Search

We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.

For both stock and currency markets, we study the return intervals τ between the daily volatilities of the price changes that are above a certain threshold q . We find that the distribution function P q (τ) scales with the mean return interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\bar {{\tau}}}\end{equation*}\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}P_{q}({\tau})={\bar {{\tau}}}^{-1}f({\tau}/{\bar {{\tau}}})\end{equation*}\end{document} . The scaling function f ( x ) is similar in form for all seven stocks and for all seven currency databases analyzed, and f ( x ) is consistent with a power-law form, f ( x ) ∼ x -γ with γ ≈ 2. We also quantify how the conditional distribution P q (τ|τ 0 ) depends on the previous return interval τ 0 and find that small (or large) return intervals are more likely to be followed by small (or large) return intervals. This “clustering” of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long-term correlations known to be present in the volatility.

Characterizing the complex atmospheric variability at all pertinent temporal and spatial scales remains one of the most important challenges to scientific research today.1–5 The main issues are to quantify, within reasonably narrow limits, the potential extent of global warming, and to downscale the global results in order to describe and quantify the regional implications of global change.

Detrended fluctuation analysis is used to test the performance of global climate models. We study the temperature data simulated by seven leading models for the greenhouse gas forcing only (GGFO) scenario and test their ability to reproduce the universal scaling (persistence) law found in the real records for four sites on the globe: (i) New York, (ii) Brookings, (iii) Tashkent and (iv) Saint Petersburg. We find that the models perform quite differently for the four sites and the data simulated by the models lack the universal persistence found in the observed data. We also compare the scaling behaviour of this scenario with that of the control run where the CO2 concentration is kept constant. Surprisingly, from the scaling point of view, the simple control run performs better than the more sophisticated GGFO scenario. This comparison indicates that the variation of the greenhouse gases affects not only trends but also fluctuations.

We study two aspects of the detrended .uctuation analysis (DFA) method, namely the scaling behavior of the leading terms of the best-0t polynomials and the detection of trends. We show analytically and numerically that the standard deviation of the leading terms of the best-0t polynomials used in DFA displays scaling behavior. Furthermore, we demonstrate that the distribution of these terms can be used to reveal the presence of trends in the data. We also argue that this distribution can be used as a sensitive tool for identifying weak trends. c 2001 Elsevier Science B.V. All rights reserved.

We revisit, by means of Monte Carlo simulations and scaling arguments, the growth model of Bunde et al. (J. Stat. Phys. 47 (1987) 1 ) where growth sites have a lifetime τ and are available with a probability p. For finite τ, the clusters are characterized by a crossover mass s×(τ)∝τφ. For masses s⪡s×, the grown clusters are percolation clusters, being compact for p>pc. For s⪢s×, the generated structures belong to the universality class of self-avoiding walk with a fractal dimension d f = 4 3 for p=1 and df≅1.28 for p=pc in d=2. We find that the number of clusters of mass s scales as N(s)=N(0) exp [−s/s × (τ)] , indicating that in contrary to earlier assumptions, the asymptotic behavior of the structure is determined by rare events which get more unlikely as τ increases.

We generate optimal paths between two given sites on a lattice representing a disordered energy landscape by applying the Dijkstra algorithm. We study the geometrical and energetic scaling properties of the optimal path under two different energy distributions that yield the weak and strong disorder limits. Our numerical results, for both two and three dimensions, suggest that the optimal paths in weak disorder are in the same universality class as the directed polymers and in the strong disorder limit are fractals with exponents similar to that found by Cieplak et al. (Phys. Rev. Lett. 72 (1994) 2320; 76 (1996) 3754).

We study the distribution $P(x;\alpha,L)$ of the relative trend $x$ in long-term correlated records of length $L$ that are characterized by a Hurst-exponent $\alpha$ between 0.5 and 1.5 obtained by DFA2. The relative trend $x$ is the ratio between the strength of the trend $\Delta$ in the record measured by linear regression, and the standard deviation $\sigma$ around the regression line. We consider $L$ between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde \cite{Lennartz2011} we show explicitely that $P$ follows the student-t distribution $P\propto [1+(x/a)^2/l]^{-(l+1)/2}$, where the scaling parameter $a$ depends on both $L$ and $\alpha$, while the effective length $l$ depends, for $\alpha$ below 1.15, only on the record length $L$. From $P$ we can derive an analytical expression for the trend significance $S(x;\alpha, L)=\int_{-x}^x P(x';\alpha,L)dx'$ and the border lines of the $95\%$ percent significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of Central West Antarctic warming.

Many real networks are embedded in space, where in some of them the links length decay as a power law distribution with distance. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations for model networks embedded on lattices of dimensions $d_e=1$ and $d_e=2$. We evaluate the dimension $d$ from the power law scaling of (a) the mass of the network with the Euclidean radius $r$ and (b) the probability of return to the origin with the distance $r$ travelled by the random walker. Both approaches yield the same dimension. For networks with $\delta < d_e$, $d$ is infinity, while for $\delta > 2d_e$, $d$ obtains the value of the embedding dimension $d_e$. In the intermediate regime of interest $d_e \leq \delta < 2 d_e$, our numerical results suggest that $d$ decreases continously from $d = \infty$ to $d_e$, with $d - d_e \sim (\delta - d_e)^{-1}$ for $\delta$ close to $d_e$. Finally, we discuss the scaling of the mass $M$ and the Euclidean distance $r$ with the topological distance $\ell$. Our results suggest that in the intermediate regime $d_e \leq \delta < 2 d_e$, $M(\ell)$ and $r(\ell)$ do not increase with $\ell$ as a power law but with a stretched exponential, $M(\ell) \sim \exp [A \ell^{\delta' (2 - \delta')}]$ and $r(\ell) \sim \exp [B \ell^{\delta' (2 - \delta')}]$, where $\delta' = \delta/d_e$. The parameters $A$ and $B$ are related to $d$ by $d = A/B$, such that $M(\ell) \sim r(\ell)^d$. For $\delta < d_e$, $M$ increases exponentially with $\ell$, as known for $\delta=0$, while $r$ is constant and independent of $\ell$. For $\delta \geq 2d_e$, we find power law scaling, $M(\ell) \sim \ell^{d_\ell}$ and $r(\ell) \sim \ell^{1/d_{min}}$, with $d_\ell \cdot d_{min} = d$., Comment: 17 pages, 11 figures

We estimate the exceedance probability $W(x,\ensuremath{\alpha};L)$ that, in a long-term correlated Gaussian-distributed (sub) record of length $L$ characterized by a fluctuation exponent $\ensuremath{\alpha}$ between 0.5 and 1.5, a relative increase $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}$ of size larger than $x$ occurs, where $\ensuremath{\Delta}$ is the total observed increase measured by linear regression and ${\ensuremath{\sigma}}_{t}$ is the standard deviation around the regression line. We consider $L$ between 500 and 2000, which is the typical length scale of monthly local and reconstructed annual global temperature records. We use scaling theory to obtain an analytical expression for $W(x,\ensuremath{\alpha};L)$. From this expression, we can determine analytically, for a given confidence probability $Q$, the boundaries $\ifmmode\pm\else\textpm\fi{}{x}_{Q}(\ensuremath{\alpha},L)$ of the confidence interval. In the presence of an external linear trend, the total observed increase is the sum of the natural and the external increase. An observed relative increase $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}$ is considered unnatural when it is above ${x}_{Q}(\ensuremath{\alpha},L)$. In this case, the size of the external relative increase is bounded by $\ensuremath{\Delta}/{\ensuremath{\sigma}}_{t}\ifmmode\pm\else\textpm\fi{}{x}_{Q}(\ensuremath{\alpha},L)$. We apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends.

In this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock. We apply this model, which contains three parameters characterizing the missing data, to the magnitude and temporal statistics of four aftershock sequences in California. We find that the observed time-dependent deviations of the frequency-magnitude scaling from the Gutenberg-Richter power law dependency can be described quantitatively by the model. We also show that, for the same set of parameters, the model is able to explain quantitatively the observed magnitude-dependent deviations of the temporal decay of aftershocks from Omori's law. In addition, we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold, as well as the violation of scaling at short and intermediate time scales.

We study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations. We find that nonlinear correlations in the record lead to a power-law (i) decay of the autocorrelation function of the return intervals, (ii) increase in the conditional return period, and (iii) decay in the probability density function of the return intervals. We show explicitly that all the observed quantities depend both on the threshold value and system size, and hence there is no simple scaling observed. We also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation.

[1] We discuss and compare the multifractal temporal scaling properties of precipitation and river discharge records on large timescales. To detect long-term correlations and multifractal behavior in the presence of trends, we apply recently developed methods (detrended fluctuation analysis (DFA) and multifractal DFA) that can systematically detect nonstationarities and overcome trends in the data at all timescales. We find that above some crossover time that usually is several weeks, the daily runoffs are characterized by an asymptotic scaling exponent that indicates a slow power law decay of the runoff autocorrelation function and varies from river to river in a wide range. Below the crossovers, pronounced short-term correlations occur. In contrast, most of the precipitation series show scaling behavior corresponding to a rapid decay of the autocorrelation function. For the multifractal characterization of the data we determine the generalized Hurst exponents and fit them by three operational models. While the fits based on the universal multifractal model describe well the scaling behavior of the positive moments in nearly all runoff and precipitation records, positive as well as negative moments are consistent with two-parameter fits from a modified version of the multiplicative cascade model for all runoff records and most of the precipitation records. For some precipitation records with weak multifractality, however, a simple bifractal characterization gives the best fit of the data.

Recent Atmosphere-Ocean Coupled General Circulation Model (AOGCM) simulations of the twentieth century climate, which account for anthropogenic and natural forcings, make it possible to study the origin of long-term temperature correlations found in the observed records. We study ensemble experiments performed with the NCAR PCM for 10 different historical scenarios, including no forcings, greenhouse gas, sulfate aerosol, ozone, solar, volcanic forcing and various combinations, such as it natural, anthropogenic and all forcings. We compare the scaling exponents characterizing the long-term correlations of the observed and simulated model data for 16 representative land stations and 16 sites in the Atlantic Ocean for these scenarios. We find that inclusion of volcanic forcing in the AOGCM considerably improves the PCM scaling behavior. The scenarios containing volcanic forcing are able to reproduce quite well the observed scaling exponents for the land with exponents around 0.65 independent of the station distance from the ocean. For the Atlantic Ocean, scenarios with the volcanic forcing slightly underestimate the observed persistence exhibiting an average exponent 0.74 instead of 0.85 for reconstructed data.

We study the multifractal temporal scaling properties of river discharge and precipitation records. We compare the results for the multifractal detrended fluctuation analysis method with the results for the wavelet transform modulus maxima technique and obtain agreement within the error margins. In contrast to previous studies, we find non-universal behaviour: On long time scales, above a crossover time scale of several months, the runoff records are described by fluctuation exponents varying from river to river in a wide range. Similar variations are observed for the precipitation records which exhibit weaker, but still significant multifractality. For all runoff records the type of multifractality is consistent with a modified version of the binomial multifractal model, while several precipitation records seem to require different models., 7 pages with 4 eps figures

In a recent letter [K. Fraedrich and R. Blender, Phys. Rev. Lett. 90, 108501 (2003)], Fraedrich and Blender studied the scaling of atmosphere and ocean temperature. They analyzed the fluctuation functions F(s) ~ s^alpha of monthly temperature records (mostly from grid data) by using the detrended fluctuation analysis (DFA2) and claim that the scaling exponent alpha over the inner continents is equal to 0.5, being characteristic of uncorrelated random sequences. Here we show that this statement is (i) not supported by their own analysis and (ii) disagrees with the analysis of the daily observational data from which the grid monthly data have been derived. We conclude that also for the inner continents, the exponent is between 0.6 and 0.7, similar as for the coastline-stations., Comment: 1 page with 2 figures

Using detrended fluctuation analysis (DFA), we study the scaling properties of the volatility time series $V_i=| T_{i+1}-T_i|$ of daily temperatures $T_i$ for ten chosen sites around the globe. We find that the volatility is long range power-law correlated with an e xponent $\gamma$ close to 0.8 for all sites considered here. We use this result to test the scaling performance of several state-of-the art global climate models and find that the models do not reproduce the observed scaling behavior., Comment: 10 pages, 3 figures. Accepted for publication in Physica A

We test the scaling performance of seven leading global climate models by using detrended fluctuation analysis. We analyse temperature records of six representative sites around the globe simulated by the models, for two different scenarios: (i) with greenhouse gas forcing only and (ii) with greenhouse gas plus aerosol forcing. We find that the simulated records for both scenarios fail to reproduce the universal scaling behavior of the observed records, and display wide performance differences. The deviations from the scaling behavior are more pronounced in the first scenario, where also the trends are clearly overestimated., Comment: Accepted for publishing in Physical Review Letters

We discuss the localization behavior of localized electronic wave functions in the one- and two-dimensional tight-binding Anderson model with diagonal disorder. We find that the distributions of the local wave function amplitudes at fixed distances from the localization center are well approximated by log-normal fits which become exact at large distances. These fits are consistent with the standard single parameter scaling theory for the Anderson model in 1d, but they suggest that a second parameter is required to describe the scaling behavior of the amplitude fluctuations in 2d. From the log-normal distributions we calculate analytically the decay of the mean wave functions. For short distances from the localization center we find stretched exponential localization ("sublocalization") in both, 1d and 2d. In 1d, for large distances, the mean wave functions depend on the number of configurations N used in the averaging procedure and decay faster that exponentially ("superlocalization") converging to simple exponential behavior only in the asymptotic limit. In 2d, in contrast, the localization length increases logarithmically with the distance from the localization center and sublocalization occurs also in the second regime. The N-dependence of the mean wave functions is weak. The analytical result agrees remarkably well with the numerical calculations., Comment: 12 pages with 9 figures and 1 table

We discuss the localization behavior of quantum particles in a one-dimensional Anderson model with self-affine random potentials, characterized by a Hurst exponent $Hg0.$ Depending on H and energy E, a new type of ``strong'' localization can occur, where all states are localized in a way different from the regular Anderson localized states. Using scaling arguments, we derive an analytical expression for the phase diagram and test it by numerical calculations. Finally, we consider a somewhat related model where the variance of the potential fluctuations is kept fixed for all system sizes L and a transition between localized and apparently extended states has been reported.

We compare scaling properties of the cardiac dynamics during sleep and wake periods for healthy individuals, cosmonauts during orbital flight, and subjects with severe heart disease. For all three groups, we find a greater degree of anticorrelation in the heartbeat fluctuations during sleep compared to wake periods. The sleep-wake difference in the scaling exponents for the three groups is comparable to the difference between healthy and diseased individuals. The observed scaling differences are not accounted for simply by different levels of activity, but appear related to intrinsic changes in the neuroautonomic control of the heartbeat., Comment: 7 pages, 4 figures; to appear in Europhysics Letters, Dec. 99

The purpose of this report is to describe some recent progress in applying scaling concepts to various systems in nature. We review several systems characterized by scaling laws such as DNA sequences, heartbeat rates and weather variations. We discuss the finding that the exponent alpha quantifying the scaling in DNA in smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the scaling exponent alpha is smaller during sleep periods compared to wake periods. We also discuss the recent findings that suggest a universal scaling exponent characterizing the weather fluctuations.

We examine the Detrended Fluctuation Analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling that appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA., Comment: 10 pages, including 8 figures

The scaling behavior of linear polymers in disordered media modelled by self-avoiding random walks (SAWs) on the backbone of two- and three-dimensional percolation clusters at their critical concentrations p_c is studied. All possible SAW configurations of N steps on a single backbone configuration are enumerated exactly. We find that the moments of order q of the total number of SAWs obtained by averaging over many backbone configurations display multifractal behavior, i.e. different moments are dominated by different subsets of the backbone. This leads to generalized coordination numbers \mu_q and enhancement exponents \gamma_q, which depend on q. Our numerical results suggest that the relation \mu_1 = p_ c \mu between the first moment \mu_1 and its regular lattice counterpart \mu is valid., Comment: 11 pages, 12 postscript figures, to be published in Phys. Rev. E

We study optimal paths in disordered energy landscapes using energy distributions of the type $P({\mathrm{log}}_{10}E)=\mathrm{const}$ that lead to the strong disorder limit. If we truncate the distribution, so that $P({\mathrm{log}}_{10}E)=\mathrm{const}$ only for ${E}_{\mathrm{min}}l~El~{E}_{\mathrm{max}},$ and $P({\mathrm{log}}_{10}E)=0$ otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length ${\mathcal{l}}_{\ifmmode\times\else\texttimes\fi{}}.$ We find that ${\mathcal{l}}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\propto}[{\mathrm{log}}_{10}{(E}_{\mathrm{max}}{/E}_{\mathrm{min}}){]}^{\ensuremath{\kappa}},$ where the exponent $\ensuremath{\kappa}$ has the value $\ensuremath{\kappa}=1.60\ifmmode\pm\else\textpm\fi{}0.03$ both in $d=2$ and $d=3.$ We show how the crossover can be understood from the distribution of local energies on the optimal paths.

Recent measurements on ion conducting glasses have revealed that conductivity spectra for various temperatures and ionic concentrations can be superimposed onto a common master curve by an appropriate rescaling of the conductivity and frequency. In order to understand the origin of the observed scaling behavior, we investigate by Monte Carlo simulations the diffusion of particles in a lattice with site energy disorder for a wide range of both temperatures and concentrations. While the model can account for the changes in ionic activation energies upon changing the concentration, it in general yields conductivity spectra that exhibit no scaling behavior. However, for typical concentrations and sufficiently low temperatures, a fairly good data collapse is obtained analogous to that found in experiment., Comment: 6 pages, 4 figures

For characterizing the inherent dynamical structure of precipitation and river runoffs, we study in representative seasonally detrended records the durations τ where the record is below a given threshold Q. When keeping the return period RQ of the threshold fixed, we find that the probability density function (PDF) of these durations shows universal features when scaled with the average duration τ. For both types of data the scaling function can be approximated by a Γ-distribution, but also a description by the corresponding PDF of long-term correlated records is possible. The parameters of the PDF depend on the kind of data (precipitation or river runoff) and on RQ but neither on the climate nor on the hydrological details of the location of the record.

We study seismic records in regimes of stationary seismic activity in Northern and Southern California. Our analysis suggests that the earthquakes are long-term power law correlated with a correlation exponent ? close to 0.4. We show explicitly that the long-term correlations can explain both the fluctuations of magnitudes and interoccurrence times (between events above a certain magnitude M) and, without any fit parameter, the scaling form of the distribution function of the interoccurrence times in the seismic records, recently obtained by Corral (Phys. Rev. Lett., 92 (2004) 108501).

We briefly review some recent attempts to achieve some genuine understanding of diffusion-limited aggregation (DLA), the paradigm model for dynamical mechanisms of disorderly growth processes. We shall see that the seminal ideas of Professor Cyril Domb have influenced to a great degree many of the recent theoretical approaches. In particular, the Domb-Hunter constant-gap scaling hypothesis becomes replaced by a continuum of gap exponents. Moreover, while the growth probabilities for the tips of the DLA structure do scale in the conventional fashion, there is evidence that the growth probabilities of the fjords do not scale. Does this competition between one part of DLA that does scale, and another that does not, underlie many of the unusual properties of this model?

The question of transport in a random two-component mixture is addressed. To this end, two models are precisely formulated that effectively extend to two-component mixtures the de Gennes ant model of one-component systems. We develop and test a scaling theory, and we solve some of the problems associated with the limit where one of the components is superconducting. Our work provides the first practical realisation of the termite model of a random superconducting network, which performs a normal walk in the normal regions of the material but performs a new and unique form of random walk in the superconducting regions. We find that the divergence of the electrical conductivity at the percolation threshold can be described by this random walk, and that the critical exponent s is given by s = 1.3 f 0.1 in d = 2. If we perturb away from the pure superconduct- ing limit, we find that the electrical properties can still be described by the random walk model, with a crossover exponent I$ = 0.4 (d = 2). Moreover, we find that the diffusion constant in this region is described by a scaling form, so that data can be made to collapse upon a single curve whose form is governed by the exponents s and I$.

The authors study the problem of transport in linear random resistor-superconductor mixtures with a random distribution of resistor strength. The superconductors with a concentration p are represented in the model as short circuits. The resistor concentration is (1-p) and their conductivity distribution is p( sigma ) approximately sigma - alpha , alpha 0 the specific conductivity scales with the linear size L of the system as L- alpha (1- alpha )/(1-p)-1(1- alpha )/. The mean square displacement (x2) of a random walker in this system scales as (x2)(2- alpha )(1- alpha/2 ) approximately (1-p)-(2- alpha )(1- alpha )/t. In the presence of a bias field they obtain (x)1(1- alpha )/ approximately (1-p)-1(1- alpha )/t. They present an exact enumeration method to study diffusion on those systems. The numerical results confirm the above scaling relations.

Models for spreading phenomena, which contain as a special case the Eden model, are reviewed. In these models, the growth sites are converted into cluster sites (with probability p) or into immune sites (with probability 1 -p). The way that the next growth site is chosen determines the kinetic features of the growth process. To characterize the process quantitatively, scaling relations and growth exponents are established which describe the asymptotic behaviour of the number of cluster sites as well as the radius of gyration as a function of time. In addition, the regime where the cluster can grow is quantified by the total number of growth sites and their spatial distribution. Scaling theories are developed to describe them.

We study diffusion of Brownian particles on random fractal structures, self-avoiding random walks and percolation clusters at critically, which serve as model systems for polymers in a good solvent and random two-component mixtures, respectively. Using numerical simulations and scaling arguments wa find that the distribution function P(r, t) of the particles is a stretched Gaussian and scales as log [P(r,t)/P(r,O)]∼−[r/R(t)] u , where R(t)∼t 1/DW is the root-mean-square displacement, u=d w /(d w −1), and d w =2d f for self-avoiding walks and d w =3d f /2 for percolation; d f is the fractal dimension of the structure. In the presence of an external constant bias field, diffusion is drastically reduced and R(t)∼logt evolves logarithmically in time, in exactly the same way in both fractal structures

The authors study diffusion in d=2 percolation clusters at criticality under the influence of an external time-dependent field E(t)=E0sin( omega t). They find that the mean displacement (x(t)) of a random walker is a periodic function with a phase shift phi approximately= pi /3 independent of omega and E0. The amplitude A(E0, omega ) of (x(t)) is linear in E0 below a crossover field E*0( omega ). Above E*0( omega ), A(E0, omega ) is strongly non-linear, showing a maximum as a function of E0. They find that both the linear and the non-linear regime can be described by a single scaling function: A(E0, omega ) approximately E0 omega -2d/F(E0/E*0( omega )) where d approximately=2.87 is the diffusion exponent and E*0( omega ) approximately omega beta with beta approximately=0.3. The linear response regime is reached for E0/ omega beta <

The authors study diffusion in percolation systems at criticality in the presence of a constant bias field E. Using the exact enumeration method they show that the mean displacement of a random walker varies as (r(t)) approximately log t/A(E) where A/(E)=In((1+E)/(1-E)) for small E. More generally, diffusion on a given configuration is characterised by the probability P(r,t) that the random walker is on site r at time t. They find that the corresponding configurational average shows simple scaling behaviour and is described by a single exponent. In contrast their numerical results indicate that the averaged moments (Pq(t))= Sigma rP q(r,t)) are described by an infinite hierarchy of exponents. For zero bias field, however, all moments are determined by a single gap exponent.

We consider a Brownian particle in a double-well potential at intermediate and high friction and study the evolution of an initial distribution which is located at the top of the barrier. The evolution equation is obtained from the Fokker-Planck-Klein-Kramers equation by an inverse friction expansion and is transformed to a time-dependent Schrodinger equation which we solve approximately using the WKB method. The result becomes exact for temperatures low compared with the barrier height. We calculate explicitly the characteristic time in which the distribution changes from a single peak to a double peak structure and show that the decay process is mainly described by a scaling-function which reduces to Suzuki's scaling solution in the limit of high friction.