Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

Collapse of two-dimensional linear polymers

We study the collapse transition of a two-dimensional, very long polymer. The model we consider is a lattice model where the chain is represented by a self-avoiding walk with nearest-neighbor attraction. By using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width. We then use finite-size scaling to determine the values of the tricritical ( point) exponentsv ₁=0.55±0.01v _u=1.15±0.15 ₁/v₁=1.80±0.05 We compare these results to the other values already proposed in the literature.     

Corrections to cluster-radius scaling for branched polymers and percolation

We report analyses of series enumerations for the mean radius of gyration for isotropic and directed lattice animals and for percolation clusters, in two and three dimensions. We allow for the leading correction to the scaling behaviour and obtain estimates of the leading correction-to-scaling exponent . We find -0.640±0.004 and =0.87±0.07 for isotropic animals in 2d, and =0.64±0.06 in 3d. For directed lattice animals we argue that the leading correction has= or= ; we also estimate =0.82±0.01 and 0.69 ±0.01 ind=2, 3 respectively. For percolation clusters at and abovep _c, we find (p _c) =0.58±0.06 and (p>p _c)=0.84±0.09 in 2d, and (p _c)=0.42±0.11 and (p>p _c)=0.41 ±0.09 in 3d.     

Metastates in Disordered Mean-Field Models II: The Superstates

We continue to investigate the size dependence of disordered mean-field models with finite local spin space in more detail, illustrating the concept of superstates as recently proposed by Bovier and Gayrard. We discuss various notions of convergence for the behavior of the paths (t_[tN]()) _{t(0, 1]} in the thermodynamic limit N. Here _n () is the Gibbs measure in the finite volume {1,..., n} and is the disorder variable. In particular we prove refined convergence statements in our concrete examples, the Hopfield model with finitely many patterns (having continuous paths) and the Curie–Weiss random-field Ising model (having singular paths).     

Critical exponents for two-dimensional tracer diffusion through a changing background at concentrationc=c p

We study the diffusion of a particle on the sites of a triangular lattice of which half the sites are occupied by a background of other particles. No two particles may occupy the same site. We carry out Monte Carlo simulations for the following model: At each Monte Carlo step the tracer attempts to move to a neighboring site, which it does if the site is unoccupied. At each step, each background particle attempts to desorb with probability. If a background particle desorbs, it is replaced at a randomly chosen site on the lattice. We define R _tr ² (t)/t=D _tr. For the case=0, we calculateD ₀t ^k and findk=0.71±0.01, wheret is the number of Monte Carlo steps. When0, we calculateD _tr ~ ^– w ₁ and findw _ad=0.24±0.02. We compare this to the model in which the background particles are constrained to move to nearest neighbor sites and findD _tr ^– w ₁ withw ₁=0.28±0.03.     

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We find=2.63820 ± 0.00004 ± 0.00030=1.352 ± 0.006 ± 0.025v=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.     

High-precision Monte Carlo test of the conformai-invariance predictions for two-dimensional mutually avoiding walks

Let _l be the critical exponent associated with the probability thatl independentN-step ordinary random walks, starting at nearby points, are mutually avoiding. Using Monte Carlo methods combined with a maximum-likelihood data analysis, we find that in two dimensions ₂=0.6240±0.0005±0.0011 and ₃=1.4575±0.0030±0.0052, where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second error bar represents statistical error (classical 95% confidence limits). These results are in good agreement with the conformal-invariance predictions ₂=5/8 and ₃=35/24.     

Self-avoiding random walks on the hexagonal lattice

We use the algorithm recently introduce by A. Berretti and A. D. Sokal to compute numerically the critical exponents for the self-avoiding random walk on the hexagonal lattice. We find=1.3509±0.0057±0.0023v=0.7580±0.0049±0.0046=0.519±0.082±0.077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error. For the effective coordination number we find=1.84779±0.00006±0.0017 The results support the Nienhuis conjecture=43/32 and provide a rough numerical check of the hyperscaling relationdv=2–. An additional analysis, taking the Nienhuis value of=(2+2^1/2)^1/2 for granted, gives=1.3459±0.0040±0.0008     

Dynamical Critical Exponent for the Majority-Vote Model

We investigate the dynamical behavior of the isotropic majority-vote model on a square lattice using a combination of damage spreading and finite-size scaling methods. For initial damage D(0)1/2, the dynamical phase diagram exhibits a chaotic-frozen phase transition at a critical noise parameter q _c =0.0818±0.0002, while for D(0)1/2 the damage does not propagate for any value of the model's parameter 0q<1/2. From simulations at q _c , we find that the dynamical critical exponent is z=0.65±0.05.     

Temperature-dependent bandstructure of ferromagnetic metals with localized moments

We consider a theoretical model for ferromagnetic metals like Gd, which takes into account the exchange interaction and the hybridization between two electronic subsystems. The first is built up by quasi-localized electrons, which take care for the existence of permanent magnetic moments, and is described by an atomic limit multiband Hubbard-Hamiltonian. The second subsystem consists of relatively broad conduction bands with more or less free electrons. We investigate the influence of electron correlations on the conduction band states in dependence of temperatureT and bandfillingn. A sensitive reaction of the band states on the magnetic ordering of the moment system leads to strong band deformations. The main goal is the determination of a quasiparticle bandstructure, which we derive in analogy to the experiment (PES, IPE) directly from the spectral density. The new aspect is a splitting of the bare dispersion _m (k) into several quasiparticle dispersion curves with stronglyT-andn-dependent spectral weights. Even forT>T _c an exchange-caused splitting is found. — In this paper we use as input for the free band energies _m (k) a nondegenerate simple cubic-tight binding expression, and in addition seven dispersionlessf-levels, in order to stress mainly the influence of many body effects. It is discussed how the model can be coupled to a LDA bandstructure calculation, in order to get quantitative results for theT-dependent electronic structure of the ferromagnetic 4f-metal Gd, the presentation of which is intended in a forthcoming paper.     

Radius of clusters at the percolation threshold: A position space renormalization group study

Using a direct position-space renormalization-group approach we study percolation clusters in the limits , wheres is the number of occupied elements in a cluster. We do this by assigning a fugacityK per cluster element; asK approaches a critical valueK _c , the conjugate variables . All exponents along the path (K–K _c ) 0 are then related to a corresponding exponent along the paths . We calculate the exponent , which describes how the radius of ans-site cluster grows withs at the percolation threshold, in dimensionsd=2, 3. Ind=2 our numerical estimate of =0.52±0.02, obtained from extrapolation and from cell-to-cell transformation procedures, is in agreement with the best known estimates. We combine this result with previous PSRG calculations for the connectedness-length exponent , to make an indirect test of cluster-radius scaling by calculating the scaling function exponent using the relation =/. Our result for is in agreement with direct Monte-Carlo calculations of , and thus supports the cluster-radius scaling assumption. We also calculate ind=3 for both site and bond percolation, using a cell of linear sizeb=2 on the simple-cubic lattice. Although the result of such small-cell calculations are at best only approximate, they nevertheless are consistent with the most recent numerical estimates.Supported in part by grants from ARO and ONR     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

Branched Polymers on the Two-Dimensional Square Lattice with Attractive Surfaces

Using the renormalization group approach, an analysis is given of the asymptotic properties of branched polymers situated on the two-dimensional square lattice with attractive impenetrable surfaces. We modeled branched polymers as site lattice animals with loops and site lattice animals without loops on the simple square lattice. We found the gyration radius critical exponent =0.6511±0.0003 and =0.6513±0.0003 for branched polymers with and without loops, respectively. Our results for the crossover exponent =0.502±0.003 for branched polymers with loops and =0.503±0.003 for branched polymers without loops satisfy the recent hyperuniversality conjecture = . In addition, we have studied partially directed site lattice animals.     

Free energy of the chiral Potts model in the scaling region

We explicitly calculate the free energy of the general solvableN-state chiral Potts model in the scaling region, forT<T _c . We do this from both of the two available results for the free energy, and verify that they are mutually consistent. Ift=T _c –T, then we find that - _c /t has a Taylor expansion in powers oft ^2/N (together with higher-order non-scaling terms of ordert, ort logt).     

Critical exponents,hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2 ₄ – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2 ₄ –. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R _g ² >/<R _e ² >=0.14026±0.00007, <R _m ² >/<R _e ² >=0.43961±0.00034, and ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent ₁=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ₁. Earlier Monte Carlo estimates ofv, which were 0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R _g ² >/<R _e ² >=0.1599±0.0002 and ^*=0.2471±0.0003; since ^*>0, hyperscaling holds. The approach to ^* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2 ₄ – for two-dimensional SAWs.     

Collapse transition and asymptotic scaling behavior of lattice animals: Low-temperature expansion

A low-temperature expansion for the free energy density of lattice animals is derived. Analysis of the series yields a collapse transition temperature ofT _c - 0.54, in close agreement with previous estimates. It is demonstrated that _p,k, the number ofp-particle,p-bond animals, obeys the asymptotic scaling law log _p,k pg(k/p) + o(p). The low-temperature series and numerical data are used to estimate the scaling function.     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

Dynamics of the formation of two-dimensional ordered structures

The growth of ordered domains in lattice gas models, which occurs after the system is quenched from infinite temperature to a state below the critical temperatureT _c, is studied by Monte Carlo simulation. For a square lattice with repulsion between nearest and next-nearest neighbors, which in equilibrium exhibits fourfold degenerate (2×1) superstructures, the time-dependent energy E(t), domain size L(t), and structure functionS(q, t) are obtained, both for Glauber dynamics (no conservation law) and the case with conserved density (Kawasaki dynamics). At late times the energy excess and halfwidth of the structure factor decrease proportional tot ^–x, whileL(t) t ^x, where the exponent x=1/2 for Glauber dynamics and x1/3 for Kawasaki dynamics. In addition, the structure factor satisfies a scaling lawS(k,t)=t ^2xS(kt^x). The smaller exponent for the conserved density case is traced back to the excess density contained in the walls between ordered domains which must be redistributed during growth. Quenches toT>T _c, T=T_c (where we estimate dynamic critical exponents) andT=0 are also considered. In the latter case, the system becomes frozen in a glasslike domain pattern far from equilibrium when using Kawasaki dynamics. The generalization of our results to other lattices and structures also is briefly discussed.     

Translational and rotational diffusion in supercooled orthoterphenyl close to the glass transition

Self diffusion coefficients in supercooled orthoterphenyl (OTP) have been determined down toD _t =3·10^–14 m²s^–1 using a¹H-NMR technique applying static field gradients up to 53T m^–1 In a range of more than two decades theD _t values agree with those of photochromic tracer molecules of the same size determined by forced Rayleigh scattering down to the glass transition temperatureT _g . A change of mechanism is found for translational diffusion atT _c 1.2T _g whereD _t is proportional to the inverse shear viscosity ^–1 atT>T _c butD _t with =0.75 atT<T _c . Rotational correlation times determined by²H-NMR stimulated echo techniques in deuterated OTP remain proportinal to ^–1 down toT _g . Our results are discussed in relation with mode coupling theory and with models of cooperative motion at the glass transition.     

Spatial fluctuations in reaction-diffusion systems: A model for exponential growth

The spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated. The model is characterized byrate constants K _ij =i+j for the clustering of ani- and aj-mer, anddiffusion constants D _j =D. It is assumed thatD1 (reaction-limited aggregation). Explicit expressions for the correlation functions at equal and at different times are calculated. The equal-time correlation functions show scaling behavior in the scaling limit. The correlation length remains finite ast, and the fluctuations becomelarge at large times (tt _D ) inany dimension. The crossover timet _D , at which the mean field theory (Smoluchowski's equation) breaks down, is given byt _D InD. These exact results imply that the upper critical dimension of this model isd _c= and, hence, that there isno unique upper critical dimension in models for the irreversible aggregation of clusters.