on the hull of two dimensional percolation clusters a recently proven equivalence between certain random walks with memory and the hull of percolation clusters in two dimensions is used to estimate the fractal dimension of the latter by means of monte carlo simulations the authors calculate the value dh 1 750 or 0 002 in agreement with a recent conjecture that dh 7 4 exactly

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

conservative nonlinear difference scheme for the cahn hilliard equation ii abstract numerical solutions for the cahn hilliard equation is considered using the crank nicolson type finite difference method existence of the solution for the difference scheme has been shown by brouwer fixed point theorem stability convergence and error analysis of the scheme are shown we also show that the scheme preserves the discrete mass even though the linearized scheme in 1 is conditionally stable and does not preserve the mass

monte carlo series analysis of irreversible self avoiding walks i the indefinitely growing self avoiding walk igsaw high precision monte carlo data are used to estimate the exponents which govern the asymptotic behaviour of the recently introduced indefinitely growing self avoiding walk in two dimensions for this walk the exponent gamma is by definition equal to one applying the same methods which are used to extract the exponents from exact series enumeration the authors give an estimate for the exponent nu of 0 567 or 0 003 the leading corrections to this asymptotic behaviour are also calculated

on scaling relations in growth models for percolating clusters and diffusion fronts employing analogies between the growth of the incipient percolating cluster and the growth of a diffusion front the authors develop a scaling theory for this latter model their scaling assumptions support the sapoval rosso gouyet conjecture dh 1 1 nu which relates in two dimensions the fractal dimension dh of the hull of the incipient percolating cluster to the correlation length exponent nu in percolation and yield relations between the static and dynamic exponents of the growth of a diffusion front

the fractal dimension of percolation cluster hulls large simulations of 2d site percolation on a square lattice provide several estimates of the fractal dimension dh of the hull or outer boundary of percolation clusters close to criticality statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield dh 1 74 or 0 02 geometrical measurements of the hull two point correlation for individual clusters with more than 105 sites give dh 1 76 or 0 01 the observed scaling behaviour is the same both slightly above and below pc

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

finite element approximation of the cahn hilliard cook equation we study the nonlinear stochastic cahn hilliard equation perturbed by additive colored noise we show almost sure existence and regularity of solutions we introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1 we also prove strong convergence without known rate

distance weighted city growth urban agglomerations exhibit complex emergent features of which zipf s law i e a power law size distribution and fractality may be regarded as the most prominent ones we propose a simplistic model for the generation of citylike structures which is solely based on the assumption that growth is more likely to take place close to inhabited space the model involves one parameter which is an exponent determining how strongly the attraction decays with the distance in addition the model is run iteratively so that existing clusters can grow together and new ones can emerge the model is capable of reproducing the size distribution and the fractality of the boundary of the largest cluster although the power law distribution depends on both the imposed exponent and the iteration the fractality seems to be independent of the former and only depends on the latter analyzing land cover data we estimate the parameter value ensuremath gamma ensuremath approx 2 5 for paris and its surroundings

percolation and self avoiding random loops presents simulations supporting the conjecture that properly defined exterior perimeters of 2d critical percolation clusters have the same fractal dimension d 4 3 as self avoiding random walks in addition the authors discuss possible generalizations

a parallel domain decomposition based implicit method for the cahn hilliard cook phase field equation in 3d we present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional cahn hilliard cook chc equation which is a fourth order stochastic partial differential equation with a noise term the equation is discretized in space and time based on a fully implicit cell centered finite difference scheme with an adaptive time stepping strategy designed to accelerate the progress to equilibrium at each time step a parallel newton krylov schwarz algorithm is used to solve the nonlinear system we discuss various numerical and computational challenges associated with the method the numerical scheme is validated by a comparison with an explicit scheme of high accuracy and unreasonably high cost we present steady state solutions of the chc equation in two and three dimensions the effect of the thermal fluctuation on the spinodal decomposition process is studied we show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise we also show the evolution of the energies and statistical moments in terms of the parallel performance it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors

a nonconforming finite element method for the cahn hilliard equation this paper reports a fully discretized scheme for the cahn hilliard equation the method uses a convexity splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable and the scheme can preserve the mass conservation and energy dissipation properties of the original problem some typical phase transition phenomena are also observed through the numerical examples

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

scaling properties of antipercolation hulls on the triangular lattice the critical properties of the perimeters or hulls of antipercolation clusters are studied in two dimensions by monte carlo simulations on the triangular lattice two different types of hulls are constructed with the help of two kinetic walk algorithms for the standard hull the authors very accurately determine the size distribution exponent tau 2 14250 0003 as well as the fractal dimension dfs 1 7500 001 the corresponding exponents for regular percolation hulls 15 7 and 7 4 respectively are within the error bars of their results for antipercolation for the reduced hull they obtain the estimate dfr 1 3340 004 for the fractal dimension a result which is again close to that found for regular percolation

structure of backbone perimeters of percolation clusters the author studies perimeters of backbones of percolation clusters at the percolation threshold in two dimensions these perimeters are self avoiding for all lattices they simulate these perimeters by three independent processes measure the average size of them and study their variation with length they find the exponent characterising this variation to be 0 744 or 0 012 it seems likely that these perimeters are in the self avoiding walk universality class

percolation and motion in a simple random environment the authors present a particularly simple model of deterministic classical motion in a two dimensional random environment as the parameters of the model are varied a transition occurs from all trajectories being localised to some being extended they construct a mean field theory for this transition and relate the model exactly to percolation models in particular parameter ranges they point out that it is a member of a new class of site percolation analogues of direct bond percolation

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

second phase spinodal decomposition for the cahn hilliard cook equation we consider the dynamics of a nonlinear partial differential equation perturbed by additive noise assuming that the underlying deterministic equation has an unstable equilibrium we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium more precisely we show that most trajectories starting at the unstable equilibrium are driven away in two stages after passing through a cylindrical region most trajectories diverge from the deterministic equilibrium through a cone shaped region which is centered around a finite dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation and on which the influence of the nonlinearity is surprisingly small this abstract result is then applied to explain spinodal decomposition in the stochastic cahn hilliard cook equation on a domain g this equation depends on a small interaction parameter e 0 and one is generally interested in asymptotic results as e 0 specifically we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e 2 dim g 2 with respect to the h 2 g norm

computationally efficient solution to the cahn hilliard equation adaptive implicit time schemes mesh sensitivity analysis and the 3d isoperimetric problem we present an efficient numerical framework for analyzing spinodal decomposition described by the cahn hilliard equation we focus on the analysis of various implicit time schemes for two and three dimensional problems we demonstrate that significant computational gains can be obtained by applying embedded higher order runge kutta methods in a time adaptive setting this allows accessing time scales that vary by five orders of magnitude in addition we also formulate a set of test problems that isolate each of the sub processes involved in spinodal decomposition interface creation and bulky phase coarsening we analyze the error fluctuations using these test problems on the split form of the cahn hilliard equation solved using the finite element method with basis functions of different orders any scheme that ensures at least four elements per interface satisfactorily captures both sub processes our findings show that linear basis functions have superior error to cost properties this strategy coupled with a domain decomposition based parallel implementation let us notably augment the efficiency of a numerical cahn hillard solver and open new venues for its practical applications especially when three dimensional problems are considered we use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions the framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3d sphere cylinder lamella doubly periodic surface with genus two lawson surface and triply periodic minimal surface p schwarz surface

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

effect of loops on the vibrational spectrum of percolation network abstract we study the effects of adding loops to a critical percolation cluster on the diffusional and equivalently scalar elastic properties of the fractal network from the numerical calculations of the eigenspectrum of the transition probability matrix we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops and we find that the additional inclusion of successively smaller loops only change these exponents little if at all this suggests that there is a new universality class associated with the loop enhanced percolation problem

accessible external perimeters of percolation clusters the authors introduce a new definition of a cluster s accessible external perimeter which is different from the usual definition of the hull this external perimeter can be measured by probing adsorbent particles coming from the cluster s exterior the size of these particles determines which perimeter sites are accessible from the outside and hence one can define different external or accessible perimeters according to this size using a monte carlo simulation they measured the fractal dimensions of these accessible perimeters de for percolation clusters at the threshold on the square and the triangular lattices the fractal dimension of the accessible perimeter is found to depend on the size of the probing particles for several sizes which are larger than some lattice dependent threshold they find de approximately 4 3 which is significantly smaller than the hull dimension dh 7 4 found with smaller probing particles

generation of percolation cluster perimeters by a random walk a type of self avoiding random walk whish generates the perimeter of two dimensional lattice percolation clusters is given the algorithm has been simulated on a computer yielding the mean perimeter length as a function of occupation probability

concentration gradient approach to continuum percolation in two dimensions the author has investigated percolation properties of a system of overlapping discs randomly distributed with a gradient of concentration fractal properties and critical exponents of this system appear to be identical to their counterparts on a lattice this is in agreement with the universality of critical exponents of percolation the concentration gradient approach permits a precise calculation of the percolation threshold corresponding to a critical area fraction of 0 6766 or 0 0005 he has investigated the part of the external perimeter of the percolation cluster which is accessible to discs of various sizes the fractal dimension of this accessible perimeter is found to decrease abruptly as a function of the radius of invading discs from 1 75 or 0 02 to 1 35 or 0 02

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

external surface of site percolation clusters in three dimensions the authors study the external surface of site percolation clusters in three dimensions by monte carlo simulations on a simple cubic lattice at p pc 0 3117 their results show that for all cluster sizes s more than 99 8 of the occupied sites are on the external surface the number of unoccupied boundary sites t is found to obey t s 1 pc pc as psi with psi 0 54 1 sigma in agreement with the prediction of a scaling argument the same argument also predicts psi 0 60 in two dimensions and it is confirmed the relevance of these results to dynamical percolation growth models is discussed

loop cluster coupling and algorithm for classical statistical models potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin the fortuin kasteleyn fk bond or the q flow loop representation we introduce a loop cluster lc joint model of bond occupation variables interacting with q flow variables and formulate an lc algorithm that is found to be in the same dynamical universality as the celebrated swendsen wang algorithm this leads to a theoretical unification for all the representations and numerically one can apply the most efficient algorithm in one representation and measure physical quantities in others moreover by using the lc scheme we construct a hierarchy of geometric objects that contain as special cases the q flow clusters and the backbone of fk clusters the exact values of whose fractal dimensions in two dimensions remain as an open question our work not only provides a unified framework and an efficient algorithm for the potts model but also brings new insights into the rich geometric structures of the fk clusters

numerical method for a singularly perturbed convection diffusion problem with delay this paper deals with the singularly perturbed boundary value problem for a linear second order delay differential equation for the numerical solution of this problem we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form it is shown that one gets first order convergence in the discrete maximum norm independently of the perturbation parameter numerical results are presented which illustrate the theoretical results

cellular automata approach to non equilibrium diffusion and gradient percolation the authors propose a deterministic approach to lattice diffusion in two dimensions this method is implemented on a cellular automaton special purpose computer in order to study the properties of the interface of particles diffusing from a source to a sink fractal properties of the diffusion front of this non equilibrium process are compared with results from percolation theory the observed agreement indicates that diffusion fronts and gradient percolation coincide asymptotically and that the cellular automata method is a viable alternative to standard simulations for this class of problems

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

spinodal decomposition for the cahn hilliard cook equation this paper gives theoretical results on spinodal decomposition for the stochastic cahn hilliard cook equation which is a cahn hilliard equation perturbed by additive stochastic noise we prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation leading to the formation of complex patterns of a characteristic size