Finite-size corrections to scaling of the magnetization distribution in the two-dimensional XY model at zero temperature.

The zero-temperature, classical XY model on an L×L square lattice is studied by exploring the distribution Φ_{L}(y) of its centered and normalized magnetization y in the large-L limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of Φ_{L}(y), and the limit distribution Φ_{L→∞}(y)=Φ_{0}(y) is obtained with high precision. The two leading finite-size corrections Φ_{L}(y)-Φ_{0}(y)≈a_{1}(L)Φ_{1}(y)+a_{2}(L)Φ_{2}(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a_{1}(L) scales as ln(L/L_{0})/L^{2} and the shape correction function Φ_{1}(y) can be expressed through the low-order derivatives of the limit distribution, Φ_{1}(y)=[yΦ_{0}(y)+Φ_{0}^{'}(y)]^{'}. Thus, Φ_{1}(y) carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude a_{2}(L)∝1/L^{2} and one finds that a_{2}Φ_{2}(y)≪a_{1}Φ_{1}(y) already for small system size (L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XY model at low temperatures, including T=0.

Order statistics of 1/fα signals.

Order statistics of periodic, Gaussian noise with 1/f(α) power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d(k) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), known for independent, identically distributed variables, remains valid for 0 ≤ α < 1. Nontrivial, α-dependent scaling exponents, d(k) k((α-3)/2), emerge for 1 < α < 5, and, finally, α-independent scaling, d(k) ~ k, is obtained for α > 5. The spectra of average ordered values ε(k) =(x(1) - x(k))~ k(ß) is also examined. The exponent ß is derived from the gap scaling as well as by relating ε(k) to the density of near-extreme states. Known results for the density of near-extreme states combined with scaling suggest that ß(α = 2) = 1/2, ß(4) = 3/2, and ß(∞) = 2 are exact values. We also show that parallels can be drawn between ε(k) and the quantum mechanical spectra of a particle in power-law potentials.

Renormalization-group theory for finite-size scaling in extreme statistics.

We present a renormalization-group (RG) approach to explain universal features of extreme statistics applied here to independent identically distributed variables. The outlines of the theory have been described in a previous paper, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.

Sugar-free, glycine-stabilized intravenous immunoglobulin prevents skin but not renal disease in the MRL/lpr mouse model of systemic lupus.

Intravenous immunoglobulin (IVIG) has a therapeutic potential in many autoimmune diseases. Based on its immune modulating and complement inhibiting effects, IVIG has been tested in systemic lupus erythematosus (SLE), but due to osmotic tubular injury caused by immunoglobulin-stabilizing sugar components, lupus nephritis had been accelerated in some patients, thus IVIG use in SLE has been abandoned. The availability of non-sugar-stabilized IVIG raised the possible re-evaluation of IVIG for SLE. We investigated high-dose, long-term non-sugar-stabilized IVIG treatment on skin and renal SLE manifestations in the MRL/lpr mouse model. Animals were treated once a week with glycine-stabilized IVIG or saline (0.2 ml/ 10 g BW) from 6 weeks until they were humanely killed at 5 months of age. IVIG diminished macroscopic cutaneous lupus compared with saline treated mice. Histology and complement-3 immunostaining also demonstrated a significant reduction of skin disease after IVIG treatment. However, renal histology and function were similar in both groups. Compared with typical osmotic tubular damage induced by 5% sucrose and 10% maltose (used for IVIG stabilization), we did not observe any osmotic tubular injury in the glycine-stabilized IVIG treated mice. Our data demonstrate a beneficial effect of IVIG on skin lupus without renal side-effects. Deeper understanding of the organ-specific pathomechanism may aid an individualized SLE therapy.

Width of reaction zones in A+B-->C type reaction-diffusion processes: effects of an electric current.

We investigate the effects of an electric current on the width of a stationary reaction zone in an irreversible A(-)+B(+)-->C reaction-diffusion process. The ion dynamics of electrolytes A identical with (A(+),A(-)) and B identical with (B(+),B(-)) is described by reaction-diffusion equations obeying local electroneutrality and the stationary state is obtained by employing reservoirs of fixed electrolyte concentrations at the opposite ends of a finite domain. We find that the width of the reaction zone decreases when the current drives the reacting ions toward the reaction zone while it increases in the opposite case. The linear response of the width to the current is estimated by developing a phenomenological theory based on conservation laws and on electroneutrality. The theory is found to reproduce numerical solutions to a good accuracy.

Coarsening of precipitation patterns in a moving reaction-diffusion front.

Precipitation patterns emerging in a two-dimensional moving front are investigated on the example of NaOH diffusing into a gel containing AlCl3 . The time evolution of the precipitate Al(OH)_{3} can be observed since the precipitate redissolves in the excess outer electrolyte NaOH and thus it exists only in a narrow optically accessible region of the reaction front. The patterns display self-similar coarsening with a characteristic length xi increasing with time as xi(t) approximately sqrt[t] . A theory based on the Cahn-Hilliard phase-separation dynamics, including redissolution, is shown to yield agreement with the experiments.

Designed patterns: flexible control of precipitation through electric currents.

Understanding and controlling precipitation patterns formed in reaction-diffusion processes is of fundamental importance with high potential for technical applications. Here we present a theory showing that precipitation resulting from reactions among charged agents can be controlled by an appropriately designed, time-dependent electric current. Examples of current dynamics yielding periodic bands of prescribed wavelength, as well as more complicated structures are given. The pattern control is demonstrated experimentally using the reaction-diffusion process 2AgNO3 + K2Cr2O7-->under Ag2Cr2O7 + 2KNO3.

Finite-size scaling in extreme statistics.

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.

RNA interference in research and therapy of renal diseases.

Significant improvements have been made during the last 20 years in therapy of renal diseases including the broadening of treatment options. Gene therapy is a potential modality for many renal diseases for which we are yet unable to offer specific treatment. Here, we introduce RNA interference (RNAi), one type of posttranscriptional gene silencing, as a novel gene therapeutic possibility and describe the mechanism and kinetics of action. We highlight the correlation between structure and efficacy of small interfering and short hairpin RNAs that are the most often used small RNAs possessing RNAi activity. Delivery is the biggest obstacle for RNAi-based gene therapy. Although hydrodynamic treatment is effective in animals, it cannot be used in human therapy. Possibilities to achieve site-specific and effective delivery are listed. Side effects of RNAi and potential solutions are also summarized. Besides the above-described world of small RNAs, we draw attention to the yet unrevealed function of human microRNAs that are localized mainly in the noncoding regions of the genome, are highly conserved among animals and possess important regulatory functions. Although there are many unanswered questions and problems to face in this new field of gene therapy, we summarize a number of experiments targeting renal diseases with the aid of RNAi. High specificity of short interfering RNAs and short hairpin RNAs raise hope for treating renal diseases.

Extreme statistics for time series: distribution of the maximum relative to the initial value.

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Guiding fields for phase separation: controlling Liesegang patterns.

Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicrometer length scales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable pattern control based on the notion of driven precipitation, meaning that the phase separation is governed by a guiding field such as, for example, a temperature or pH field. The phase separation is modeled through a nonautonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front that separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a one-dimensional system with a diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity-local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields.

Maximal height statistics for 1/f(alpha) signals.

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 01 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.

Fisher waves and front roughening in a two-species invasion model with preemptive competition.

We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits an effective power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of nonequilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of O(t(-2/3)).

Synchronization landscapes in small-world-connected computer networks.

Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by "simulating the simulations," based on the exact algorithmic rules, supported by coarse-grained arguments.

Formation of Liesegang patterns in the presence of an electric field.

The effects of an external electric field on the formation of Liesegang patterns are investigated. The patterns are assumed to emerge from a phase separation process in the wake of a diffusive reaction front. The dynamics is described by a Cahn-Hilliard equation with a moving source term representing the reaction zone, and the electric field enters through its effects on the properties of the reaction zone. We employ our previous results [I. Bena, F. Coppex, M. Droz, and Z. Rácz, J. Chem. Phys. 122, 024512 (2005)] on how the electric field changes both the motion of the front, as well as the amount of reaction product left behind the front, and our main conclusion is that the number of precipitation bands becomes finite in a finite electric field. The reason for the finiteness in case when the electric field drives the reagents towards the reaction zone is that the width of consecutive bands increases so that, beyond a distance l(+), the precipitation is continuous (plug is formed). In case of an electric field of opposite polarity, the bands emerge in a finite interval l(-), since the reaction product decreases with time and the conditions for phase separation cease to exist. We give estimates of l(+/-) in terms of measurable quantities and thus present an experimentally verifiable prediction of the "Cahn-Hilliard equation with a moving source" description of Liesegang phenomena.

Front motion in an A+B-->C type reaction-diffusion process: effects of an electric field.

We study the effects of an external electric field on both the motion of the reaction zone and the spatial distribution of the reaction product, C, in an irreversible A- + B+ -->C reaction-diffusion process. The electrolytes A identical with (A+,A-) and B identical with (B+,B-) are initially separated in space and the ion-dynamics is described by reaction-diffusion equations obeying local electroneutrality. Without an electric field, the reaction zone moves diffusively leaving behind a constant concentration of C's. In the presence of an electric field which drives the reagents towards the reaction zone, we find that the reaction zone still moves diffusively but with a diffusion coefficient which slightly decreases with increasing field. The important electric field effect is that the concentration of C's is no longer constant but increases linearly in the direction of the motion of the front. The case of an electric field of reversed polarity is also discussed and it is found that the motion of the front has a diffusive as well as a drift component. The concentration of C's decreases in the direction of the motion of the front, up to the complete extinction of the reaction. Possible application of the above results to the understanding of the formation of Liesegang patterns in an electric field is briefly outlined.

Dynamic scaling of fronts in the quantum XX chain.

The dynamics of the transverse magnetization in the zero-temperature XX chain is studied with emphasis on fronts emerging from steplike initial magnetization profiles. The fronts move with fixed velocity and display a staircase like internal structure whose dynamic scaling is explored both analytically and numerically. The front region is found to spread with time subdiffusively with the height and the width of the staircase steps scaling as t(-1/3) and t(1/3), respectively. The areas under the steps are independent of time; thus the magnetization relaxes in quantized "steps" of spin flips.

Statistics of extremal intensities for Gaussian interfaces.

The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e., roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the nondispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in nonconventional scenarios.

Magnetization distribution in the transverse Ising chain with energy flux.

The zero-temperature transverse Ising chain carrying an energy flux j(E) is studied with the aim of determining the nonequilibrium distribution functions, P(M(z)) and P(Mx) of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M(z)) is a Gaussian both at j(E)=0 and at j(E) not equal to 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(Mx), is evaluated numerically for spin chains of up to 20 spins. For the equilibrium case (j(E)=0), we find the expected Gaussian fluctuations away from the critical point, while the critical order-parameter fluctuations are shown to be non-Gaussian with a scaling function Phi(x)=Phi(M(x)/)=P(Mx) strongly dependent on the boundary conditions. When j(E) not equal to 0, the system displays long-range, oscillating correlations but P(Mx) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j(E). In particular, we find that, at critical transverse field, the width has a j(-3/8)(E) asymptotic in the j(E)-->0 limit.

Roughness distributions for 1/f alpha signals.

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f(alpha) noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, 1/f noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions (alpha < or = 1/2, 1/2 < alpha < or = 1, and 1< alpha), distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for alpha < or = 1/2 the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for alpha > 1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha = 2 and alpha-->infinity, in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of 1/f(alpha) processes.