Contour lines of the discrete scale invariant rough surfaces

We study the fractal properties of the 2d discrete scale invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale invariant rough surfaces. The fractal properties of the 2d DSI rough surfaces apart from possessing the discrete scale invariance property follow the properties of the contour lines of the corresponding scale invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents some other new fractal exponents are also calculated.Comment: 9 Pages, 12 figure

Fractal Analysis of Protein Potential Energy Landscapes

The fractal properties of the total potential energy V as a function of time t are studied for a number of systems, including realistic models of proteins (PPT, BPTI and myoglobin). The fractal dimension of V(t), characterized by the exponent \gamma, is almost independent of temperature and increases with time, more slowly the larger the protein. Perhaps the most striking observation of this study is the apparent universality of the fractal dimension, which depends only weakly on the type of molecular system. We explain this behavior by assuming that fractality is caused by a self-generated dynamical noise, a consequence of intermode coupling due to anharmonicity. Global topological features of the potential energy landscape are found to have little effect on the observed fractal behavior.Comment: 17 pages, single spaced, including 12 figure

Golden gaskets: variations on the Sierpi\'nski sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor \la\in(0,1). As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are "overlaps" in \S_\la as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic \la's (so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these special values by showing that \S_\la is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique ``address'' is self-similar, and compute its dimension. For ``non-multinacci'' values of \la we show that if \la is close to 2/3, then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.Comment: 27 pages, 10 figure

Surface Hardening and Self-Organized Fractality Through Etching of Random Solids

When a finite volume of etching solution is in contact with a disordered solid, complex dynamics of the solid-solution interface develop. If the etchant is consumed in the chemical reaction, the dynamics stop spontaneously on a self-similar fractal surface. As only the weakest sites are corroded, the solid surface gets progressively harder and harder. At the same time it becomes rougher and rougher uncovering the critical spatial correlations typical of percolation. From this, the chemical process reveals the latent percolation criticality hidden in any random system. Recently, a simple minimal model has been introduced by Sapoval et al. to describe this phenomenon. Through analytic and numerical study, we obtain a detailed description of the process. The time evolution of the solution corroding power and of the distribution of resistance of surface sites is studied in detail. This study explains the progressive hardening of the solid surface. Finally, this dynamical model appears to belong to the universality class of Gra dient Percolation.Comment: 14 pages, 15 figures (1457 Kb

Determining mean first-passage time on a class of treelike regular fractals

Relatively general techniques for computing mean first-passage time (MFPT) of random walks on networks with a specific property are very useful, since a universal method for calculating MFPT on general graphs is not available because of their complexity and diversity. In this paper, we present techniques for explicitly determining the partial mean first-passage time (PMFPT), i.e., the average of MFPTs to a given target averaged over all possible starting positions, and the entire mean first-passage time (EMFPT), which is the average of MFPTs over all pairs of nodes on regular treelike fractals. We describe the processes with a family of regular fractals with treelike structure. The proposed fractals include the $T$ fractal and the Peano basin fractal as their special cases. We provide a formula for MFPT between two directly connected nodes in general trees on the basis of which we derive an exact expression for PMFPT to the central node in the fractals. Moreover, we give a technique for calculating EMFPT, which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials. Making use of the proposed methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on the fractals and show how they scale with the number of nodes. In addition, to exhibit the generality of our methods, we also apply them to the Vicsek fractals and the iterative scale-free fractal tree and recover the results previously obtained.Comment: Definitive version published in Physical Review

The Fractal Geometry of Critical Systems

We investigate the geometry of a critical system undergoing a second order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=Tc, we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite size effects we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.Comment: 1 LaTeX file, 4 figures in ps-files. Accepted for publication in Physical Review

Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices

The family of Vicsek fractals is one of the most important and frequently-studied regular fractal classes, and it is of considerable interest to understand the dynamical processes on this treelike fractal family. In this paper, we investigate discrete random walks on the Vicsek fractals, with the aim to obtain the exact solutions to the global mean first-passage time (GMFPT), defined as the average of first-passage time (FPT) between two nodes over the whole family of fractals. Based on the known connections between FPTs, effective resistance, and the eigenvalues of graph Laplacian, we determine implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical results. The obtained closed-form solution shows that the GMFPT approximately grows as a power-law function with system size (number of all nodes), with the exponent lies between 1 and 2. We then provide both the upper bound and lower bound for GMFPT of general trees, and show that leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size. We also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs. This study provides a comprehensive understanding of random walks on the Vicsek fractals and general treelike networks.Comment: Definitive version accepted for publication in Physical Review

Inter-Intra Molecular Dynamics as an Iterated Function System

The dynamics of units (molecules) with slowly relaxing internal states is studied as an iterated function system (IFS) for the situation common in e.g. biological systems where these units are subjected to frequent collisional interactions. It is found that an increase in the collision frequency leads to successive discrete states that can be analyzed as partial steps to form a Cantor set. By considering the interactions among the units, a self-consistent IFS is derived, which leads to the formation and stabilization of multiple such discrete states. The relevance of the results to dynamical multiple states in biomolecules in crowded conditions is discussed.Comment: 7 pages, 7 figures. submitted to Europhysics Letter

Self-similar sets: projections, sections and percolation

We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.Postprin