Position Space Renormalization Group Approach Research Articles

A non-equilibrium Monte Carlo renormalization-group approach based upon the microscopic master equation applied to the three-state driven lattice gas

We present a general position-space renormalization-group approach for systems in steady states far from equilibrium and illustrate its application to the three-state driven lattice gas. The method is based upon the possibility of a closed form representation of the parameters controlling transition rates of the system in terms of the steady state probability distribution of small clusters, arising from the application of the master equations to small clusters. This probability distribution on various length scales is obtained through a Monte Carlo algorithm on small lattices, which then yields a mapping between parameters on different length scales. The renormalization-group flows indicate the phase diagram, analogous to equilibrium treatments. For the three-state driven lattice gas, we have implemented this procedure and compared the resulting phase diagrams with those obtained directly from simulations. Results in general show the expected topology with one exception. For high densities, an unexpected additional fixed point emerges, which can be understood qualitatively by comparing it with the fixed point of the fully asymmetric exclusion process.

Position-space renormalization-group approach for driven diffusive systems applied to the asymmetric exclusion model.

This paper introduces a position-space renormalization-group approach for nonequilibrium systems and applies the method to a driven stochastic one-dimensional gas with open boundaries. The dynamics are characterized by three parameters: the probability alpha that a particle will flow into the chain to the leftmost site, the probability beta that a particle will flow out from the rightmost site, and the probability p that a particle will jump to the right if the site to the right is empty. The renormalization-group procedure is conducted within the space of these transition probabilities, which are relevant to the system's dynamics. The method yields a critical point at alpha(c)=beta(c)=1/2, in agreement with the exact values, and the critical exponent nu=2.71, as compared with the exact value nu=2.00.

Novel position-space renormalization group for bond directed percolation in two dimensions

A new position-space renormalization group approach is investigated for bond directed percolation in two dimensions. The threshold value for the bond occupation probabilities is found to be p c =0.6443. Correlation length exponents on time (parallel) and space (transverse) directions are found to be ν ∥=1.719 and ν ⊥=1.076, respectively, which are in very good agreement with the best-known series expansion results.

Double transitions in the fully frustrated XY model

The fully frustrated XY model is studied via the position-space renormalization group approach. The model is mapped into two coupled XY models, for which the scaling equations are derived. By integrating directly the scaling equations, we observe that there exists a narrow temperature range in which both the vortex and coupling charge fugacities grow large, suggesting double transitions in the system. While the transition at lower temperature is identified to be of the Kosterlitz-Thouless type, the higher-temperature one appears not to be of the Ising universality class.

On position-space renormalization group approach to percolation

In a position-space renormalization group (PSRG) approach to percolation one calculates the probabilityR(p,b) that a finite lattice of linear sizeb percolates, wherep is the occupation probability of a site or bond. A sequence of percolation thresholdsp c (b) is then estimated fromR(p c ,b)=p c (b) and extrapolated to the limitb→∞ to obtainp c =p c (∞). Recently, it was shown that for a certain spanning rule and boundary condition,R(p c ,∞)=R c is universal, and sincep c is not universal, the validity of PSRG approaches was questioned. We suggest that the equationR(p c ,b)=α, where α isany number in (0,1), provides a sequence ofp c (b)'s thatalways converges top c asb→∞. Thus, there is anenvelope from any point inside of which one can converge top c . However, the convergence is optimal if α=R c . By calculating the fractal dimension of the sample-spanning cluster atp c , we show that the same is true aboutany critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.

Influence of morphology on the optical properties of nanocermets and granular inhomogeneous media

The macroscopic dielectric function of 2-dimensional (2D) and 3-dimensional (3D) inhomogeneous media depends on the morphology of the sample. Usual effective medium theories (namely the Maxwell Garnett theory and the mean field theories of Bruggeman, Ping Sheng) do not take into account the actual morphology of the clusters. We proposed earlier a position space renormalization group approach, based on the Kadanoff blocks method, to calculate the optical properties of cermets and granular materials. The aim of this work is to show the influence of the morphology of these media on their optical properties by applying this model to various samples that only differ in their morphological parameters (same materials, same concentration and same thickness).

Diffusion-Limited Aggregation on Percolating Cluster: Crossover and Multifractal Structure

Viscous fingering in porous media is considered as the diffusion-limited aggregation (DLA) on the percolating cluster. The crossover between percolation and DLA is studied by using a three-parameter position-space renormalization-group approach. The global flow diagram in the two-parameter space is obtained. It is found that there are two nontrivial fixed points, the percolation point and the DLA point. Above the percolation threshold, the system crosses eventually over to the DLA fractal on the perfect lattice. The fractal nature and the multifractal structure of the growth probability distribution are derived from the position-space renormalization-group method. The multifractal structure at the percolation threshold is compared with that at the perfect lattice.

Optical properties of two-dimensional and three-dimensional inhomogeneous media: a renormalization approach

The authors propose a position space renormalization group approach based on the Kadanoff block method, for the calculation of an effective dielectric function of cermet-type films (3D) or island films (2D). These models have been applied to simulated lattices and to digitized transmission electron micrographs of 2D and 3D systems. The results account for both the resonant absorption and the optical crossover corresponding to the metal-to-non-metal transition. The effective dielectric function is in very good quantitative agreement with most of the experimental data, even near the percolation threshold where it is found to obey scaling laws with critical exponents very close to the theoretical values.

Crossover and thermodynamic representation in the extended eta model for fractal growth.

The \ensuremath{\eta} model for the dielectric breakdown is extended to the case where double power laws apply. It is shown that a crossover phenomenon between the diffusion-limited aggregation (DLA) fractal and the \ensuremath{\eta} fractal occurs in the extended \ensuremath{\eta} model. Through the use of the dimensional analysis, a dimensionless parameter is found to govern the crossover. It is shown that when \ensuremath{\eta}1 the crossover from the DLA fractal to the \ensuremath{\eta} fractal occurs with increasing size, and if \ensuremath{\eta}>1 the inverse crossover from the \ensuremath{\eta} fractal to the DLA fractal appears. It is also shown that the crossover radius is controlled by changing the applied field. The global flow diagram in the two-parameter space is obtained by using a two-parameter position-space renormalization-group approach. The crossover exponent and the crossover radius are calculated. The crossover phenomenon is described in terms of a thermodynamic representation of the two-phase equilibrium.

Fractal-to-nonfractal crossover for viscous fingers.

We propose a position-space renormalization-group approach to the problem of viscous fingering in the absence of surface tension, with arbitrary viscosity ratio between the injected and displaced fluid. We find there are only two fixed points, the Eden and the diffusion-limited aggregation (DLA) points. The Eden point, which corresponds to a compact cluster with nonfractal surface, is stable in all directions, while the DLA fixed point is a saddle point. Hence if the viscosity of the injected fluid is not zero, the system must eventually cross over to a compact cluster. We also calculate the crossover exponent \ensuremath{\varphi} and crossover radius ${\mathit{R}}_{\ifmmode\times\else\texttimes\fi{}}$, and discuss possible experimental measurements.

Position-space renormalization group for SAWs on the L lattice

The self-avoiding walk problem on the L lattice is treated with a position-space renormalization group approach using a 'centre-rule' weight function.

Calculation of the optical dielectric function of 2D and 3D inhomogeneous media by real space renormalization

We propose a position space renormalization group approach based on Kadanoff's block method, for the calculation of an effective dielectric function of cermet type films (3D) or island films (2D). The results obtained on simulated lattices account for both the resonant absorption and the optical cross-over corresponding to the metal-nonmetal transition. The effective dielectric function is found to obey scaling laws with critical exponents s 3D = 0.75 and s 2D = 1.35 for polarization terms and t 3D = 2.2, t 2D = 1.52 for the conduction, very close to the theoretical estimates. These models have also been applied to digitized transmission electron micrographs of 2D and 3D systems, and we obtain a very good agreement with most of the experimental data.

Position space renormalisation group approach to fractal dimensions of an infinite cluster, its backbone and cutting bonds for percolation

A position space renormalisation group method is presented to study the cluster structure of an infinite cluster at the percolation threshold. Fractal dimensions of an infinite cluster, its backbone and cutting bonds are determined by the small-cell method for two-dimensional bond percolation. It is shown that the fractal dimension of its cutting bonds agrees with the inverse of the connectedness length exponent at the fixed point.

Ising model at an edge: a position space renormalisation group approach

The critical behaviour of a three-dimensional Ising model, bounded by two plane surfaces meeting at an angle alpha , is studied using the Migdal-Kadanoff and two-terminal cluster renormalisation methods. The authors obtain expressions for the edge fixed point and for the independent edge exponent yhe at the various transitions. They find that yhe is not completely universal but depends on the angle alpha , and that it shows some unexpected features.

The Ising antiferromagnet in a uniform field

The antiferromagnetic Ising model in a uniform field on a square lattice is studied by a position-space renormalisation group approach. A self-dual cluster that preserves sublattice symmetry is employed: at each site of the cluster, the coupling to the field is chosen to be proportional to the coordination number of that site. The author obtains the physically correct result that the magnetic field is irrelevant (in the renormalisation group sense) near the zero-field fixed point. The phase diagram he obtains compares well with experimental results for CoCs3Br5 and with the conjectured analytic form postulated by Muller-Hartmann and Zittartz. He discusses possible applications of this approach to other situations of interest.

Spiral bond animals-ratio approach

Bond animals with a constraint of a given winding direction on the square lattice are enumerated up to 14 bonds. Numerical evidence further confirms the previous conjecture using the position space renormalisation group approach, that they belong to a new universality class.

Position-space renormalization-group approach to the resistance of random walks

Position-space renormalization-group approach to the resistance of random walks

A direct renormalization group study of loops in polymers

A direct position space renormalization group approach is developed and used to study the effects of loops on the universal properties of models of ring polymers, linear polymers, branched polymers and gels. It is found that the presence or absence of loops does not change the critical exponents and other universal properties of the models studies.

Randomly directed bond percolation: a position-space renormalisation group approach

The position-space renormalisation group technique is applied to the randomly directed bond percolation problem. There are three types of bonds in the lattice considered. The bond is undirected with probability u. The bond randomly oriented along one of the two possible directions is present with probability q. The bond is broken with probability 1-u-q. A recursion relation fixed point different from that for the usual undirected bond percolation problem is found. The critical exponents and the percolation threshold probabilities are obtained for percolation in the square lattice.

Directed and diode percolation

We study the novel percolation phenomena that occur in random-lattice networks consisting of resistor-like and diode-like bonds. Resistor bonds connect or "transmit information" in either direction along their length, while diodes connect in one direction only. We first treat the special case of directed bond percolation, in which the diodes are aligned along a preferred axis. Mean-field theory shows that clusters become extremely anisotropic near the percolation transition and that their shapes are characterized by two correlation lengths, one parallel and one transverse to the preferred axis. These lengths diverge with exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=\frac{1}{2}$, respectively, from which we can show that the upper critical dimension for this system must be five. We also treat a more general random network on the square lattice containing resistors and diodes of arbitrary orientation. Duality arguments are applied to obtain exact results for the location of phase transitions in this system. We then use a position-space renormalization-group approach to map out the phase diagram and calculate critical exponents. This system has an isotropic percolating phase, and phases which percolate in only one direction. Novel types of transitions occur between these phases, in which the diode orientation plays a fundamental role. These percolating phases meet with the nonpercolating phase along a line of multicritical points, where concentration and orientational fluctuations are simultaneously critical.