renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

statistical hydrodynamics of lattice gas automata we investigate the space and time behavior of spontaneous thermohydrodynamic fluctuations in a simple fluid modeled by a lattice gas automaton and develop the statistical mechanical theory of thermal lattice gases to compute the dynamical structure factor i e the power spectrum of the density correlation function a comparative analysis of the theoretical predictions with our lattice gas simulations is presented the main results are i the spectral function of the lattice gas fluctuations is fully compatible with the spectrum obtained from experimental measurements performed in real fluids ii in the long wavelength limit the correlations of lattice gas fluctuations are well described by the landau placzek theory iii at short wavelengths and or at low densities good agreement is obtained between the lattice gas simulation results and the boltzmann theory these results provide solid support to the validity of the thermal lattice gas automaton as a consistent model system for real fluids

on generalized harmonic number sums we evaluate generalized harmonic number sums with parameter in terms of values of polylogarithm functions and several examples are given instances of such sums occur in diverse areas including analytic number theory and in calculations of high energy nuclear and atomic physics

diffusion lattice boltzmann scheme on a orthorhombic lattice we present a diffusion lattice boltzmann dlb scheme which is derived from first principles as opposed to the traditional lattice bgk schemes the dlb is valid for orthorhombic lattices and it has two eigenvalues of the collision operator it is shown that the diffusion coefficient depends only on one eigenvalue of the collision operator hence the dlb scheme can be optimized with means of the additional eigenvalue of the collision operator and with different lattice spacing along the principal axes the properties of the dlb scheme concerning consistency stability and accuracy are studied with eigenmode analysis this analysis shows that the dlb scheme is consistent with diffusion for a wide range of diffusion coefficients it has unconditional stability and that it has third order accuracy furthermore it is shown that accuracy is improved by setting the additional eigenvalue to zero and by densifying the lattice spacing along the direction of the density gradient

on some series containing p x p y and 9 p x p y 2 for certain values of x and y abstract in this article we discuss series containing the factor k 1 k 1 2 1 2 k 1 2 and 1 2 k 1 case1 2 k with k n here x d dxlog x and x is the gammafunction

hydrodynamic behavior of lattice boltzmann and lattice bhatnagar gross krook models we present a numerical analysis of the ranges of validity of classical and generalized hydrodynamics for lattice boltzmann and lattice bhatnagar gross krook bgk methods in two and three dimensions as a function of the collision parameters of these models our analysis is based on the wave number dependence of the evolution operator good ranges of validity are found as long as the bgk relaxation time is chosen smaller than or equal to unity the additional degree of freedom in a two parameter bgk model can give further improvements

a lattice boltzmann equation for diffusion the formulation of lattice gas automata lga for given partial differential equations is not straightforward and still requires some sort of magic lattice boltzmann equation lbe models are much more flexible than lga because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements here a lbe is presented for diffusion in an arbitrary number of dimensions the model is probably the simplest lbe which can be formulated it is shown that the resulting algorithm with relaxation parameter 1 is identical to an explicit finite difference efd formulation at its stability limit underrelaxation 0 1 allows stable integration beyond the stability limit of efd the time step of the explicit lbe integration is limited by accuracy and not by stability requirements

global invariants and equilibrium states in lattice gases it is now well known that in addition to the physical conserved quantities lattice gases also have other unphysical ones related to the discretization of their phase space from an abstract point of view a lattice gas can be considered like a full discrete markov processl and these spurious conserved quantities yield the existence of a nonspatially homogeneous equilibrium state forlk we show that a particular set of these conserved quantities is of special interest its elements will be called regular these regular invariants are simply built from the local ones and their projection on each node is always a locally conserved quantity moreover for most models they are one to one related to the gibbs states oflk which remain factorized it turns out that all the classical known spurious invariants are regular and one can exhibit simple conditions to build models with only regular invariants for the latter it is then justified to determine the transport coefficients of the locally conserved densities with the green kubo procedure

lattice gas automata with time dependent collision rules abstract particle scatterer models with time dependent collision rules in two dimensional square and triangular lattices are studied very good agreement is seen between analytical and molecular dynamics results for the diffusion coefficient at all scatterer concentrations the results differ significantly from the corresponding analysis for stochastic dynamics in all square lattice lorentz models there is a spurious diffusion mode its transport coefficient has the same value as the usual diffusion coefficient the long time tails with a high frequency modulation 1 t observed in computer simulations are explained in terms of this spurious mode

generalized boltzmann equation for lattice gas automata in this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi detailed balance starting from a coupled bbgky hierarchy for then particle distribution functions cluster expansion techniques are used to derive approximate kinetic equations in zeroth approximation the standard nonlnear boltzmann equation is obtained the next approximation yields the ring kinetic equation similar to that for hard sphere systems describing the time evolution of pair correlations the ring equation is solved to determine the nonvanishing pair correlation functions in equilibrium for two models that violate semidetailed balance one is a model of interacting random walkers on a line the other one is a two dimensional fluid type model on a triangular lattice the numerical predictions agree very well with computer simulations

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

quadratic and cubic harmonic number sums abstract this paper develops an approach to evaluation of euler related sums the approach is based on simple integral computations by the approach we can obtain some closed form representations of sums of quadratic and cubic harmonic numbers and reciprocal binomial coefficients the given representations are new

velocity autocorrelation function in lattice gases from the ring kinetic theory comparison with numerical simulations we obtain the complete time dependence of the velocity autocorrelation function vacf for lattice gas cellular automata usingring kinetic theory this theory accounts for the simplest correlated collisions that improve on the molecular chaos approach and yields a closed equation for the vacf that we evaluate for both infinite and finite systems we compare our analytical results with numerical simulations at all times as well as with long time results of the mode coupling theories finding a very good agreement for all times at all densities

lattice gas automata with self organization abstract self organization or dynamic phase transitions occur in computer simulations of a lattice gas model with strictly local collision dynamics conserving number of particles momentum and lattice symmetries the microscopic collision processes are biased i e they have asymmetric transition rates in this model a spatially uniform initial state is unstable at the onset of instability long wavelength sound modes drive the system into a state with very long range order with dynamics similar to spinodal decomposition the structure factor s k t and the selected wavelength for maximal growth max t are measured as a function of time the state with long range order is highly organized into moving stable spatial patterns of triangles or parallel strips of macroscopic size with two or one broken symmetry respectively the entropy suddenly drops at self organization both the onset of metastability and the detailed structure of the final patterns are explained by mean field theory

on harmonic numbers and nonlinear euler sums abstract in this paper we are interested in euler type sums with products of harmonic numbers stirling numbers and bell numbers we discuss the analytic representations of euler sums through values of polylogarithm function and riemann zeta function moreover we provide explicit evaluations for almost all euler sums with weight 5 which can be expressed in terms of zeta values and polylogarithms furthermore we give explicit formula for several classes of euler related sums in terms of zeta values and harmonic numbers and several examples are given the given representations are new

stress stress correlation functions in lattice gases beyond boltzmann s approximation the complete time dependence of the stress stress correlation functions in lattice gas cellular automata is calculated from the ring kinetic theory using numerical and analytical methods this provides corrections typically of 10 20 to the usual molecular chaos calculations where correlation functions decay exponentially the resulting correlation function crosses over from an initial exponential decay to the long time behavior calculated from mode coupling theory the present theory applied to the viscosity accounts for a substantial part of the observed difference between the boltzmann theory and simulations

integrals of polylogarithmic functions recurrence relations and associated euler sums we show that integrals of the form 0 1 x m li p x li q x dx m 2 p q 1 and 0 1 log r x li p x li q x x dx p q r 1 satisfy certain recurrence relations which allow us to write them in terms of euler sums from this we prove that in the first case for all m p q and in the second case when p q r is even these integrals are reducible to zeta values in the case of odd p q r we combine the known results for euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight p q r the proofs are constructive giving a method for the evaluation of these and other similar integrals and we present a selection of explicit evaluations in the last section

phase transitions in a probabilistic cellular automaton growth kinetics and critical properties we investigate a discrete time kinetic model without detailed balance which simulates the phase segregation of a quenched binary alloy the model is a variation on the rothman keller cellular automaton in which particles of type a b move toward domains of greater concentration of a b modifications include a fully occupied lattice and the introduction of a temperature like parameter which endows the system with a stochastic evolution using computer simulations we examine domain growth kinetics in the two dimensional model for long times after a quench from disorder we find that the average domain sizer t t1 3 in agreement with the prediction of lifshitz slyozov wagner theory using a variety of methods we analyze the critical properties of the associated second order transition our analysis indicates that this model does not fall within either the ising or mean field classes

implementation of the fchc lattice gas model on the connection machine the 4 dimensional fchc lattice gas model has been implemented on a connection machine cm 2 with 16k processors symmetries are used to reduce the collision table to a size that fits into local memory this method avoids the degradation of the reynolds coefficientr but at the price of increased computing time bit shuffling between parallel lattices is introduced to reduce the discrepancy between measured viscosities and those predicted from the boltzmann approximation thereby a model with a negative shear viscosity is obtained a fluid having a uniform initial velocity is unstable and organized nonuniform motions develop because of the buildup of very strong correlations between the parallel lattices the discrepancy with the boltzmann values decreases only very slowly with the number of parallel lattices

new properties of multiple harmonic sums modulo p and p analogues of leshchiner s series in this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences 1 a c 1 b 2 a c 2 b and prove a number of congruences for these sums modulo a prime p the congruences obtained allow us to find nice p analogues of leshchiner s series for zeta values and to refine a result due to m hoffman and j zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p moreover we are also able to provide a new proof of zagier s formula for zeta 2 a 3 2 b based on a finite identity for partial sums of the zeta star series

staggered diffusivities in lattice gas cellular automata the majority of lgcas has spurious conservation laws the so called staggered invariants first discovered by kadanoff mcnamara and zanetti consequently there are additional hydrodynamic modes of diffusive type which modify mode coupling theories and the nonlinear fluid dynamic equations the diffusivities of these staggered modes are evaluated in the mean field approximation for lgcas on triangular lattices starting from the green kubo formulas for the staggered diffusivities

self diffusion in ca fluids tagged particle properties such as the diffusion coefficients and velocity autocorrelation function are calculated in the mean field approximation for all standard lattice gas cellular automata defined on hyper cubic square and triangular lattices and on a line tagged particle dynamics is introduced through maximally random or minimally random collision rules for a completely filled lattice the former reduces to a random walk the latter to ballistic motion

evaluations of nonlinear euler sums of weight ten abstract in this paper we present a new family of identities for euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series then we apply it to obtain the closed forms of all quadratic euler sums of weight ten furthermore we also establish some relations between multiple zeta star values and nonlinear euler sums as applications of these relations we give new closed form representations of several cubic euler sums through single zeta values and linear sums finally with the help of numerical computations of mathematica or maple we evaluate several other euler sums of weight ten

mode coupling theory and tails in ca fluids abstract after a summary of important effects of mode coupling theory in continuous fluids the theory is extended to ca fluids and the long time tails calculated for the velocity correlation function and for the stress stress correlation function including the spurious contributions from the staggered momentum densities at finite densities the fermi statistics strongly suppresses the long time singularities arising from two fluid modes but not those from tagged particle diffusion

hydrodynamics and time correlation functions for cellular automata hydrodynamic excitations in lattice gas cellular automata are described in terms of equilibrium time correlation functions for the local conserved variables for large space and time scales the linearized hydrodynamic equations are obtained to navier stokes order exact expressions for the associated susceptibilities and transport coefficients are identified in terms of correlation functions the general form of the time correlation functions for conserved densities in the hydrodynamic limit is given and illustrated by some examples suitable for comparison with computer simulation the transport coefficients are related to time correlation functions for the conserved fluxes in a way analogous to the green kubo expressions for continuous fluids the general results are applied for a one component fluid and several types of binary diffusion also discussed are the effects of unphysical slow modes such as staggered particle or momentum densities

propagating staggered waves in cellular automata fluids the majority of lattice gases have apart from the physical conserved quantities of particle number momentum and energy spurious ones usually staggered in space and time at the level of linear excitations these staggered modes may be purely diffusive or damped propagating waves in the eight and nine bit model on the square lattice one finds a large number of new spurious modes and the authors derive green kubo relations for the diffusivities and damping constants and calculate them in the boltzmann approximation

collective diffusion in a lattice gas automaton we report on non mean field and ring kinetic theory calculations of both the momentum autocorrelation function and the collective diffusion coefficient in a diffusive lattice gas automaton for both quantities the ring approximation is calculated exactly a saddle point method yields a leadingt 2 and a subleadingt 5 2 long time tail in the momentum autocorrelation function the ring kinetic corrections to the mean field value of the diffusion coefficient are in good agreement with computer simulations

relaxation and transport in fchc lattice gases fchc lattice gases are the basic models for studying flow problems in three dimensional systems this paper presents a self contained theoretical analysis and some computer simulations of such lattice gases extended to include an arbitrary number of rest particles with special emphasis on non semi detailed balance nsdb models the special fchc lattice symmetry guarantees isotropy of the navier stokes equations and enumerates the 12 spurious conservation laws staggered momenta the kinetic theory is based on the mean field approximation or the nonlinear boltzmann equation it is shown how calculation of the eigenvalues of the linearized boltzmann equation offers a simple alternative to the chapman enskog method or the multi time scale methods for calculating transport coefficients and relaxation rates the simulated values for the speed of sound in nsdb models slightly disagree with the boltzmann prediction

nonlinear lattice gas hydrodynamics starting from a microdynamical description we derive the equations governing the evolution of the hydrodynamic variables in a lattice gas automaton the essential features are i the local collision rules satisfy semi detailed balance this condition guarantees that a factorized local equilibrium distribution of the fermi dirac form is invariant under the collision step but not under propagation ii particles entering a collision are uncorrelated boltzmann hypothesis and iii the system can be arbitrarily far from global equilibrium we do not not assume linear response as usually imposed to obtain the dissipative contributions linearization of the resulting hydrodynamic equations leads to green kubo formulas for the transport coefficients the main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice boltzmann approximation these equations have a validity domain extending beyond the region close to equilibrium

long time tails in lattice lorentz gases we consider a lorentz gas on a square lattice with a fraction c of scattering sites the collision laws are deterministic fixed mirror model or stochastic with transmission reflection and deflection probabilities and respectively if all mirrors are parallel the mirror model is exactly solvable for the general case a self consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function 0 t and the tensor correlation q 0 q t withq x y both functions showt 2 tails as opposed to the continuous lorentz gas where the tails are respectivelyt 2 andt 3 inclusion of the self consistent ring collisions increases the low density coefficient of the tail in 0 t by 30 100 as compared to the simple ring collisions depending on the model parameters

some evaluation of parametric euler sums abstract in this paper by using the method of contour integral representations and the theorem of residues and integral representations of series we discuss the analytic representations of parametric euler sums that involve harmonic numbers through zeta values and rational function series either linearly or nonlinearly furthermore we give explicit formulae for several parametric quadratic and cubic sums in terms of zeta values and rational series moreover some interesting new consequences and illustrative examples are considered

critical couplings of mixed spin 1 2 spin s ising model a free fermion approximation a free fermion approximation procedure is applied to the mixed spin 1 2 spin s ising models the author s results for the critical coupling kc are in very satisfactory agreement with the high temperature series results of yousif and bowers 1984

thermodynamical properties of the random field mixed spin ising model abstract a theoretical study of a mixed spin ising system consisting of spin 1 2 and spin 1 with a random field is studied by the use of the finite cluster approximation within the framework of a single site cluster theory in this approach the phase diagrams and magnetizations are investigated for the simple cubic lattice when the random field is bimodally and trimodally distributed the internal energy specific heat and susceptibility are also calculated for both kinds of the probability distributions

biased lattice gases with correlated equilibrium states the approach to and structure of the equilibrium state is studied for a 7 bit lattice gas with biased forward and backward transition rates by means of mean field theory and computer simulations if the rate constants obey the factorizability and the detailed balance conditions the occupations of different velocity directions are uncorrelated anh theorem is valid and a nonuniversal equilibrium state exists that depends explicitly on the transition rates in case the above conditions are not satisfied theh theorem is no longer valid and mean field theory also predicts nontrivial velocity correlations in postcollision states the simulations are mainly concentrated on the time dependence of pre and postcollision velocity correlations on a single node and on slowly increasing fluctuations that might indicate metastable behavior

parametric euler sum identities we consider some parametrized classes of multiple sums first studied by euler identities between meromorphic functions of one or more variables in many cases account for reduction formulae for these sums

four parametric linear euler sums abstract in this paper firstly we aim to investigate analytic continuations of altogether four types of parametric linear euler sums by using the euler maclaurin summation formula and the euler boole summation formula secondly we show how nicely three shuffle relations among the four parametric linear euler sums can be derived by using an l summing formula due to hassani and rahimpour thirdly using one of the shuffle relations we present certain series representations of product of riemann zeta functions some special cases and relevant connections of the results presented here with those involving known identities are also demonstrated and indicated

some evaluation of cubic euler sums abstract p flajolet and b salvy 15 prove the famous theorem that a nonlinear euler sum s i 1 i 2 i r q reduces to a combination of sums of lower orders whenever the weight i 1 i 2 i r q and the order r are of the same parity in this article we develop an approach to evaluate the cubic sums s 1 2 m p and s 1 l 1 l 2 l 3 by using the approach we establish some relations involving cubic quadratic and linear euler sums specially we prove the cubic sums s 1 2 m m and s 1 2 l 1 2 2 l 1 are reducible to zeta values quadratic and linear sums moreover we prove that the two combined sums involving multiple zeta values of depth four i j 1 2 i j m i m j 1 1 and i j k 1 2 3 i j k m i m j m k 1 can be expressed in terms of multiple zeta values of depth 3 here 2 m 1 m 2 m 3 n finally we evaluate the alternating cubic euler sums s 1 3 2 r 1 and show that it are reducible to alternating quadratic and linear euler sums the approach is based on tornheim type series computations

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

low viscosity lattice gases a class of lattice gas models are studied which are variants of the fchc model the aim is to achieve the highest possible reynolds coefficient inverse dimensionless viscosity for efficient simulations of the three dimensional incompressible navier stokes equations the models include an arbitrary number of rest particles and violation of semi detailed balance within the framework of the boltzmann approximation exact expressions are obtained for the reynolds coefficients the minimization of the viscosity is done by solving a hitchcock type optimization problem for the fine tuning of the collision rules when the number of rest particles exceeds one there is a range of densities at which the viscosity takes negative values various optimal models with up to 26 bits per node have been implemented on a cray 2 and their true transport coefficients have been measured with good accuracy fairly large discrepancies with boltzmann values are observed when semi detailed balance is violated in particular no negative viscosity is obtained still the best model has a reynolds coefficient of 13 5 twice that of the best previously implemented model and thus is about 16 times more efficient computationally suggestions are made for further improvements it is proposed to use models with very high reynolds coefficients for sub grid scale modeling of turbulent flows

mixed spin 1 2 and spin 1 ising model in transverse random fields abstract we investigate the critical properties of a mixed spin 1 2 and spin 1 ising model in a transverse random field in this study we use two approaches the pair approximation and the mean field renormalization group each combined with the discretized path integral representation it is found that both methods predict quantitatively the same phase diagram critical exponents of the system are also calculated