Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers

Nakanishi, Hisao; Stanley, H. Eugene

doi:10.1103/physrevb.22.2466

Cited by 111 publications

(45 citation statements)

References 41 publications

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“…Additionally, also root labels can be recycled, when they are not present in the current hyperplane of investigation. These methods are known as Nakanishi recycling [10].…”

Section: Recyclingmentioning

confidence: 99%

Simulation of Percolation on Massively-Parallel Computers

Tiggemann¹

2001

Int. J. Mod. Phys. C

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A novel approach to parallelize the well-known Hoshen-Kopelman algorithm has been chosen, suitable for simulating huge lattices in high dimensions on massively-parallel computers with distributed memory and message passing. This method consists of domain decomposition of the simulated lattice into strips perpendicular to the hyperplane of investigation that is used in the Hoshen-Kopelman algorithm. Systems of world record sizes, up to L = 4000256 in two dimensions, L = 20224 in three, and L = 1036 in four, gave precise estimates for the Fisher exponent τ , the corrections to scaling ∆1, and for the critical number density nc.Percolation is a thoroughly studied model in statistical physics [2]. The algorithm invented by Hoshen and Kopelman in 1976 allows for examining large percolation lattices using Monte Carlo methods [3]. Unfortunately, this algorithm was invented for traditional sequential computers, and implementation on modern parallel computers is far from trivial.The normal way to parallelize an algorithm which works on regular data structures is domain decomposition. In this case, the investigated lattice is cut into strips, and each processor is assigned one such strip for investigation. As the Hoshen-Kopelman algorithm investigates the lattice hyperplane by hyperplane (line by line in two dimensions; plane by plane in three dimensions), various ways of domain decomposition can be characterized by this hyperplane: for example, strips parallel or perpendicular to that plane.The easiest way would be to choose parallel strips, as in that case the simulation of each strip can be carried out locally on each processor, only after that communication is neccessary (exchanging the borders). Unfortunately, this means that each processor has to store a full hyperplane ofsystem. In two dimensions, this no problem at all, but in higher dimensions this limits the size of systems that can be simulated. When decomposing the system in perpendicular strips, each processor has to store only a part of the hyperplane, and thus larger system sizes can be 1

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“…Additionally, also root labels can be recycled, when they are not present in the current hyperplane of investigation. These methods are known as Nakanishi recycling [10].…”

Section: Recyclingmentioning

confidence: 99%

Simulation of Percolation on Massively-Parallel Computers

Tiggemann¹

2001

Int. J. Mod. Phys. C

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“…Consequently the scaling function f (x) = f * (x) + a 0 + ... + a n * x n * σ must have a maximum f (x max >0) > f (0) in the phase t < t c [2]. This asymmetry of the scaling function is observed, for example, for ordinary percolation at dimension d < 6 [25], and in explosive percolation [6,14,21]. In these examples, 0 < β < 1.…”

Section: Relation Between the Singularity Of S And The Scaling Functimentioning

confidence: 97%

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

et al. 2015

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We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely before the emergence of the percolation cluster, the system is in a "critical phase" with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that susceptibility in this situation does not obey the Curie-Weiss law.

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“…where n A N A ¢ A 0 is the number of droplets N A of mass A, normalized to the system size A 0 ; q 0 is a normalization constant depending only on the value of τ [49]; τ is the topological critical exponent; ∆µ µ ¡ µ l , and µ and µ l are the actual and liquid chemical potentials respectively; c 0 εA σ is the surface free energy of a droplet of size A; c 0 is the zero temperature surface energy coefficient; σ is the critical exponent related to the ratio of the dimensionality of the surface to that of the volume; and ε § Recently, multifragmentation data from the Indiana Silicon Sphere (ISiS) Collaboration was shown to exhibit both reducibility and thermal scaling [50,51], thus it may be interesting to determine if Fisher's model describes the ISiS data as well. In order to find if this is the case, the ISiS charge yields from AGS experiment E900a of 8 GeV/c π ¤ Au fragmentation data (see Fig.…”

Section: Experimental Properties Of the Saturated Nuclear Vapor And Tmentioning

confidence: 99%

Theoretical approaches and experimental evidence for liquid-vapor phase transitions in nuclei

Moretto¹,

Elliott²,

Phair³

et al. 2002

AIP Conference Proceedings

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Abstract. The leptodermous approximation is applied to nuclear systems for T ¢ 0. The introduction of surface corrections leads to anomalous caloric curves and to negative heat capacities in the liquid-gas coexistence region. Clusterization in the vapor is described by associating surface energy to clusters according to Fisher's formula. The three-dimensional Ising model, a leptodermous system par excellence, does obey rigorously Fisher's scaling up to the critical point. Multifragmentation data from several experiments including the ISiS and EOS Collaborations, as well as compound nucleus fragment emission at much lower energy follow the same scaling, thus providing the strongest evidence yet of liquid-vapor coexistence.

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Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers

Cited by 111 publications

References 41 publications

Simulation of Percolation on Massively-Parallel Computers

Simulation of Percolation on Massively-Parallel Computers

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

Theoretical approaches and experimental evidence for liquid-vapor phase transitions in nuclei

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