Anomalous Power Law Behavior Research Articles

Anomalous power-law behavior in the electrical impedance of endothelial cellular networks

In this paper, we report on the electrical impedance measurement of human endothelial cellular networks and show the existence of emergent power law behavior in its admittance. In particular, we find that the admittance scales with the frequency ω as ωα, with the exponent that varies with the degree of the disruption caused by the inflammation in the endothelial cellular network. We demonstrate that the power law of the measured electrical admittance can be understood by a simple percolation model of a large R–C (resistor–capacitor) network, which allows us to relate quantitatively and the intensity of inflammation. Our results suggest that the electrical properties of heterogeneous biomaterials, like living tissues, behave as a complex microstructural network.

Magnetic quantum phase diagram of magnetic impurities in two-dimensional disordered electron systems

The quantum phase diagram of disordered electron systems as function of the concentration of magnetic impurities nm and the local exchange coupling J is studied in the dilute limit. We take into account the Anderson localisation of the electrons by a nonperturbative numerical treatment of the disorder potential. The competition between RKKY interaction JRKKY and the Kondo effect, as governed by the temperature scale TK, is known to gives rise to a rich magnetic quantum phase diagram, the Doniach diagram. Our numerical calculations show that in a disordered system both the Kondo temperature TK and JRKKY are widely distributed. Accordingly, also their ratio, JRKKY /TK is widely distributed as shown in Fig. 1 (a). However, we find a sharp cutoff of that distribution, which allows us to define a critical density of magnetic impurities nc below which Kondo screening wins at all sites of the system above a critical coupling Jc, forming the Kondo phase[see Fig. 1 (b)]. As disorder is increased, Jc increases and a spin coupled phase is found to grow at the expense of the Kondo phase. From these distribution functions we derive the magnetic susceptibility which show anomalous power law behavior. In the Kondo phase that power is determined by the wide distribution of the Kondo temperature, while in the spin coupled phase it is governed by the distribution of JRKKY. At low densities and small J < Jc we identify a paramagnetic phase. We also report results on a honeycomb lattice, graphene, where we find that the spin coupled phase is more stable against Kondo screening, but is more easily destroyed by disorder into a PM phase.

Theory of Double Potential Step Chronoamperometry at Rough Electrodes: Reversible Redox Reaction and Ohmic Effects

Theory is developed for the DPSC (double potential step chronoamperometry) response for finite fractal and nonfractal electrode roughness with and without uncompensated solution resistance. Mathematical equation for the statistically averaged current transient and its relation to surface structure factor of random roughness is highlighted for reversible charge transfer. Result shows the enhancement of current transient upto outer crossover time for the rough electrode. Current response after application of each potential steps can show upto three regimes, i.e. short, intermediate and long time regimes. In short time, when diffusion layer thickness is small, current response is controlled by the roughness factor. Crossover between short and intermediate time regimes is controlled by diffusion layer thickness scaled with mean square surface curvature. Current behavior in long time (after outer crossover time), when diffusion layer thickness is large compared to width of roughness, becomes identical to smooth electrode. Crossover between intermediate and long time regimes is controlled by mean square width scaled with diffusion layer thickness. For fractal roughness, current has anomalous power law behavior (in intermediate time regime) while for nonfractal roughness current shows crossover between two regimes. Increase in the ratio of reverse and forward current is usually understood as kinetic complications but here we show that this can also emerge out of electrode roughness. Uncompensated solution resistance correction is also included in our theory to account for realistic experimental conditions. Finally, results are compared with the experimental data of a reversible system in viscous medium.

From Anderson to anomalous localization in cold atomic gases with effective spin–orbit coupling

We study the dynamics of a spin–orbit (SO)-coupled Schrödinger particle with two internal degrees of freedom moving in a one-dimensional random potential. Numerical calculation of the density of states reveals the emergence of a Dyson-like singularity at zero energy when the system approaches the quasi-relativistic limit of the random-mass Dirac model for large SO coupling. Simulations of the expansion of an initially localized wave-packet show a crossover from an exponential (Anderson) localization to an anomalous power-law behavior reminiscent of the zero-energy (mid-gap) state of the random-mass Dirac model. We discuss conditions under which the crossover is observable in an experiment and derive the zero-energy state, thus proving its existence under proper conditions. Finally we describe a possible experimental realization using an ensemble of cold 87Rb-atoms interacting with external control lasers and speckle fields.

Anomalous Warburg Impedance: Influence of Uncompensated Solution Resistance

We present the theoretical results elucidating the influence of uncompensated solution resistance on anomalous Warburg’s impedance. Here, we obtain the mathematical expression which incorporates the diffusion at the rough electrode/electrolyte interface and bulk solution resistance via phenomenological length scales—diffusion length ((D/ω)1/2) and pseudoreaction penetration length (LΩ). The roughness at the interface is contained in the expression through the surface structure factor, which can be used to describe statistically any random surface morphology. Detailed analysis for realistic fractal electrodes, characterized as a finite self-affine scaling property with two lateral cutoff lengths, is presented. Limiting behavior in higher frequency is attributed to resistive effects of the electrolyte, and surface roughness is seen through the roughness factor. In the intermediate frequency regime, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness (marking impedance loss) along with the interplay of LΩ and at lower frequencies the response crossover to classical Warburg’s behavior. This crossover frequency is dependent on the smallest of root-mean-square width (h) or lateral correlation length (L), while high-frequency crossover is dependent largely on LΩ. Phase response also shows the maximum phase gain for intermediate frequencies owing to the roughness features and is a signature to the fractal or nonfractal roughness at the interface. Our results show that owing to solution resistance the impedance response can mimic pseudo-quasireversibilty inducing a delay in the onset of diffusion-controlled regime and can hack the anomalous response due to roughness partially or completely.

Theory of Absorbance Transients of an Optically Transparent Rough Electrode

Theory for the potentiostatic absorbance transient for a species generated at the rough electrode/electrolyte interface is developed. Absorbance transients are strongly affected by electrode geometries, particularly the effect of electrode roughness which is the least understood area, is dealt here. A general operator structure between concentration, surface absorbance density, absorbance transients and arbitrary roughness profile of electrode is emphasized. The statistically averaged absorbance transients is obtained by ensemble averaging over all possible surface roughness configurations. An elegant mathematical formula between the average spectroelectrochemical absorbance transient and surface structure factor or power spectrum of roughness is obtained. This formula is used to obtain an explicit equation for the absorbance on an approximately self-affine (or realistic) fractal and nonfractal electrode. Our results are also applicable to chronocoulometric response as its directly related to chronoabsorptometric response. Limiting behavior in short time, is attributed to the electrode roughness factor and surface curvatures. Limiting behavior in long time is like a smooth electrode response but the transition region is attributed to the mean square width of roughness. In the intermediate time, the absorbance has anomalous power law behavior which is attributed to the diffusion length weighted spatial frequency features of roughness and becomes almost time independent for large roughness electrodes. Finally, this theory provides a quantitative description of the role of roughness on potential step transmission chronoabsorptometry and chronocoulometric measurements, therefore opening up the way to explore and understand various anomalies in their response.

Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects

A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface.

Theory of Generalized Gerischer Admittance of Realistic Fractal Electrode

We developed a theory for the generalized Gerischer admittance for an irregular interface, operating under diffusion and homogeneous kinetics coupled with a fast heterogeneous charge transfer reaction. The generalized admittance expressions are obtained for a deterministic surface (viz, the exact mathematical function of roughness profile is known) as well as for a stochastic surface (viz, the statistical properties of roughness profiles are known). A roughness power spectrum or structure factor is sufficient to characterize the statistical properties of stochastic geometrical irregularities. An elegant expression for the generalized Gerischer admittance is obtained as a functional of the roughness power spectrum. This equation is applicable for fractal as well as nonfractal stochastic roughness. Realistic fractal electrodes have a fractal nature over limited length scales and possess an approximate self-affine property, which is characterized in terms of a band-limited power law function for the power spectrum. The generalized Gerischer impedance shows three frequency regimes, viz, (i) low frequency region, which has a kinetic-controlled frequency (ω) independent impedance and phase angle that follows constraint, 0 ≤ ϕ(ω) < 45°, (ii) anomalous power law behavior for intermediate frequency and their phase angle ϕ(ω) > 45° and shows an approximately constant phase angle region, and (iii) high frequency limiting Warburg impedance behavior.

Anomalous Self-Energy Features in the 2D Hubbard Model

The electronic self-energy of the two-dimensional Hubbard model, computed in the non-crossing approximation within the composite operator method, is studied as a function of frequency as well as temperature at the Fermi surface, for a value of doping at which the pseudogap is already well developed. For values of momentum belonging to the phantom arc of the Fermi surface, both dependences show anomalous power law behavior in contrast to that expected from the Fermi liquid theory.

Diffusion-controlled potentiostatic current transients on realistic fractal electrodes

We analyze the problem of diffusion to irregular electrode whose irregularity is characterized as statistically isotropic self-affine fractals on limited length scales. The power spectrum of a surface fractal is approximated in terms of a white noise for low wave-numbers and a power law function for the intermediate wave-numbers. This power spectrum has four fractal morphological parameters. They are fractal dimension ( D H ), lower ( ℓ ) and upper ( L) cutoff length scales of fractality, and the proportionality factor ( μ ) related to topothesy or strength of roughness. Our explicit results for the potentiostatic current transient and its limiting laws are presented. These limiting laws are: (i) short time expansion, (ii) long time expansion and (iii) intermediate time expansion. The intermediate time limiting law for the current transient captures the classical anomalous power law behavior which is usually observed in experimental data. Our results show that the scaling exponent of anomalous region is dependent on D H as well as on ℓ and μ . These results also unravel the connection between the crossover times and the roughness characteristics of realistic fractal surfaces. We demonstrate an excellent comparison between the theoretical results and the experimental potentiostatic current transient. Finally, we also show the localization of current density on a rough corrugated electrode where surface corrugation is taken as a band-limited Weierstrass–Mandelbrot random function.

Anomalous Coulomb diamonds and power-law behavior sensitive to back-gate voltages in carbon nanoscale peapod quantum dots

We report anomalous charging effect of single electrons (Coulomb diamonds) observed in carbon nanoscale peapod quantum dots that encapsulate a series of ${\mathrm{C}}_{60}$ molecules. We find that behaviors of diamonds are anomalously sensitive to back-gate voltages $({V}_{\mathrm{bg}})$, exhibiting two evidently different ${V}_{\mathrm{bg}}$ regions and a large polarity on ${V}_{\mathrm{bg}}$. In particular, we find only a sequence of one large diamond followed by three smaller ones existing around ground state. Magnetic-field dependence indicates the presence of shell filling by spin singlet to doubly degenerate electronic levels for these. The encapsulated-${\mathrm{C}}_{60}$ molecules indirectly affect this shell filling at low ${V}_{\mathrm{bg}}$ possibly via nearly free electrons. In contrast, they act as individual quantum dots coupled in series in high ${V}_{\mathrm{bg}}$ region. It directly contributes to highly overlapped very large diamonds. Moreover, we report power-law behaviors on conductance versus energy relationships observed in the same carbon nanoscale peapods. We find that the values of powers are also highly sensitive to applied ${V}_{\mathrm{bg}}$ with three different regions and anomalously high at high source-drain voltages. Because the power laws are found at voltages, which are the nearest outside of the above-mentioned fourfold Coulomb diamonds, correlation of the anomalous powers with orbital-related Tomonaga-Luttinger liquid is discussed.

Magnetic response and quantum critical behavior in the doped two-leg extended Hubbard ladder

We have investigated quantum critical behavior in the doped two-leg extended Hubbard ladder, by using a weak-coupling bosonization method. In the ground state, the dominant fluctuation changes from the conventional d-wave-like superconducting (SCd) state into density-wave states, with increasing nearest-neighbor repulsions and/or decreasing doping rate. The competition between the SCd state and the charge-density-wave state coexisting with the p-density-wave state becomes noticeable on the critical point, at which the gap for magnetic excitations vanishes. Based on the Majorana-fermion description of the effective theory, we calculate the temperature dependence of the magnetic response such as the spin susceptibility and the NMR relaxation rate, which exhibit unusual properties due to two kinds of spin excitation modes. On the quantum critical point, the spin susceptibility shows paramagnetic behavior with logarithmic corrections and the NMR relaxation rate also exhibits anomalous power-law behavior. We discuss the commensurability effect due to the umklapp scattering and relevance to the two-leg ladder compounds Sr_{14-x}Ca_xCu_{24}O_{41}.

Quantum criticality in the frustrated Laves phase compound [formula omitted

Geometric frustration in nearly magnetic metals provides a promising and relatively little explored path to obtaining high electronic densities of states—the hatching condition of novel quantum order. Various geometrically frustrated compounds close to the border of magnetism can be found amongst the intermetallic C14 and C15 Laves phases and we have selected NbFe 2 as one of the more interesting examples. We report on measurements of the DC magnetisation, AC magnetic susceptibility, specific heat capacity and electrical resistivity in off-stoichiometric Nb 1 - y Fe 2 + y ( y ≃ - 0.04 ). Our sample is classified as a strongly enhanced Pauli paramagnet with anomalous resistivity power-law behaviour: the low-temperature electrical resistivity follows the form ρ = ρ 0 + AT α with 1 < α < 1.5 over a wide range and returns to the quadratic temperature dependence expected from Fermi Liquid theory in its simplest form only below 500 mK. The absolute magnitude of the measured low-temperature heat capacity is inconsistent with a model based solely on proximity to ferromagnetism and suggests a significant contribution from antiferromagnetic fluctuations.

Fluctuations of conformational states in biological molecules: theory for anomalous spectral diffusion dynamics

An anomalous power law behavior of spectral diffusion broadening of persistent holes is found in large biological molecules dissolved in a glassy host at very low temperature. We argue that this is caused by the internal degrees of freedom of the biomolecule itself rather than by excitations of the glassy host. To explain the observed universal time dependence of the hole width w∼ t 1/4, we propose a stochastic model of protein dynamics close to the equilibrium, which describes this process in terms of the quasi-one-dimensional diffusion of proteins in conformation space. Assuming that each step of diffusive motion changes the electronic excitation energy randomly, we derive the observed time behavior of the spectral hole. The physical mechanisms involved are discussed.

Low-energy behavior of a random classical Heisenberg chain

Low-energy magnons in a classical Heisenberg chain with random nearest-neighbor interactions are investigated using the negative eigenvalue counting technique. A detailed study is made for a range of concentrations of ``wrong sign'' bonds (i.e., $c&lt;~0.1$ and $c&gt;~0.9)$. It is found that low-energy magnons have an anomalous power-law (2/3) behavior in the spectrum for random sign-fixed magnitude exchange interactions. Combining these and previous results, a phenomenological fit is established describing the concentration dependence of the spectrum for a wide range of values of $c$. A discussion based on scaling arguments is presented to account for the anomalous power law for arbitrary concentrations. The general case where the magnitudes and the signs of the exchange interactions and the magnitudes of the spins are all random quantities is further investigated to check whether the anomalous power-law behavior holds. For uniform (flat), non-singular distributions of the random quantities, the 2/3 power in the spectrum is obtained but for singular distributions and mixed distributions (e.g., the magnitude of the spin has a singular distribution while the magnitude of the exchange interaction has a regular distribution) significant deviations from the 2/3 power law occur.

Subdiffusion and Anomalous Local Viscoelasticity in Actin Networks.

We use magnetic tweezers to study local viscoelastic response in filamentous actin networks. The choice of magnetic, colloidal particles of varying size allows us to explore properties on the relevant micron and submicron scales. At these scales the mechanical response is determined by the bending properties of individual filaments and described by an anomalous power-law behavior. In the absence of external forces the particles exhibit a subdiffusive motion.

A ferromagnetic chain in a random weak field

The harmonic magnon modes in a Heisenberg ferromagnetic chain in a random weak field are studied. The Lyapunov exponent for the uniform ( k = 0) mode is computed using the coherent potential approximation (CPA) in the weak-disorder limit. The CPA results are compared with the numerical and weak-disorder expansions of various random systems. We have found that the inverse localization length and the integrated density of states have anomalous power law behaviour as reported earlier. The CPA also reproduces the dispersion law for the same system, calculated by Pimentel and Stinchcombe using the real space renormalization scaling technique. A brief comment is also made for the uniform weak-field case.

Magnons in a one-dimensional spin glass: The high-field and zero-field limits.

The high-field and zero-field behavior of the linearized magnetic excitations (harmonic magons) in a one-dimensional \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass are studied. In the high-field limit---a field strong enough to ensure complete alignment of the ground state---the density of states (DOS), the inverse localization length (ILL), and the dynamic structure factor are calculated over the interval -4JE-H4J (H is applied field) by employing the coherent-exchange approximation (CEA), negative-eigenvalue counting, and matrix diagonalization. In the low-energy regime (0.001E-H0.1J), the CEA closely approximates the exact results reproducing, in particular, the anomalous power-law behavior of the DOS \ensuremath{\rho}(E)\ensuremath{\sim}(E-H${)}^{1/3}$ and the ILL 1/L(E)\ensuremath{\sim}(E-H${)}^{2/3}$ for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)${\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}$ and 1/L(E)=g(c)${\mathit{E}}^{2/3}$.

Renormalisation group theory for two problems in linear continuum mechanics

We consider the problems of: (1) the stress field in an infinite wedge when a moment is applied at the tip, and (2) the flow of an inviscid incompressible fluid past an infinite wedge. Both problems are linear, but nevertheless exhibit anomalous dimensions for wedge angles, α, larger than a critical value α c. In (1) it is the stress field and in (2) it is the velocity potential which can have anomalous power law behaviour in distance from the tip. We use these problems as simple examples to illustrate how partial differential equations can be solved using the renormalisation group.

The impedance of a self-affine surface

A family of self-affine surfaces is used to model the interface between a rough blocking electrode and an electrolyte. It is shown that the surface impedance scales as (i omega )-p where the exponent p varies from 0 to 1. An expression for p is found in terms of the surface geometry. It is shown that p depends on both the Holder exponent, H, and the degree of sparseness of the structure. For surfaces whose structure is everywhere dense, or very sparse, this anomalous power-law behaviour is no longer seen.