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Showing papers in "Physica A-statistical Mechanics and Its Applications in 1999"


Journal ArticleDOI
TL;DR: A mean-field method is developed to predict the growth dynamics of the individual vertices of the scale-free model, and this is used to calculate analytically the connectivity distribution and the scaling exponents.
Abstract: Random networks with complex topology are common in Nature, describing systems as diverse as the world wide web or social and business networks. Recently, it has been demonstrated that most large networks for which topological information is available display scale-free features. Here we study the scaling properties of the recently introduced scale-free model, that can account for the observed power-law distribution of the connectivities. We develop a mean-eld method to predict the growth dynamics of the individual vertices, and use this to calculate analytically the connectivity distribution and the scaling exponents. The mean-eld method can be used to address the properties of two variants of the scale-free model, that do not display power-law scaling. c 1999 Elsevier Science B.V. All rights reserved.

2,167 citations


Journal ArticleDOI
Y.C. Zhou1, Bryan D Wright1, Runyu Yang1, B.H. Xu1, Aibing Yu1 
TL;DR: In this article, a rolling friction model is proposed to avoid arbitrary treatments or unnecessary assumptions, and its validity is confirmed by the good agreement between the simulated and experimental results under comparable conditions, which suggest that the angle of repose increases significantly with the rolling friction coefficient and decreases with particle size.
Abstract: The contact between spheres results in a rolling resistance due to elastic hysteresis losses or viscous dissipation. This resistance is shown to be important in the three-dimensional dynamic simulation of the formation of a heap of spheres. The implementation of a rolling friction model can avoid arbitrary treatments or unnecessary assumptions, and its validity is confirmed by the good agreement between the simulated and experimental results under comparable conditions. Numerical results suggest that the angle of repose increases significantly with the rolling friction coefficient and decreases with particle size.

583 citations


Journal ArticleDOI
TL;DR: In this article, a lattice gas model with biased random walkers is presented to mimic the pedestrian counter flow in a channel under the open boundary condition of constant density, and the transition point is given by pc=0.45±0.01, not depending on the system size.
Abstract: A lattice gas model with biased random walkers is presented to mimic the pedestrian counter flow in a channel under the open boundary condition of constant density. There are two types of walkers without the back step: the one is the random walker going to the right and the other is the random walker going to the left. It is found that a dynamical jamming transition from the freely moving state at low density to the stopped state at high density occurs at the critical density. The transition point is given by pc=0.45±0.01, not depending on the system size. The transition point depends on the strength of drift and decreases with increasing drift. Also, we present the extended model to take into account the traffic rule in which a pedestrian walks preferably on the right-hand side of the channel.

487 citations


Journal ArticleDOI
TL;DR: This work focuses on interbeat interval variability as an important quantity to help elucidate possibly non-homeostatic physiologic variability because (i) the heart rate is under direct neuroautonomic control, and (ii) analysis of these heart rate dynamics may provide important practical diagnostic and prognostic information not obtainable with current approaches.
Abstract: Even under healthy, basal conditions, physiologic systems show erratic fluctuations resembling those found in dynamical systems driven away from a single equilibrium state. Do such "nonequilibrium" fluctuations simply reflect the fact that physiologic systems are being constantly perturbed by external and intrinsic noise? Or, do these fluctuations actually, contain useful, "hidden" information about the underlying nonequilibrium control mechanisms? We report some recent attempts to understand the dynamics of complex physiologic fluctuations by adapting and extending concepts and methods developed very recently in statistical physics. Specifically, we focus on interbeat interval variability as an important quantity to help elucidate possibly non-homeostatic physiologic variability because (i) the heart rate is under direct neuroautonomic control, (ii) interbeat interval variability is readily measured by noninvasive means, and (iii) analysis of these heart rate dynamics may provide important practical diagnostic and prognostic information not obtainable with current approaches. The analytic tools we discuss may be used on a wider range of physiologic signals. We first review recent progress using two analysis methods--detrended fluctuation analysis and wavelets--sufficient for quantifying monofractual structures. We then describe recent work that quantifies multifractal features of interbeat interval series, and the discovery that the multifractal structure of healthy subjects is different than that of diseased subjects.

358 citations


Journal ArticleDOI
TL;DR: In this article, the authors review progress in the understanding of gravitational thermodynamics and pinpoint the error in the proof that all systems have positive specific heat and say when it can occur and discuss the development of the thermal runaway in both the gravothermal catastrophe and its inverse.
Abstract: Starting from Antonov's discovery that there is no maximum to the entropy of a gravitating system of point particles at fixed energy in a spherical box if the density contrast between centre and edge exceeds 709, we review progress in the understanding of gravitational thermodynamics. We pinpoint the error in the proof that all systems have positive specific heat and say when it can occur. We discuss the development of the thermal runaway in both the gravothermal catastrophe and its inverse. The energy range over which microcanonical ensembles have negative heat capacity is replaced by a first order phase transition in the corresponding canonical ensembles. We conjecture that all first order phase transitions may be viewed as caused by negative heat capacities of units within them. We find such units in the theory of ionization, chemical dissociation and in the Van der Waals gas so these concepts are applicable outside the realm of stars, star clusters and black holes.

300 citations


Journal ArticleDOI
TL;DR: The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality as discussed by the authors, which allows exact calculation of the critical exponents characterizing the distribution of avalanche sizes in all dimensions.
Abstract: The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The Abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing the distribution of avalanche-sizes in all dimensions. For the undirected case, the model is related to q → 0 state Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a nondashAbelian stochastic variant, which lies in a different universality class, related to directed percolation.

285 citations


Journal ArticleDOI
TL;DR: In this article, a clear power law distribution consistent with the Zipf's law can be confirmed for Japanese companies over more than three decades in income scale, and it is confirmed that such power laws hold in most of job categories with slightly modified exponents.
Abstract: Distribution functions of annual income of companies are analyzed based on two company databases. A clear power law distribution consistent with the Zipf's law can be confirmed for Japanese companies over more than three decades in income scale. Similar distributions can be confirmed in some other countries. It is confirmed that such power laws hold in most of job categories with slightly modified exponents. An annual income of a company is about two orders of magnitude smaller than its total assets, and the growth rate distribution of income is nearly independent of the income size in contrast to the case of growth rate of assets.

279 citations


Journal ArticleDOI
TL;DR: In this paper, the lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena.
Abstract: The lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena. They are the lattice versions of the hydrodynamic model of traffic. Two lattice models are presented: one is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other is the difference equation in which both time and space variables are discrete. We apply the linear stability theory and the nonlinear analysis to the lattice models. It is shown that the time-dependent Ginzburg–Landau (TDGL) equation is derived to describe the traffic flow near the critical point. A thermodynamic theory is formulated for describing the phase transitions and critical phenomena. It is also shown that the perturbed modified Korteweg-de Vries (MKdV) equation is derived to describe the traffic jam.

224 citations


Journal ArticleDOI
TL;DR: In this paper, the conditional entropy concept is used to identify a small probabilistic edge that smart speculators can exploit, and a perfect random walk has this entropy maximized, and departure from the maximal value represents a price history's predictability.
Abstract: Empirical evidence suggests that even the most competitive markets are not strictly efficient. Price histories can be used to predict near future returns with a probability better than random chance. Many markets can be considered as favorable games , in the sense that there is a small probabilistic edge that smart speculators can exploit. We propose to identify this probability using conditional entropy concept. A perfect random walk has this entropy maximized, and departure from the maximal value represents a price history's predictability. We propose that market participants should be divided into two categories: producers and speculators. The former provides the negative entropy into the price, upon which the latter feed. We show that the residual negative entropy can never be arbitraged away: infinite arbitrage capital is needed to make the price a perfect random walk.

205 citations


Journal ArticleDOI
TL;DR: In this paper, the authors focus on the entropic depletion interaction as a means to tune the range of attraction between colloids and show that meta-stable critical fluctuations may enhance the rate of crystal nucleation.
Abstract: Increase in visible order can be associated with an increase in microscopic disorder. This phenomenon leads to many counter-intuitive phenomena such as entropy driven crystallization and phase separation. I devote special attention to the entropic depletion interaction as a means to tune the range of attraction between colloids. The range of the intermolecular potential determines whether or not stable liquid-vapor coexistence is possible. For short range attraction, the liquid-vapor transition may be located below the sublimation line. Under those conditions, meta-stable critical fluctuations may enhance the rate of crystal nucleation

192 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed that clusters shatter and aggregate continuously as the concentration evolves randomly, reflecting the incessant time evolution of groups of opinions and market moods, and this model spontaneously exhibits reasonable power-law statistics for the distribution of price changes and accounts for the other important stylized facts of stock market price fluctuations.
Abstract: In the Cont–Bouchaud model [ cond-mat /9712318] of stock markets, percolation clusters act as buying or selling investors and their statistics controls that of the price variations. Rather than fixing the concentration controlling each cluster connectivity artificially at or close to the critical value, we propose that clusters shatter and aggregate continuously as the concentration evolves randomly, reflecting the incessant time evolution of groups of opinions and market moods. By the mechanism of “sweeping of an instability” [ Sornette, J. Phys. I 4 , 209 (1994)] , this market model spontaneously exhibits reasonable power-law statistics for the distribution of price changes and accounts for the other important stylized facts of stock market price fluctuations.

Journal ArticleDOI
TL;DR: The 3/4-power law extends over almost 27 orders of magnitude ranging from the largest mammal down to the molecular complex catalyzing metabolism at the most fundamental level!

Journal ArticleDOI
TL;DR: In this paper, two lattice models are presented to simulate the traffic flow on a two-lane highway and the jamming transitions among the freely moving phase, the coexisting phase, and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation.
Abstract: The two lattice models are presented to simulate the traffic flow on a two-lane highway. They are the lattice versions of the hydrodynamic model of traffic: the one (model A) is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other (model B) is the difference equation in which both time and space variables are discrete. The jamming transitions among the freely moving phase, the coexisting phase, and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation. The modified Korteweg–de Vries (MKdV) equations are derived from the lattice models near the critical point. The traffic jam is described by a kink–antikink solution obtained from the MKdV equation. It is found that the critical point, the coexisting curve, and the neutral stability line decrease with increasing the rate of lane changing. Also, the computer simulation is performed for the model B. It is shown that the coexisting curves obtained from the MKdV equation are consistent with the simulation result.

Journal ArticleDOI
TL;DR: In this article, the detrended fluctuation analysis (DFA) statistical method is applied to microwave radiometer data to sort out correlations and decorrelations in stratus cloud formation, persistence and breakup.
Abstract: A method to sort out correlations and decorrelations in stratus cloud formation, persistence and breakup is introduced. The detrended fluctuation analysis (DFA) statistical method is applied to microwave radiometer data. The existence of long-range power-law correlations in stratus cloud liquid water content fluctuations is demonstrated over a 2-h period. Moreover using a finite size (time) interval window, a change from Brownian to non-Brownian fluctuation regimes is clearly shown to define the cloud structure changes. Such findings are similar to those found in DNA and financial data sequences when mosaics of persistent and antipersistent patches are present. The occurrence of these statistics in stratus cloud liquid water content suggests the usefulness of similar studies on cloud-resolving model output and for better meteorological predictability.

Journal ArticleDOI
TL;DR: It is shown that anticipation of drivers reduces the influence of slow cars drastically and the formation of platoons at low densities in two-lane traffic models.
Abstract: For single-lane traffic models it is well known that particle disorder leads to platoon formation at low densities. Here we discuss the effect of slow cars in two-lane systems. Surprisingly, even a small number of slow cars can initiate the formation of platoons at low densities. The robustness of this phenomenon is investigated for different variants of the lane-changing rules as well as for different variants on the single-lane dynamics. It is shown that anticipation of drivers reduces the influence of slow cars drastically.

Journal ArticleDOI
TL;DR: The second law of thermodynamics is observed in individual macroscopic systems as discussed by the authors, and it can be understood as arising naturally from time-symmetric microscopic laws in accord with the ideas of Thompson, Maxwell and Boltzmann.
Abstract: Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems. It can be understood as arising naturally from time-symmetric microscopic laws in accord with the ideas of Thompson, Maxwell and Boltzmann. Alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles (probability distributions) of certain systems having only a few degrees of freedom are, in my opinion, unnecessary, misguided and misleading.

Journal ArticleDOI
Serge Galam1
TL;DR: In this paper, the concept and technics of real space renormalization group are applied to study majority rule voting in hierarchical structures and it is found that democratic voting can lead to totalitarianism by keeping in power a small minority.
Abstract: The concept and technics of real space renormalization group are applied to study majority rule voting in hierarchical structures. It is found that democratic voting can lead to totalitarianism by keeping in power a small minority. Conditions of this paradox are analyzed and singled out. Indeed majority rule produces critical thresholds to absolute power. Values of these thresholds can vary from 50% up to at least 77%. The associated underlying mechanism could provide an explanation for both former apparent eternity of communist leaderships and their sudden collapse.

Journal ArticleDOI
TL;DR: A simple new cellular automation is introduced, in which the interaction between cars is Galilei-invariant, and it is shown that this type of interaction accounts for metastable states in a very natural way.
Abstract: Traffic phenomena such as the transition from free to congested flow, lane inversion and platoon formation can be accurately reproduced using cellular automata. Being computationally extremely efficient, they simulate large traffic systems many times faster than real time so that predictions become feasible. A riview of recent results is given. The presence of metastable states at the jamming transition is discussed in detail. A simple new cellular automation is introduced, in which the interaction between cars is Galilei-invariant. It is shown that this type of interaction accounts for metastable states in a very natural way.

Journal ArticleDOI
TL;DR: In this article, the authors studied the anomalous diffusion under the influence of a random force modeled as Gaussian colored noise with arbitrary correlation and with/without external field, and obtained the variances of displacement, velocity and cross variance between displacement and velocity, their asymptotic and crossover behavior.
Abstract: In this paper, from the unifying viewpoint we will cover our recent work on the nonequilibrium statistical description of anomalous diffusion and application of this theory to explaining late experiment. We will study the motion of a particle under the influence of a random force modeled as Gaussian colored noise with arbitrary correlation and with/without external field. In the very general case, the generalized Langevin equation is presented. We obtain the variances of displacement, velocity and cross variance between displacement and velocity, their asymptotic and crossover behavior. The exact equations for the joint and marginal probability density functions, and their solutions are obtained. Finally the anomalous diffusion is described in the framework of nonequilibrium statistical mechanics. The experimental results (Skjeltorp et al., Phys. Rev. E 58 (1998) 4229) can well be explained by our theory presented in this paper.

Journal ArticleDOI
TL;DR: A cellular automata model developed to study the evolution of an infectivity nucleus in several conditions and for two kinds of epidemiologically different diseases is presented and analogies to endemic situations and pandemics are found.
Abstract: We present a cellular automata model developed to study the evolution of an infectivity nucleus in several conditions and for two kinds of epidemiologically different diseases. We analyse the role of the model parameters, concerning the epidemiological and demographic aspects of the problem, and of the evolution rules in relation to the spread of such infectious diseases, the arising of periodic temporal modulations related to the infectivity and recovery fronts, and the evolution of travelling waves. Among the obtained results we find analogies to endemic situations and pandemics.

Journal ArticleDOI
TL;DR: It is found that cytosine-guanine (CG) concentration does have a strong "background" effect on redundancy, and for the purine-pyrimidine binary mapping rule, the Shannon redundancy for the set of analyzed sequences is larger for noncoding regions compared to coding regions.
Abstract: We review evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene, and utilize this fact to build a Coding Sequence Finder Algorithm, which uses statistical ideas to locate the coding regions of an unknown DNA sequence. Finally, we describe briefly some recent work adapting to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the "redundancy" of a linguistic text in terms of a measurable entropy function, and reporting that noncoding regions in eukaryotes display a larger redundancy than coding regions. Specifically, we consider the possibility that this result is solely a consequence of nucleotide concentration differences as first noted by Bonhoeffer and his collaborators. We find that cytosine-guanine (CG) concentration does have a strong "background" effect on redundancy. However, we find that for the purine-pyrimidine binary mapping rule, which is not affected by the difference in CG concentration, the Shannon redundancy for the set of analyzed sequences is larger for noncoding regions compared to coding regions.

Journal ArticleDOI
TL;DR: The first results for large-scale flocking in the presence of noise in three dimensions are presented, and it is shown that depending on the control parameters both disordered and long-range ordered phases can be observed.
Abstract: We study a model of flocking in order to describe the transitions during the collective motion of organisms (e.g., birds) in three dimensions. In this model the particles representing the organisms are self-propelled, i.e., they move with the same absolute velocity. In addition, the particles locally interact by choosing at each time step the average direction of motion of their neighbours and the effects of fluctuations are taken into account as well. We present the first results for large-scale flocking in the presence of noise in three dimensions. We show that depending on the control parameters both disordered and long-range ordered phases can be observed. The corresponding phase diagram has a number of features which are qualitatively different from those typical for the analogous equilibrium models.

Journal ArticleDOI
TL;DR: In this article, it is shown that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension.
Abstract: It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

Journal ArticleDOI
TL;DR: In this paper, random quantum systems that exhibit unusual behavior associated with "infinite randomness" fixed points are discussed, focusing on the random quantum Ising model, which undergoes a transition at zero temperature from a phase with infinite susceptibility and continuously variable exponents to a ferromagnetic phase via a quantum critical point characterized by tunneling scaling with energy Ω and length scales, related by 1 n Ω ∼ L ψ.
Abstract: Random quantum systems that exhibit unusual behavior associated with “infinite randomness” fixed points are discussed, focusing on the random quantum Ising model. This system undergoes a transition at zero temperature from a phase with infinite susceptibility and continuously variable exponents to a ferromagnetic phase via a quantum critical point characterized by “tunneling scaling” with energy Ω and length scales, L, related by 1 n Ω ∼ L ψ . Exact results in one dimension and a scaling picture in higher dimensions are derived from a simple renormalization group. Other random quantum critical points and quantum disordered phases that can exhibit similar features are discussed briefly.

Journal ArticleDOI
TL;DR: The scaling exponent characterizing the long-range correlations in heartbeat time series as well as the multifractal features recently discovered in heartbeat rhythm are described.
Abstract: We present several recent studies based on statistical physics concepts that can be used as diagnostic tools for heart failure. We describe the scaling exponent characterizing the long-range correlations in heartbeat time series as well as the multifractal features recently discovered in heartbeat rhythm. It is found that both features, the long-range correlations and the multifractility, are weaker in cases of heart failure.

Journal ArticleDOI
TL;DR: In this paper, a centripetal packing of mono-sized spherical particles is simulated by means of the granular dynamic or discrete element method, where the packing has a limit packing density of 0.637-0.645, an overall mean coordination number of around 6.0 and a radial distribution function of clear split second peak.
Abstract: This paper presents a study of the centripetal packing of mono-sized spherical particles simulated by means of the granular dynamic or discrete element method. A packing is formed by imposing an assumed centripetal force on particles randomly generated in a spherical space. Different from the conventional simulation techniques, dynamic information of individual particles including transient forces and trajectory is traced in the present simulation. Structural properties, such as packing density, radial distribution function, coordination number distribution and homogeneity, are analyzed, with particular reference to the effects of the magnitude of the centripetal force and the number of particles. Comparison with the literature results suggests that such a dynamic model can satisfactorily simulate the dynamics of forming a packing and produce more realistic structural information. In particular, it is confirmed that a centripetal packing is not homogeneous in structure, becoming looser as its size or the number of particles increases. The packing has a limit packing density of 0.637–0.645, an overall mean coordination number of around 6.0 and a radial distribution function of clear split second peak. The centripetal force affects the rate of densification and the mean coordination number but not packing density and radial distribution function.

Journal ArticleDOI
TL;DR: In this article, the existence or not of long-, medium-, short-range power-law correlations in economic systems as well as to the presence of financial cycles are investigated, and the possibility of crash predictions is indicated.
Abstract: Problems in economy and finance have started to attract the interest of statistical physicists. Fundamental problems pertain to the existence or not of long-, medium-, short-range power-law correlations in economic systems as well as to the presence of financial cycles. Methods like the extended detrended fluctuation analysis, and the multi-affine analysis are recalled emphasizing their value in sorting out correlation ranges and predictability. Among spectacular results, the possibility of crash predictions is indicated. The well known financial analyst technique, the so-called moving average, is shown to raise questions about fractional Brownian motion properties. Finally, the (m,k)-Zipf method and the i-variability diagram technique are presented for sorting out short range correlations. Analogies with other fields of modern applied statistical physics are also presented in view of some universal openess.

Journal ArticleDOI
TL;DR: In this article, a quantitative comparison between the experimental microstructure of a sedimentary rock and three theoretical models for the same rock is presented, and quantitative dierences and similarities between the various microstructures by a method based on local porosity theory.
Abstract: A quantitative comparison between the experimental microstructure of a sedimentary rock and three theoretical models for the same rock is presented. The microstructure of the rock sample (Fontainebleau sandstone) was obtained by microtomography. Two of the models are stochastic models based on correlation function reconstruction, and one model is based on sedimentation, compaction and diagenesis combined with input from petrographic analysis. The porosity of all models closely match that of the experimental sample and two models have also the same two point correlation function as the experimental sample. We compute quantitative dierences and similarities between the various microstructures by a method based on local porosity theory. Dierences are found in the degree of anisotropy, and in uctuations of porosity and connectivity. The stochastic models dier strongly from the real sandstone in their connectivity properties, and hence need further renement when used to model transport. c 1999 Elsevier Science B.V. All rights reserved.

Journal ArticleDOI
Sumiyoshi Abe1
TL;DR: In this article, the generalized variance, covariance and correlation coefficient regarding the particle energies are calculated and their properties discussed using the classical ideal gas model, and it is shown that the correlation is suppressed for a large number of particles.
Abstract: In Tsallis’ generalized statistical mechanics, correlation is induced by nonextensivity even if the microscopic degrees of freedom are dynamically independent. Here, using the classical ideal gas model, the generalized variance, covariance and correlation coefficient regarding the particle energies are calculated and their properties discussed. It is shown that the correlation is suppressed for a large number of particles. This demonstrates the validity of the independent particle picture for a dense gas rather than for a dilute gas. It is also found that, in the thermodynamic limit, the correlation again vanishes and the generalized variance exhibits a power-law behavior with respect to the particle number density. Relevance of these results to the zeroth law of thermodynamics in nonextensive statistical mechanics is pointed out.

Journal ArticleDOI
TL;DR: In this article, the authors present high statistics simulations for 2-d percolation clusters in the bus bar geometry at the critical point, for site and for bond percolations, and measured their backbone sizes and electrical conductivities.
Abstract: We present high statistics simulations for 2-d percolation clusters in the “bus bar” geometry at the critical point, for site and for bond percolation. We measured their backbone sizes and electrical conductivities. For all sets of measurements we find large corrections to scaling, most of which do not seem to be described by single powers. Using single power terms for the corrections to scaling of the backbone masses, we would obtain fractal dimensions which are different for site and bond percolation, while the correct result is D b =1.6432±0.0008 for both. For the conductivity, the corrections to scaling are strongly non-monotonic for bond percolation. The exponent t′=t/ν is measured as 0.9826±0.0008 , in disagreement with the Alexander–Orbach and other conjectures.