time dependent critical properties of ising models by damage spreading abstract we use the method of damage spreading to study the time t taken for a spin at a distance r to be damaged for the first time in ising models with heat bath dynamics we report time dependent scaling exponents d t 2 0 1d 2 52 2d 2 26 3d and 2 0 4d if we assume that t r dt we also analyze our data assuming various forms of logarithmic corrections and we find that d t 2 25 or less 2d and 2 0 3d we argue that it is extremely plausible that d t is equal to z the dynamic critical exponent and thus our estimate of z in two dimensions is higher than the consensus value of 2 14 but we find good agreement in one three and four dimensions with the exact and expected values

monte carlo renormalization of the three dimensional ising model abstract we have investigated the simple cubic ising model by means of the monte carlo renormalization technique the emphasis of our study concerns the influence of truncation i e the dimensionality of the coupling subspace in which the analysis of the correlation functions generated by the monte carlo and spin blocking algorithms is performed to this purpose we have included up to 36 even and 21 odd couplings in our analysis we find that the increase in the number of couplings has a significant influence on the largest eigenvalues of the linearized renormalization transformation matrices these eigenvalues serve to estimate the renormalization exponents yi and yh remarkably we find no significant finite size effect on these eigenvalues when the maximum number of couplings is included in the analysis except for the smallest system sizes 83 43 after a suitable extrapolation to the fixed point we find good agreement with existing results for the critical exponents we have determined the critical point of the simple cubic ising model as k 0 221652 6 also in agreement with existing results

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

surface tension and universality in the ising model abstract we report on a new numerical computation of the surface tension between domains of opposite magnetisation in the ising model in two and three dimensions the method is quite general and can be applied to any statistical mechanics model it is based on a partition of the lattice into two halves which are slowly driven from one magnetization state to the other the free energy of the interface is the result of the difference between the total free energies of the homogeneous and of the mixed phases we first test the method in the two dimensional ferromagnetic ising model by comparing with the onsager solution we then compare the results in three dimensions with previous monte carlo simulations of thermal equilibrium and of decays via nucleation the three dimensional results are also used in the computation of the universal amplitude combination that involves the correlation length and the surface tension near the critical temperature the numerical results agree well with measurements performed in the laboratory

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

a second multivariable normal form for model reduction of discrete time systems the normal form derived by mansour i for single input single output time invariant linear discrete systems will be extended to the multivariable case an algorithm will be used to achieve simple computation of this form starting with the luenberger first canonical form the construction of this form will be illustrated by examples applied to model reduction of multivariable discrete time systems

universal surface tension and critical isotherm amplitude ratios in three dimensions the universal surface tension and critical isotherm amplitude ratios are studied numerically for three dimensional ising models modern estimates of the critical temperature and exponents allow reliable evaluation of the critical surface tension amplitude k using recent monte carlo data for the simple cubic lattice likewise the amplitudes cc for the susceptibility and 1c for the second moment correlation length on the critical isotherm have been re estimated using existing series expansions the method of inhomogeneous differential approximants also yields a direct estimate of the correction to scaling exponent c on the critical isotherm which via scaling corresponds to the thermal correction exponent 0 55 5 this supports previous estimates and the stronger conclusion 0 54 3 for the universal ratios we estimate k 1 2 0 0965 2 cc b 1c 1 0 93 25 and c cc 1c 1 2 1 17 2 where b 1 and c are the amplitudes of the spontaneous magnetization and second moment correlation length and of the susceptibility above tc

the 2 1 d ising model on a triangular lattice the 2 1 d ising model on a triangular lattice is explored for lattices of up to 5 5 sites finite size scaling techniques are used to estimate the critical parameters xc 0 2096 2 nu 0 627 4 beta 0 322 6 gamma 1 236 8 these agree with previous estimates within errors and with the hypotheses of universality and hyperscaling the behaviour at low temperatures is also studied and theoretical predictions of finite size scaling at a first order transition are confirmed

the sum of like powers of the zeros of the riemann zeta function on analyse une methode de calcul de la somme r p r ou r est un entier superieur a 1 la somme etant prise sur tous les zeros complexes de la fonction zeta de riemann

critical slowing down in local dynamics simulations the author reports some universal and non universal aspects of the critical dynamics of the three dimensional ising model obtained in extensive monte carlo finite size simulations the author shows that the time dependence of the magnetization of finite lattices is composed of two kinds of fluctuations at the critical point i phase fluctuations from one metastable minimum of the free energy to the other dominating the long time behaviour of the magnetization ii critical fluctuations inside each minimum decaying on a comparatively short timescale both kinds of fluctuations show up the same critical exponent z 2 10 or 0 02 which is also in excellent agreement with the exponent z of energy fluctuations

on model reduction of discrete time systems abstract in this paper further justification of model reduction by badreddin mansour s method is presented in particular connections between polynomial root location properties coefficient properties and schur cohn coefficient properties are established lattice realizations of a discrete system and its reduced model are given which illustrate certain relationships between the original model and its reduced form these relationships give insight into the validity of the method of reduction and offer insight on the order of approximation

the dynamical critical exponent of the two three and five dimensional kinetic ising model abstract we present monte carlo simulation results for the dynamical critical exponent z of the two three and five dimensional kinetic ising model we used glauber as well as metropolis dynamics both type of dynamics lead to the same z values in two and three dimensions the z values were calculated from the magnetization relaxation from an ordered state into the equilibrium state at tc for very large systems up to 169984 2 3072 3 and 48 5 for the two dimensional case we obtain z 2 21 0 03 and for the three dimensional case z 2 08 0 03 for the five dimensional glauber case the requirement z 2 gives a critical point j kbtc 0 11391 from monte carlo simulations whereas new series expansions with four more terms give 0 113935 0 000015 for this ratio

statistical and systematic errors in monte carlo sampling we have studied the statistical and systematic errors which arise in monte carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used we find that depending on the degree of self averaging exhibited by the quantities measured the statistical errors can increase decrease or stay the same as the system size is increased the systematic underestimation of response functions due to the finite number of measurements made is also studied we develop a scaling formalism to describe the size dependence of these errors as well as their dependence on the bin length size of the statistical sample both at and away from a phase transition the formalism is tested using simulations of thed 3 ising model at the infinite lattice transition temperature we show that for a 96 96 96 system noticeable systematic errors systematic underestimation of response functions are still present for total run lengths of 106 monte carlo steps site mcs with measurements taken at regular intervals of 10 mcs

on the critical polynomial of the simple cubic ising model in 1985 rosengren conjectured that the critical point of the symmetric simple cubic sc ising model is given by nu c identical to tanh j kbtc nu r identical to square root 5 2 cos pi 8 this guess is examined in the context of attempting to construct the fully critical polynomial p3 nu x nu y nu z with a root nu c jx jy jz for the anisotropic sc ising model with couplings jx jy and jz it transpires that nu r is a surd which satisfies r nu r2 0 where r x is a quartic polynomial with integral coefficients but r nu 2 is a poor candidate for p3 nu nu nu since it does not display various nice properties embodied in the critical polynomial p2 nu x nu y for the square 2d ising lattices methods for constructing nice polynomials qk nu x nu y nu z that provide excellent approximations for nu c and for nu r are demonstrated however scaling arguments etc for the dimensional crossover induced when say jz to 0 cast doubt on the existence and nature of the sought for critical polynomial p3

on the dynamic exponent of the two dimensional ising model the authors use the method of damage spreading to measure the time taken for the equilibrium damage the magnetization to disappear in finite systems 10 or 2 27

analytic continuation at first order phase transitions we study the analytic structure of thermodynamic functions at first order phase transitions in systems with short range interactions and in particular in the two dimensional ising model we analyze the nature of the approximation of the d 2 system by ann strip investigation of the structure of the eigenvalues of the transfer matrix in the vicinity of h 0 in the complexh plane allows us to define a new function which provides rapidly convergent approximations to the stable free energyf and its derivatives for allh 0 this new function is used for numerical calculation of the coefficients cn in the power series expansions of the magnetizationm in the form m h 1 cn h h 0 n for various h0 0 the resulting series are studied by conventional methods we confirm recent series analysis results on the existence of the droplet model type essential singularity at h 0 evidence is found for a spinodal at h hsp ti 0

a proof of a discrete stability test via the lyapunov theorem in this note a discrete stability test is proved by using the lyapunov theorem the proof is simple and elegant and is similar to the celebrated proof of the routh hurwitz criterion by parks 1

comparison of monte carlo results for the 3d ising interface tension and interface energy with extrapolated series expansions abstract we compare monte carlo results for the interface tension and interface energy of the 3 dimensional ising model with pade and inhomogeneous differential approximants of the low temperature series that was recently extended by arisue to 17th order in u exp 4 the series is expected to suffer from the roughening singularity at u 0 196 the comparison with the monte carlo data shows that the pade and inhomogeneous differential approximants fail to improve the truncated series result of the interface tension and the interface energy in the region around the roughening transition the monte carlo data show that the specific heat displays a peak in the smooth phase neither the truncated series nor the pade approximants find this peak we also compared monte carlo data for the energy of the asos model with the corresponding low temperatures series that we extended to order u12

critical dynamics of the three dimensional ising model a monte carlo study extensive monte carlo simulations have been performed to analyze the dynamical behavior of the three dimensional ising model with local dynamics we have studied the equilibrium correlation functions and the power spectral densities of odd and even observables the exponential relaxation times have been calculated in the asymptotic one exponential time region we find that the critical exponentz 2 09 0 02 characterizes the algebraic divergence with lattice size for all observables the influence of scaling corrections has been analyzed we have determined integrated relaxation times as well their dynamical exponentzint agrees withz for correlations of the magnetization and its absolute value but it is different for energy correlations we have applied a scaling method to analyze the behavior of the correlation functions this method verifies excellent scaling behavior and yields a dynamical exponentzscal which perfectly agrees withz

master singular behavior for the sugden factor of one component fluids near their gas liquid critical point we present the master i e unique behavior of the squared capillary length the so called sugden factor as a function of the temperaturelike field along the critical isochore asymptotically close to the gas liquid critical point of about twenty one component fluids this master behavior is obtained using the scale dilatation of the relevant physical fields of the one component fluids the scale dilatation method introduces the fluid dependent scale factors in a manner analog to the linear relations between physical fields and scaling fields needed by the renormalization theory applied to any physical system belonging to the ising like universality class the master behavior for the sugden factor satisfies hyperscaling it can be asymptotically fitted by the leading terms of the theoretical crossover functions for the correlation length and the susceptibility in the homogeneous domain recently obtained from massive renormalization in field theory in the absence of corresponding estimation of the theoretical crossover functions for the interfacial tension we define the range of the temperaturelike field where the master leading power law can be practically used to predict the singular behavior of the sugden factor in conformity with the theoretical description provided by the massive renormalization scheme within the extended asymptotic domain of the one component fluid subclass

finite size scaling study of a first order temperature driven symmetry breaking structural phase transition we present a study of finite size effects in a model exhibiting a first order temperature driven symmetry breaking structural phase transition in thel cylindrical geometry in thel limit exact studies demonstrate the applicability of our scaling ansatz even in the one dimensional limit making this model ideal for studying finite size effects the scaling ansatz similar to the previously developed ansatz for field driven transitions demonstrates that latent heat is crucial in driving these transitions this ansatz is supported by a 2 2 phenomenological transfer matrix based upon the symmetries of the system this produces an analytic free energy which has the scaling form order parameter probability distributions show that the high and low temperature phases coexist only in a small finite size affected regime near the bulk transition temperature this regime vanishes exponentially fast asl diverges

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

duality symmetry strong coupling expansion and universal critical amplitudes in two dimensional phi 4 field models we show that the exact beta function g in the continuous two dimensional g4 model possesses the kramers wannier duality symmetry the duality symmetry transformation d g such that d g d g g is constructed and the approximate values of g computed from the duality equation d g g are shown to agree with the available numerical results the calculation of the beta function g for the 2d scalar g4 field theory based on the strong coupling expansion is developed and the expansion of g in powers of g 1 is obtained up to order g 8 the numerical values calculated for the renormalized coupling constant g are in reasonably good agreement with the best modern estimates recently obtained from the high temperature series expansion and with those known from the perturbative four loop renormalization group calculations the application of cardy s theorem for calculating the renormalized isothermal coupling constant gc of the 2d ising model and the related universal critical amplitudes is also discussed

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

geometrical properties of swendsen wang clusters abstract several geometrical properties of swendsen wang clusters are studied numerically for the 2d ising and 3 state potts models at the critical temperature the fractal dimensions of the total mass backbone elastic backbone shortest path and red bonds are determined the results agree with some theoretical predictions

field theory of finite size effects for systems with a one component order parameter the field theoretic renormalization group approach is used to describe finite size effects near the critical point of the 4 model with a one component order parameter problems of previous perturbation approaches for t tc are discussed and an improved perturbation theory is employed that is applicable both above and below tc the susceptibility order parameter specific heat and a cumulant ratio are calculated for fixed d 4 in one loop order for the case of a cube with periodic boundary conditions finite size scaling functions are evaluated in three dimensions without using the e 4 d expansion quantitative agreement with monte carlo mc data of the three dimensional ising model is found in most cases additional mc data of larger systems would be desirable in order to test the detailed predictions of the finite size field theory more conclusively in the asymptotic region

cluster monte carlo algorithms abstract the swendsen wang and wolff monte carlo algorithms are described in some detail using the potts model as an example various generalizations are then reviewed and some applications are discussed two complete fortran programs for the algorithms are provided

infinite range mean field percolation transfer matrix study of longitudinal correlation length two infinite range directed percolation models equivalent also to epidemic models are considered for a finite population finite number of sites n at each time directed axis step the general features of the transfer matrix spectrum evolution operator spectrum for the epidemic are studied numerically and compared with analytical predictions in the limitn one of the models is devised to allow numerical results to be obtained forn as high as nearly 800 for the largest longitudinal percolation correlation length relaxation time for epidemic the finite n behavior of this correlation length is studied in detail including scaling near the percolation transition exponential divergence withn above the percolation transition as well as other noncritical and critical point properties

finite size corrections for inclined interfaces in two dimensions exact results for ising and solid on solid models analysis of finite size corrections for the surface tension and surface stiffness coefficients in two dimensional models with inclined interfaces is presented we obtain a universal leading contribution proportional to lnl l for the 2d system of sizel by explicit calculations for restricted and unrestricted solid on solid models and the square lattice ising model we demonstrate the gaussian nature of rough interfaces with fixed ends and derive the leading 1 l type corrections for appropriate surface quantities

non equilibrium critical relaxation of the three dimensional ising model abstract the non equilibrium relaxation process at the critical point is studied using monte carlo simulation for the ferromagnetic ising model on a cubic lattice it was observed that the time dependent magnetization decayed following a power law t from the all up initial condition three kinds of single spin flip dynamics were studied for several lattices up to 1536 3 the system size dependences are also studied the value of was estimated to be 0 250 2 this value of leads to the estimation of the dynamical exponent z to be 2 06 2 if we assume z where and denote the critical exponents for spontaneous magnetization and correlation length in equilibrium state respectively and 0 515 the result of the non equilibrium relaxation in the ordered phase is also given

critical dynamics of finite ising model the glauber kinetics of ising spins is considered as a queueing process and simulated event by event as first proposed by bortz kalos and lebowitz the advantage of this algorithm compared to the standard single flip monte carlo method is discussed for the situation of slowing down of dynamics this process is used to generate fluctuations of magnetization and energy in the critical regimet tc of two dimensional ising models the analysis of these fluctuations leads to numerical determination of the critical exponents for dynamics for the size dependence of correlation time attc and for frequency dependence of the power spectrums from the finite size scaling hypothesis scaling relations are settled which are confirmed by this numerical experiment

asymptotic degeneracy of the transfer matrix spectrum for systems with interfaces relation to surface stiffness and step free energy two and three dimensional ising type systems are considered in the finite cross section cylindrical geometry an interface is forced along the cylinder strip in 2d by the antiperiodic or boundary conditions detailed predictions are presented for the largest asymptotically degenerate set of the transfer matrix eigenvalues for rough interfaces i e for 0 t tc in 2d tr t tc in 3d the eigenvalues are split algebraically and the spectral gaps are governed by thesurface stiffness coefficient for rigid interfaces i e 0 t tr in 3d the eigenvalues are split exponentially with the gaps determined by thestep free energy

crossover finite size scaling at first order transitions in a recent paper we developed a method which allows one to control rigorously the finite size behavior in long cylinders near first order phase transitions at low temperature here we apply this method to asymmetric transitions with two competing phases and to theq state potts model as a typical model of a temperature driven transition whereq low temperature phases compete with one high temperature phase we obtain the finite size scaling of the firstn eigenvalues wheren is the number of competing phases of the transfer matrix in a periodic box of volumel l t and as a corollary the finite size scaling of the shape of the order parameter in a hypercubic box t l the infinite cylinder t and the crossover regime from hypercubic to cylindrical scaling for the two phase case n 2 we find that the crossover length l is given by o lw exp lv where is the inverse temperature is the surface tension and w 1 2 if v 1 2 whilew 0 if v 1 2 for the standard ising model we also consider free boundary conditions showing that l exp lv o lv 1 for any dimension v 1 2 for v 1 2 we finally discuss a class of boundary conditions which interpolate between free corresponding to the interpolating parameter g 0 and periodic boundary conditions corresponding to g 1 finding that l o lw exp lv withw 0 forg 0 andw 1 2 for 0 g 1

finite size scaling of the interfacial tension the interfacial free energy appropriate to a finite system in the ising universality class is calculated to one loop order consideration is given to the behavior in both three and four dimensions via an epsilon expansion the results corresponding to three dimensions are compared with the simulation data of mon and jasnow

dynamical properties of the two and three dimensional ising models by damage spreading the method of damage spreading is used to measure the dynamic exponent z for both the two and three dimensional ising models with heat bath dynamics the authors also measure a new exponent alpha describing the rate at which a damage cloud grows they report values of z 2 24 or 0 04 and alpha 0 77 or 0 11 in two dimensions and z 2 202 or 0 03 and alpha 0 92 or 0 10 in three dimensions

difference equations in statistical mechanics ii solid on solid models in two dimensions a review and some new results are presented for the solid on solid models of wetting in two dimensions i e line interfaces with an emphasis on the difference equations arising in the transfer matrix calculations for these models methods for solving the appropriate difference equations exactly or approximately are surveyed including specific results for short range long range power law and applied field like binding external potentials

non equilibrium relaxation and interface energy of the ising model abstract from the non equilibrium critical relaxation study of the two dimensional ising model the dynamical critical exponent z is estimated to be 2 165 0 010 for this model the relaxation in the ordered phase of this model is consistent with exp t behavior the interface energy of the three dimensional ising model is studied and the critical exponent of the correlation length and the critical amplitude of the surface tension 0 are estimated to be 0 6250 0 0025 and 1 42 0 04 respectively a dynamic monte carlo renormalization group method is successfully applied to the equilibrium properties of the three dimensional ising model

on the alexandrowicz relation between static and dynamic ising exponents abstract the cluster growth equation known from nucleation theory is found to reproduce the alexandrowicz relation if the growth probability is assumed to be proportional to the volume per site of the critical cluster supposing weak q dependence of the fractal dimension of the chemical distance in critical clusters of the q state potts model the relation is roughly fulfilled

properties of metastable ising models evolving under the swendsen wang dynamics the behavior of the metastable nearest neighbor ising model governed by swendsen wang dynamics sw is investigated ind 2 the results are compared to those obtained in standard metropolis dynamics both the sw and metropolis systems are observed to decay from the metastable state via the formation of nucleating droplets nucleation rates are measured and found to agree with those predicted by classical nucleation theory the growth rates of the droplets are observed to differ between the two dynamics in addition the dynamic critical exponentz is measured in a mean field curie weiss metastable ising model at the spinodal it is found that for sw dynamics z 2 since this is the same value as that obtained in the metropolis case this result shows that sw does not change the dynamical universality class at the spinodal

a note on the stability of linear discrete systems and lyapunov method in a publication by y p harn and c t chen 1 a proof of a discrete stability test by using the lyapunov method was given in this note the main results obtained by the author in this direction in 1965 6 7 and the extensions and applications obtained partially with other authors since then and published in different journals 8 10 are surveyed the similarity to the results in 1 is obvious

single cluster monte carlo dynamics for the ising model we present an extensive study of a new monte carlo acceleration algorithm introduced by wolff for the ising model it differs from the swendsen wang algorithm by growing and flipping single clusters at a random seed in general it is more efficient than swendsen wang dynamics ford 2 giving zero critical slowing down in the upper critical dimension monte carlo simulations give dynamical critical exponentszw 0 33 0 05 and 0 44 0 10 ind 2 and 3 respectively and numbers consistent withzw 0 ind 4 and mean field theory we present scaling arguments which indicate that the wolff mechanism for decorrelation differs substantially from swendsen wang despite the apparent similarities of the two methods

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 0 025 v 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

optimization of monte carlo simulations abstract monte carlo simulations of phase transitions and critical phenomena are hampered by long relaxation times and large fluctuations the computational effort for the collection of data points in a simulation often varies from data point to data point by orders of magnitude the optimal investment of computing time at different data points is a priori not clear we propose a method to calculate distributions of statistical weight and computing time which minimize the error of the parameters to be determined the method is discussed for the example of a finite size simulation in statistical physics we calculate explicitly the optimal distributions for algebraic dependencies it is shown that the optimal distributions depend on d z x where d is the dimension of the system z is the dynamical critical exponent and x is the scaling exponent of the relative variance of the data depending on one may save 60 80 of the total computing time compared to distributions with equal relative weight

critical dynamics a consequence of a random stepwise growth of clusters critical dynamics of correlated particles here glauber dynamics of singly flipping ising spins is explained by a random stepwise growth and contraction of clusters as follows at equilibrium a cluster of size s is described by its length l a random walk like path connecting a sequence of neighbor spins the length scales as l s where constitutes a new static critical exponent we assume that on the average the random growth of a cluster from zero to size s and length ls requires a sequence of l2 spin flips this gives for dynamic critical exponent z 2 where and are the usual static exponents exact results at dimension d 1 and 4 and simulation results at d 2 and 3 support the theory

critical dynamics of cluster algorithms in the dilute ising model autocorrelation times for thermodynamic quantities att c are calculated from monte carlo simulations of the site diluted simple cubic ising model using the swendsen wang and wolff cluster algorithms our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100 down to 40 this is of crucial importance when estimating static properties of the model since the variances of these estimators increase with autocorrelation time the dynamical critical exponents are calculated for both algorithms observing pronounced finite size effects in the energy autocorrelation data for the algorithm of wolff we conclude that when applied to the dilute ising model cluster algorithms become even more effective than local algorithms for whichincreasing autocorrelation times are expected

the shape of the van der waals loop and universal critical amplitude ratios assuming that a van der waals loop for a fluid ferromagnet etc can be defined its shape is analysed on the basis of universal critical amplitude ratios for three dimensional ising type systems selected estimates for these ratios are tabulated near criticality the loop attains a universal scaled form given optimally by a new parametric extended sine model for the bulk asymptotic equation of state but a particular low order interpolating pade approximant provides a reasonable fit among other universal features the reduced spinodal magnetization density and conjugate field are found to be in contrast to the van der waals values thus the spinodal lies appreciably closer to the binodal or coexistence curve than predicted classically

critical endpoints interfaces and walls abstract the behavior of thermodynamic properties at critical endpoints is discussed and the first results of studies underway in collaboration with p j upton and with m c barbosa are sketched new universal aspects beyond normal bulk criticality appear in singularities of the phase boundary to the spectator phase and in the surface tensions of the corresponding noncritical interfaces mean field theory and more general local free energy functionals can be applied some quantitative estimates are presented for universal surface tension amplitude ratios for two distinct classes of endpoint symmetric and nonsymmetric a promising parametric trigonometric model for the scaled equation of state is mentioned and other approaches are touched upon

simulation of time exponent in 2d glauber kinetic ising model abstract simulations of the magnetization relaxation of large square lattices at the curie point give a dynamical exponent near 2 2 somewhat higher than expected

surface tension from finite volume vacuum tunneling in the 3d ising model we measure the surface tension in the broken phase of the 3d ising model at a temperaturet 0 955t c with two different methods which are taken from quantum field theory in finite volumes both methods rely on finite size effects close to the phase transition the first one measures from the size dependence of the vacuum tunneling energy which is determined by the decay of a correlation giving 0 030 the second one extracts from the size dependence of the rate of flip events and its corresponding correlation time it leads to 0 027 both values agree reasonably with other calculations

the dynamic critical exponent of dilute and pure ising systems a massive dynamic field theory is employed to calculate the critical exponent z of the three dimensional dilute ising model to two loop order without expansion in square root 4 d with result z approximately 2 191 applying the same method to the pure ising model at three loop order we find z 2 022 for d 3 and z 2 124 for d 2

geometry and dynamics of randomly connected fractal clusters abstract the dynamics of critical ising droplets and suitable other clusters is argued to come from growth and decay in small steps not via aggregation of large clusters the resulting critical exponent z for metropolis glauber kinetics is related to the exponent for the chain lengths of the cluster branches simulations of 60960 2 and 1409 3 ising models confirm roughly the exponent z but throw doubts on the small step approach when interpreted with the help of the becker doring equation

formulas for higher derivatives of the riemann zeta function the functional equation for c s is used to obtain formulas for all derivatives t k s a closed form evaluation of t k 0 is given and numerical values are computed to

a cluster model of critical dynamics abstract critical relaxation exponent z is described by a reversible growth of clusters particles of a cluster are ordered into connected sequences such that a stepwise variation of their length describes relaxation we derive z z where is a scaling exponent relating cluster size to connected length at equilibrium available from simulation of ising clusters furthermore a flory like equation gives and hence z z in d 2 and 3 theoretical z an simulation aided z agree reasonably well with each other and with direct simulations of z in d 4 e z agrees with the exact rg

critical temperatures of the d 3 s 1 2 ising model the effect of confluent corrections to scaling much attention has been paid to the d 3 s 1 2 ising model on the bcc lattice in recent years because the best high temperature series expansions were available for this lattice however in order to compare series estimates of critical exponents and temperatures with independent evaluations via the monte carlo method lattices of lower coordination number must be considered since monte carlo studies are usually made on these a study of extant susceptibility series on the fcc sc diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent monte carlo analyses of the d 3 s 1 2 ising model on the sc and diamond lattices

the dynamic critical exponent of the three dimensional ising model we measure the dynamic exponent of the three dimensional ising model using a damage spreading monte carlo approach as described by macisaac and jan we simulate systems froml 5 tol 60 at the critical temperature tc 4 5115 we report a dynamic exponent z 2 35 0 05 a value much larger than the consensus value of 2 02 whereas if we assume logarithmic corrections we find thatz 2 05 0 05

self consistent screening approximation for critical dynamics we generalize bray s self consistent screening approximation to describe the critical dynamics of the theory in order to obtain the dynamical exponent z we have to make an ansatz for the form of the scaling functions which fortunately can be restricted by general arguments numerical values of z for d 3 and n 1 10 are obtained using two different ansatz and differ by a very small amount in particular the value of obtained for the three dimensional ising model agrees well with recent monte carlo simulations

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

crossover critical behavior in the three dimensional ising model the character of critical behavior in physical systems depends on the range of interactions in the limit of infinite range of the interactions systems will exhibit mean field critical behavior i e critical behavior not affected by fluctuations of the order parameter if the interaction range is finite the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class a variety of systems including fluids and anisotropic ferromagnets belongs to the three dimensional ising universality class recent numerical studies of ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation induced critical behavior and mean field type critical behavior in this work we compare these numerical results with a crossover landau model based on renormalization group matching for this purpose we consider an application of the crossover landau model to the three dimensional ising model without fitting to any adjustable parameters the crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range ten orders of the scaled distance to the critical temperature the dependence of the coupling constant on the interaction range governing the crossover critical behavior is discussed

description of critical dynamics by static geometry of clusters abstract critical slowing down of relaxation exponent z is modelled by clusters which grow and recede at the end of connected sequences of particles consequently the relaxation time of a cluster of size s is proportional to the number of back and forth steps needed to travel its typical connected length ls this and an assumption on mutual screening of ends lead to z 2 5 0 25 where is a static geometric exponent in ls s ising exponents measured elsewhere with the help of simulation give z in addition a flory like approximation is derived here it relates to and analytically giving z in d 2 3 and 4 respectively we findz 2 14 2 05 and 2 275 1 while z 2 17 4 and 2 04 4 both sets of values are in fair agreement with the literature