1/f noise in random resistor networks: Fractals and percolating systems

Spanning Network Gels from Nanoparticles and Graph Theoretical Analysis of Their Structure and Properties

Gels self-assembled from colloidal nanoparticles (NPs) translate the size-dependent properties of nanostructures to materials with macroscale volumes. Large spanning networks of NP chains provide high interconnectivity within the material necessary for a wide range of properties from conductivity to viscoelasticity. However, a great challenge for nanoscale engineering of such gels lies in being able to accurately and quantitatively describe their complex non-crystalline structure that combines order and disorder. The quantitative relationships between the mesoscale structural and material properties of nanostructured gels are currently unknown. Here, it is shown that lead telluride NPs spontaneously self-assemble into a spanning network hydrogel. By applying graph theory (GT), a method for quantifying the complex structure of the NP gels is established using a topological descriptor of average nodal connectivity that is found to correlate with the gel's mechanical and charge transport properties. GT descriptions make possible the design of non-crystalline porous materials from a variety of nanoscale components for photonics, catalysis, adsorption, and thermoelectrics.

Analysis of temporal correlation in heart rate variability through maximum entropy principle in a minimal pairwise glassy model

In this work we apply statistical mechanics tools to infer cardiac pathologies over a sample of M patients whose heart rate variability has been recorded via 24 h Holter device and that are divided in different classes according to their clinical status (providing a repository of labelled data). Considering the set of inter-beat interval sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\mathbf {r}(i) \} = \{ r_1(i), r_2(i), \ldots , \}$$\end{document}{r(i)}={r1(i),r2(i),…,}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,M$$\end{document}i=1,…,M, we estimate their probability distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\mathbf {r})$$\end{document}P(r) exploiting the maximum entropy principle. By setting constraints on the first and on the second moment we obtain an effective pairwise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r_n,r_m)$$\end{document}(rn,rm) model, whose parameters are shown to depend on the clinical status of the patient. In order to check this framework, we generate synthetic data from our model and we show that their distribution is in excellent agreement with the one obtained from experimental data. Further, our model can be related to a one-dimensional spin-glass with quenched long-range couplings decaying with the spin–spin distance as a power-law. This allows us to speculate that the 1/f noise typical of heart-rate variability may stem from the interplay between the parasympathetic and orthosympathetic systems.

Low-Frequency Dynamics of Strongly Correlated Electrons in (BEDT-TTF)2X Studied by Fluctuation Spectroscopy

Fluctuation spectroscopy measurements of quasi-two-dimensional organic charge-transfer salts (BEDT-TTF) 2 X are reviewed. In the past decade, the method has served as a new approach for studying the low-frequency dynamics of strongly correlated charge carriers in these materials. We review some basic aspects of electronic fluctuations in solids, and give an overview of selected problems where the analysis of 1 / f -type fluctuations and the corresponding slow dynamics provide a better understanding of the underlying physics. These examples are related to (1) an inhomogeneous current distribution due to phase separation and/or a percolative transition; (2) slow dynamics due to a glassy freezing either of structural degrees of freedom coupling to the electronic properties or (3) of the electrons themselves, e.g., when residing on a highly-frustrated crystal lattice, where slow and heterogeneous dynamics are key experimental properties for the vitrification process of a supercooled charge-liquid. Another example is (4), the near divergence and critical slowing down of charge carrier fluctuations at the finite-temperature critical endpoint of the Mott metal-insulator transition. Here also indications for a glassy freezing and temporal and spatial correlated dynamics are found. Mapping out the region of ergodicity breaking and understanding the influence of disorder on the temporal and spatial correlated fluctuations will be an important realm of future studies, as well as the fluctuation properties deep in the Mott or charge-ordered insulating states providing a connection to relaxor or ordered ferroelectric states studied by dielectric spectroscopy.

Linear polymers in disordered media: the shortest, the longest, and the mean self-avoiding walk on percolation clusters

Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters and a lot can be learned about the statistics of these polymers by studying the length distribution of SAWs on percolation clusters. This distribution encompasses 2 distinct averages, viz., the average over the conformations of the underlying cluster and the SAW conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, this disorder is redundant in the sense the renormalization group; i.e., differences to the ordered case appear merely in nonuniversal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. Our main focus, however, lies on strong disorder, i.e., the medium being close to the percolation point, where disorder is relevant. Employing a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of the corresponding Feynman diagrams, we calculate the scaling exponents for the shortest, the longest, and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced widespread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that the Meir-Harris model leads back up to 2-loop order to the renormalizable real-world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant in the course of the calculation.

Scaling behaviors of the voltage distribution in dielectric breakdown networks

We study the distribution of voltage drops across bonds in dielectric breakdown networks and its qth moments in the two-dimensional Euclidean space. Performing numerical simulations, we find that the distribution is composed of three different power-law regimes which are distinguished by two crossover voltages V(1) and V(2). The scaling behaviors of these crossover voltages with respect to the system size govern those of the qth moments of the voltage distribution. This feature differs from the multifractal behavior of the qth moment in random resistor networks. We discuss the implications of these scaling behaviors in relation to the application of the dielectric breakdown network to memory devices.

Tunneling and Percolation Behavior in Granular Metals

The nature of the percolation process in granular metals is examined for the first time by a computer simulation of a system of metallic grains embedded in an insulating matrix. Assuming that the intergrain conduction is due to quantum mechanical tunneling it is found that a percolation-like critical behavior of the conductivity is obtained, but that a percolation universal behavior will be found only in a very special case. In contrast, the behavior of the electrical noise does not deviate substantially from the universal one. Comparison of these results with recent experimental observations suggests that in the metallic range, both transport properties are controlled by the continuous metallic network rather than by intergrain tunnelin.. We propose that the metallic network resembles the previously studied system of ‘inverted random voids’.

Multifractality of self-avoiding walks on percolation clusters

We consider self-avoiding walks on the backbone of percolation clusters in space dimensions d=2,3,4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the peculiarities of the model. We obtain estimates for the set of critical exponents that govern scaling laws of higher moments of the distribution of percolation cluster sites visited by self-avoiding walks, in a good correspondence with an appropriately summed field-theoretical epsilon=6-d expansion [H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)10.1103/PhysRevE.75.020801].

A network model to correlate conformational change and the impedance spectrum of single proteins

Integrated nanodevices based on proteins or biomolecules are attracting increasing interest in today's research. In fact, it has been shown that proteins such as azurin and bacteriorhodopsin manifest some electrical properties that are promising for the development of active components of molecular electronic devices. Here we focus on two relevant kinds of protein: bovine rhodopsin, prototype of G-protein-coupled-receptor (GPCR) proteins, and the enzyme acetylcholinesterase (AChE), whose inhibition is one of the most qualified treatments of Alzheimer's disease. Both these proteins exert their function starting with a conformational change of their native structure. Our guess is that such a change should be accompanied with a detectable variation of their electrical properties. To investigate this conjecture, we present an impedance network model of proteins, able to estimate the different impedance spectra associated with the different configurations. The distinct types of conformational change of rhodopsin and AChE agree with their dissimilar electrical responses. In particular, for rhodopsin the model predicts variations of the impedance spectra up to about 30%, while for AChE the same variations are limited to about 10%, which supports the existence of a dynamical equilibrium between its native and complexed states.

Flows on graphs with random capacities

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold that depends on the distribution of capacities. We then examine the maximal total flux from the root to the leaves. Our methods generalize to simple graphs with loops, e.g., to hierarchical lattices and to complete graphs.

Multifractality in a broad class of disordered systems

We study multifractality in a broad class of disordered systems which includes, e.g., the diluted x-y model. Using renormalized field theory we analyze the scaling behavior of cumulant averaged dynamical variables (in case of the x-y model the angles specifying the directions of the spins) at the percolation threshold. Each of the cumulants has its own independent critical exponent, i.e., there are infinitely many critical exponents involved in the problem. Working out the connection to the random resistor network, we determine these multifractal exponents to two-loop order. Depending on the specifics of the Hamiltonian of each individual model, the amplitudes of the higher cumulants can vanish and in this case, effectively, only some of the multifractal exponents are required.

Resistance and resistance fluctuations in random resistor networks under biased percolation

We consider a two-dimensional random resistor network (RRN) in the presence of two competing biased processes consisting of the breaking and recovering of elementary resistors. These two processes are driven by the joint effects of an electrical bias and of the heat exchange with a thermal bath. The electrical bias is set up by applying a constant voltage or, alternatively, a constant current. Monte Carlo simulations are performed to analyze the network evolution in the full range of bias values. Depending on the bias strength, electrical failure or steady state are achieved. Here we investigate the steady state of the RRN focusing on the properties of the non-Ohmic regime. In constant-voltage conditions, a scaling relation is found between <R>/<R>(0) and V/V(0), where <R> is the average network resistance, <R>(0) the linear regime resistance, and V0 the threshold value for the onset of nonlinearity. A similar relation is found in constant-current conditions. The relative variance of resistance fluctuations also exhibits a strong nonlinearity whose properties are investigated. The power spectral density of resistance fluctuations presents a Lorentzian spectrum and the amplitude of fluctuations shows a significant non-Gaussian behavior in the prebreakdown region. These results compare well with electrical breakdown measurements in thin films of composites and of other conducting materials.

Multifractal current distribution in random-diode networks

Recently it has been shown analytically that electric currents in a random-diode network are distributed in a multifractal manner [O. Stenull and H. K. Janssen, Europhys. Lett. 55, 691 (2001)]. In the present paper we investigate the multifractal properties of a random diode network at the critical point by numerical simulations. We analyze the currents running on a directed percolation cluster and confirm the field-theoretic predictions for the scaling behavior of moments of the current distribution. It is pointed out that a random diode network is a particularly good candidate for a possible experimental realization of directed percolation.

Multifractal properties of resistor diode percolation

Focusing on multifractal properties we investigate electric transport on random resistor diode networks at the phase transition between the nonpercolating and the directed percolating phase. Building on first principles such as symmetries and relevance we derive a field theoretic Hamiltonian. Based on this Hamiltonian we determine the multifractal moments of the current distribution that are governed by a family of critical exponents [psi(l)]. We calculate the family [psi(l)] to two-loop order in a diagrammatic perturbation calculation augmented by renormalization group methods.

Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution

We study the multifractal moments of the current distribution in randomly diluted resistor networks near the percolation threshold. When an external current is applied between two terminals x and x(') of the network, the lth multifractal moment scales as M((l))(I)(x,x(')) approximately equal /x-x'/(psi(l)/nu), where nu is the correlation length exponent of the isotropic percolation universality class. By applying our concept of master operators [Europhys. Lett. 51, 539 (2000)] we calculate the family of multifractal exponents [psi(l)] for l>or=0 to two-loop order. We find that our result is in good agreement with numerical data for three dimensions.

Multifractal properties of the random resistor network

We study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. We consider two ways of applying the voltage difference: (i) two parallel bars, and (ii) two points. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small i as P(i) approximately 1/i, where i is the current. As a consequence, the moments of i of order q</=q(c)=0 do not exist and all currents of value below the most probable one have the fractal dimension of the backbone. The backbone can thus be described in terms of only (i) blobs of fractal dimension d(B) and (ii) high current carrying bonds of fractal dimension going from 1/nu to d(B).