronts. Specifically, we found that after depinning the wetting front moves with decreasing velocity η ɑ (hp/hD)ɣ, where hD is the front elevation with respect to its pinned position at lower humidity hp, while ɣ = 1/3. The spatiotemporal maps of depinned front activity are established. The front motion is controlled by the dynamics of local avalanches directed at 30° to the balk flow direction. Although the roughness of the pinned wetting front is self-affine and the avalanche size distribution displays a power-law asymptotic, the roughness of the moving front becomes multiaffine a few minutes after depinning. [ABSTRACT FROM AUTHOR]

We point out that the chemical space of a totally disconnected Cantor dust Kn ⊂ En is a compact metric space Cn with the spectral dimension ds = dl = n > D, where D and dl = n are the fractal and chemical dimensions of Kn, respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in Cn into Kn ⊂ En defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on Kn ⊂ En, are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow ds is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided. [ABSTRACT FROM AUTHOR]

We study the effect of weak vibrations on the imbibition of water in granular media. In our experiments, we have observed that as soon as the vibration is applied, an initially pinned wetting front advances in the direction of imbibition. We found that the front motion is governed by the avalanches of localized intermittent advances directed at 45° to the imbibition direction. When the rescaled gravitational acceleration of vertical vibrations is in the range of 0.81 < G 0.95, we observed an almost steady motion of wetting front with a constant velocity &ngr;cr(G) oc exp(--1/G) during more than 20 min, whereas at lower accelerations (0.5 < G 0.8) the front velocity decreases in time asvars . We suggest that the steady motion of an imbibition front in a weakly vibrated granular medium can be treated as a creep motion associated with nonthermal temporal fluctuations of packing density in a weakly vibrated granular medium. [ABSTRACT FROM AUTHOR]

A model of fractal continuum flow employing local fractional differential operators is suggested. The generalizations of the Green-Gauss divergence and Reynolds transport theorems for a fractal continuum are suggested. The fundamental conservation laws and hydrodynamic equations for an anisotropic fractal continuum flow are derived. Some physical implications of the long-range correlations in the fractal continuum flow are briefly discussed. It is noteworthy to point out that the fractal (quasi)metric defined in this paper implies that the flow of an isotropic fractal continuum obeying the Mandelbrot rule of thumb for intersection is governed by conventional hydrodynamic equations. [ABSTRACT FROM AUTHOR]

Abstract: We analyze the effect of spatial heterogeneity in the initial spin distribution on spin dynamics in a three-state square lattice divided into spatial cells (districts). In the spirit of the statistical mechanics of social impact, we introduce a dominant influence rule (DIR), according to which, in a single update step, a chosen node adopts the state determined by the influence of its discussion group formed by the node itself and its neighbors within one or more coordination spheres. In contrast to models based on some form of majority rule (MR), a system governed by the DIR is easily trapped in a stable non-consensus state, if all nodes of the discussion group have the same weight of influence. To ensure that a consensus in the DIR system is necessarily reached, we need to put a stochastic process in the update rule. Further, the stochastic DIR model is used as a starting point for understanding the effect of spatial heterogeneity of active agent (non-zero spin) distribution on the exit probabilities. Initially, the positive and negative spins (active agents) are assigned to some nodes with non-uniform spatial distributions; while the rest of the nodes remain in the state with spin zero (uncommitted voters). By varying the relative means and skewness of the initial spin distributions, we observe critical behaviors of exit probabilities in finite size systems. The critical exponents are obtained by Monte Carlo simulations. The results of numerical simulations are discussed in the context of social dynamics. [Copyright &y& Elsevier]

Stress and strain relaxation in randomly folded paper sheets under axial compression is studied both experimentally and theoretically. Equations providing the best fit to the experimental data are found. Our findings suggest that, in an axially compressed ball folded from an elastic or elasto-plastic material, the relaxation dynamics is ruled by activated processes of an energy foci rearrangement in the crumpling network. The dynamics of relaxation is discussed within a framework of Edwards's statistical mechanics. The functional forms of the activation barrier between admissible jammed folding configurations of the crumpling network under axial compression are derived. It is shown that relaxation kinetics can be mapped to activated dynamics of depinning and creep of elastic interface in a disordered medium. [ABSTRACT FROM AUTHOR]

We study the kinetics of water escape from balls folded from square aluminum foils of different thickness and edge size. We found that the water discharge rate obeys the scaling relation Q α Vp(M - Mr)α with the universal scaling exponents α = 3 ± 0. 1, where Vp is the volume of pore space, M(t) is the actual mass of water in the ball, and Mr is the mass of residual water. The last is found to be a power-law function of Vp. The relation of these findings to the fractal geometry of randomly folded matter is discussed. [ABSTRACT FROM AUTHOR]

We study theoretically and experimentally the effect of long-range correlations in the material microstructure on the stress concentration in the vicinity of the notch tip. We find that while in a fractal continuum the notch-tip displacements obey the classic asymptotic for a linear elastic continuum, the power-law decay of notch-tip stresses is controlled by the long-range density correlations. The corresponding notch-size effect on fracture strength is in good agreement with the experimental tests performed on notched sheets of different kinds of paper. In particular, we find that there is no stress concentration if the fractal dimension of the fiber network is D ⩽ d - 0.5, where d is the topological dimension of the paper sheet. [ABSTRACT FROM AUTHOR]

Abstract: We study the scaling properties of forced folding of thin materials of different geometry. The scaling relations implying the topological crossovers from the folding of three-dimensional plates to the folding of two-dimensional sheets, and further to the packing of one-dimensional strings, are derived for elastic and plastic manifolds. These topological crossovers in the folding of plastic manifolds were observed in experiments with predominantly plastic aluminum strips of different geometry. Elasto-plastic materials, such as paper sheets during the (fast) folding under increasing confinement force, are expected to obey the scaling force–diameter relation derived for elastic manifolds. However, in experiments with paper strips of different geometry, we observed the crossover from packing of one-dimensional strings to folding two dimensional sheets only, because the fractal dimension of the set of folded elasto-plastic sheets is the thickness dependent due to the strain relaxation after a confinement force is withdrawn. [Copyright &y& Elsevier]

Abstract: This study evaluates the morphology of borided interfaces by means of the fractal theory. The boride layers were formed in the AISI M2 steel by applying the paste boriding treatment at temperatures of 1253 and 1273K and treatment times of 2 and 6h, while a boron carbide paste thickness of 4 or 5mm covered the samples surface in order to produce the boron diffusion. The morphology of interfaces formed between FeB and Fe2B layers and between Fe2B layer and steel substrate was analyzed by the rescaled-range (R/S), root-mean-square (RMS), and Fourier power spectrum (FPS) methods. Moreover, the multi-affine spectra of roughness exponent were obtained by calculating the q-order height–height correlation functions. We found that both interfaces are multi-affine, rather than self-affine. The multi-affine spectra of roughness exponents are found to be different for FeB/Fe2B and Fe2B/substrate interfaces, but independent on the treatment parameters (boron carbide paste thickness, temperature, and boriding time). Furthermore, we found that the multi-affine spectra of both interfaces behave as it is expected for “universal multi-fractals” with the Lévy index γ =1, associated with the multiplicative cascades with a log-Cauchy distribution. Furthermore, our data suggest a great homogeneity of the boron diffusion field, characterized by universal fractal dimension D diff =2.90±0.01. These findings provide a novel insight into the nature of phase formation during the boriding treatment. [Copyright &y& Elsevier]

Abstract: In the paper wetting experiments at low evaporation rate, after a short Washburn regime the film flow of filtered water overtakes the main impregnation front. Accordingly, we study the kinetic roughening dynamics and pinning of two strongly coupled fronts moving in different papers. We find that the kinetic roughening dynamics of precursor and main fronts belongs to different universality classes, nevertheless, at the final stage the distance between the fronts decrease until both fronts are pinned in the same configuration , the scaling properties of which are determined by the long-range correlations in the pore network. [Copyright &y& Elsevier]

Abstract: Fractal mechanics and probabilistic methods are applied to an ideal porous material (gypsum rock). The behavior of the material fracture is brittle in essence, concluding that the lines of fracture propagation have self-affine invariance, due to the high rugosity of the crack developed within a wide but limited length interval of scale in the cracks. The validity of the mechanic equations of self-affine cracks are verified by a comparison between the predictions of self-affine fracture mechanics and the results of standardized mechanical tests, obtaining the fractal fracture toughness of gypsum rock. [Copyright &y& Elsevier]

Abstract: In a system far from equilibrium the system size can play the role of control parameter that governs the spatiotemporal dynamics of the system. Accordingly, the kinetic roughness of interfaces in systems far from equilibrium may depend on the system size. To get an insight into this problem, we performed a detailed study of rough interfaces formed in paper combustion experiments. Using paper sheets of different width λ, we found that the turbulent flame fronts display anomalous multi-scaling characterized by non-universal global roughness exponent α and by the system size dependent spectrum of local roughness exponents, , whereas the burning fronts possess conventional multi-affine scaling characterized by the universal spectrum of roughness exponent . The structure factor of turbulent flame fronts also exhibits unconventional scaling dependence on λ. These results are expected to apply to a broad range of far from equilibrium systems when the kinetic energy fluctuations exceed a certain critical value. [Copyright &y& Elsevier]

The stress-strain behavior of fractal networks possesses self-affine scaling associated with statistical scale invariance of damage patterns. We found that the damage part of stress-strain curve of fractal network and the corresponding rupture line are characterized by the same scaling (Hurst) exponent, H, which is not universal, rather it depends on the network fractal dimension as H=DB−1. Furthermore, the same exponent governs the changes in the stress-strain behavior as the strain rate increases. These results were reproduced by Monte Carlo simulations using a fractal version of fiber bundle model. [Copyright &y& Elsevier]

The height fluctuations in the flame front during slow combustion of paper display an unconventional anomalous scaling behaviour, whereas the burning front roughness is statistically self-affine. The anomalous roughness is characterized by different scaling exponents in the local (ζ1) and the global (α > ζ[sub 1]) length scales. Furthermore, we find that the local roughness exponent, as well as the power spectrum S (structure factor of flame front), decrease as the sheet width W increases as ζ[sub 1]∝ S ∝ W[sup -v], where ν>0, despite the fact that the global roughness exponent does not depend on W. This distinguishes the unconventional anomalous roughening from the scaling behaviour expected in the cases of super-roughening and intrinsic anomalous roughening. The physical implications of these points are discussed. [ABSTRACT FROM AUTHOR]

Some new observations concerning the spatial-temporal dynamics of selfaffine interfaces formed in paper-wetting experiments are reported. We find that the motion of the wet front in a porous medium has a stepwise nature. The height of the wetted area, as a function of time, displays a Devil's-staircase-like behaviour with scaling exponent delta, whereas the front width oscillates erratically with time. These erratic oscillations possess a statistical self-affine invariance in time with the scaling exponent chi, which is found to be equal to the growth exponent beta. We also note that the values of chi, beta and delta, as well as the interface roughness (Hurst) exponent H vary from one experiment to another in wide ranges, and that their distributions obey a normal distribution. The mean values of all the exponents are dependent on the paper structure, but not on the environmental conditions. Furthermore, we find that the mean value of beta depends on the interface front orientation with respect to the fibre direction in the paper, whereas the mean values of H and delta do not depend on it. [ABSTRACT FROM AUTHOR]

The statistical topography of rupture lines in paper is studied. A new distribution function is introduced to characterize the crack roughness statistics. This function is employed to fit experimental data on the crack-length and crackwidth sample distributions for four kinds of paper. In all cases, the chi-squared test for the goodness of fit yields a significance level better than 0.97. The physical implications of experimental results are discussed. [ABSTRACT FROM AUTHOR]

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d . In this regard, we point out that the constitutive inequality D > d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined. [ABSTRACT FROM AUTHOR]