Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.     

Percolation clusters above criticality form Kardar-Parisi-Zhang surfaces

This study is concerned with the characteristics of regular (isotropic) percolation clusters above the critical threshold p{c}. Analytic arguments for the general dimension case, and numerical results for the two-dimensional case, lead to the conclusion that the characteristics of the shortest paths (defined as the chemical distance l) between given two sites on a percolation cluster are similar to the characteristics of optimal paths in the directed polymer model. A corollary which should be valid for the general dimension case, and verified by numerical results for the two-dimensional case, is that a cluster whose sites are at chemical distance l from a given site forms a Kardar-Parisi-Zhang surface.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

On the growth of percolation clusters: The effects of time correlations

We discuss the effects of the time correlations in the choice of growth sites for percolation clusters in two dimensions. To this end, we study two well-defined models: (i) FIFO (First-In, First-Out), in which the next-cluster growth site is theoldest, and (ii) FILO (First-In, Last-Out), where the next cluster growth site to be chosen is thenewest. We find that FIFO and FILO have dramatically differentkinetic exponents, even though thestatic exponents are the same (viz., percolation exponents). We find that the percolation thresholdp _c is analogous to the point of a linear polymer, and we develop the corresponding tricritical point scaling relations.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Centrality dependence of charged hadron transverse momentum spectra in Au+Au collisions from sqrt[s(NN)]=62.4 to 200 GeV

We have measured transverse momentum distributions of charged hadrons produced in Au+Au collisions at sqrt[s(NN)]=62.4 GeV. The spectra are presented for transverse momenta 0.25相似文献   

Sodium ion ordering and vacancy cluster formation in NaxCoO2 (x=0.71 and 0.84) single crystals by synchrotron X-Ray diffraction

In the rich phase diagram of NaxCoO2, x=0.71 enjoys special stability and is called the Curie-Weiss metal due to its anomalous properties. Similarly, x=0.84 prepared from high temperature melt is a special end point beyond which the system phase separates. Using synchrotron x-ray diffraction on single crystals, we discovered sqrt[12]a and sqrt[13]a superlattice structures which we interpret as the ordering of Na (vacancy) clusters. These results lead to a picture of coexisting local moments and itinerant carriers and form the first step towards understanding the many anomalous properties of cobaltates.     

Lateral diffusion and percolation in membranes

An algorithm based on Voronoi tessellation and percolation theory is presented to study the diffusion of model membrane components (solutes) in the plasma membrane. The membrane is modeled as a two-dimensional space with integral membrane proteins as static obstacles. The Voronoi diagram consists of vertices, which are equidistant from three matrix obstacles, joined by edges. An edge between two vertices is said to be connected if solute particles can pass directly between the two regions. The percolation threshold, pc, determined using this passage criterion is pc approximately equal to 0.53. This is smaller than if the connectivity of edges were assigned randomly, in which case the percolation threshold pr=2/3, where p is the fraction of connected edges. Molecular dynamics simulations show that diffusion is determined by percolation of clusters of edges.     

Collapse of percolation clusters — A transfer matrix study

Exact calculations using transfer matrices on finite strips are performed to study the two-dimensional problem of site percolation clusters with an attractive nearest neighbor interaction. Thermodynamic quantities such as free energy per site and specific heat are calculated. Finite-size scaling with two strips of different widths yields very accurate approximations of the critical line and the correlation length exponent. The result shows clearly a site percolation fixed point at very high temperatures, a random animal fixed point at intermediate temperatures, a point for the collapse of lattice animals at lower temperatures, and a compact-cluster fixed point at the lowest temperatures.On leave from Institute of Theoretical Physcis, Chinese Academy of Sciences, Beijing, China.     

Azimuthal anisotropy and correlations in the hard scattering regime at RHIC

Azimuthal anisotropy (v(2)) and two-particle angular correlations of high p(T) charged hadrons have been measured in Au+Au collisions at sqrt[s(NN)]=130 GeV for transverse momenta up to 6 GeV/c, where hard processes are expected to contribute significantly. The two-particle angular correlations exhibit elliptic flow and a structure suggestive of fragmentation of high p(T) partons. The monotonic rise of v(2)(p(T)) for p(T)<2 GeV/c is consistent with collective hydrodynamical flow calculations. At p(T)>3 GeV/c, a saturation of v(2) is observed which persists up to p(T)=6 GeV/c.     

The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

The percolation transition in filling a nanoporous body by a nonwetting liquid

The paper presents the results of an experimental study of the percolation transition in filling by nonwetting liquids of nanoporous bodies of various natures with different specific surface areas and mean pore and granule sizes. The liquid that we used was an aqueous solution of ethylene glycol. The hysteresis and non-outflow phenomena observed in this transition at various (known) surface energies of liquids were studied by varying the concentration of ethylene glycol. This helped us explain the mechanism of the percolation transition in filling nanoporous bodies with nonwetting liquids. It was shown that, to quantitatively describe the observed dependences in terms of percolation theory taking into account energy barriers to filling, we must use a non-scaling distribution function of clusters of accessible and filled pores that admits the formation of pore clusters of arbitrary dimensions.     

Elliptic flow of charged particles in Pb-Pb collisions at sqrt[S(NN)] = 2.76 TeV

We report the first measurement of charged particle elliptic flow in Pb-Pb collisions at sqrt[S(NN)] =2.76 TeV with the ALICE detector at the CERN Large Hadron Collider. The measurement is performed in the central pseudorapidity region (|η|<0.8) and transverse momentum range 0.2

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Distributions and moments of structural properties for percolation clusters

We study numerically and by scaling methods the distributions and moments of several structural properties of percolation clusters in two and three dimensions. The clusters are generated at criticality and properties such as the distribution of the mass as a function of linear size or chemical distance are studied. Our results suggest that the hierarchy of moments can be represented by a single gap exponent. Using a scaling approach, we obtain analytical forms for the different distribution functions which agree very well with the numerical data.     

New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor \(\sqrt{t}\) are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by \(\sqrt {t\ln t}\), with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.     

Absence of suppression in particle production at large transverse momentum in sqrt[s(NN)]=200 GeV d+Au collisions

Transverse momentum spectra of charged hadrons with p(T)<8 GeV/c and neutral pions with p(T)<10 GeV/c have been measured at midrapidity by the PHENIX experiment at BNL RHIC in d+Au collisions at sqrt[s(NN)]=200 GeV. The measured yields are compared to those in p+p collisions at the same sqrt[s(NN)] scaled up by the number of underlying nucleon-nucleon collisions in d+Au. The yield ratio does not show the suppression observed in central Au+Au collisions at RHIC. Instead, there is a small enhancement in the yield of high momentum particles.     

Transition from Kardar-Parisi-Zhang to tilted interface critical behavior in a solvable asymmetric avalanche model

We use a discrete-time formulation of the asymmetric avalanche process (ASAP) [Phys. Rev. Lett. 87, 084301 (2001)]] of p particles on a finite ring of N sites to obtain an exact expression for the average avalanche size as a function of toppling probabilities and particle density rho=p/N. By mapping the model onto driven interface problems, we find that the ASAP incorporates the annealed Kardar-Parizi-Zhang and quenched tilted interface dynamics for rhorho(c), respectively, with rho(c) being the critical density for given toppling probabilities and N--> infinity. We analyze the crossover between two regimes and show which parameters are relevant near the transition point.     

Multifractality of self-avoiding walks on percolation clusters

We consider self-avoiding walks on the backbone of percolation clusters in space dimensions d=2,3,4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the peculiarities of the model. We obtain estimates for the set of critical exponents that govern scaling laws of higher moments of the distribution of percolation cluster sites visited by self-avoiding walks, in a good correspondence with an appropriately summed field-theoretical epsilon=6-d expansion [H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)10.1103/PhysRevE.75.020801].     

Clusters and Ising droplets in the antiferromagnetic lattice gas

A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if P_b = 1 - exp(-|K|/2). Along this line these clusters called “droplets” diverge with Ising exponents.     

Strong disorder fixed point in absorbing-state phase transitions

The effect of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class is studied by a strong disorder renormalization group approach and by density matrix renormalization group calculations. We show that for sufficiently strong disorder the critical behavior is controlled by a strong disorder fixed point and in one dimension the critical exponents are conjectured to be exact: beta=(3-sqrt[5])/2 and nu( perpendicular )=2. For disorder strengths outside the attractive region of this fixed point, disorder dependent critical exponents are detected. Existing numerical results in two dimensions can be interpreted within a similar scenario.