rotators periodicity and absence of diffusion in cyclic cellular automata cyclic cellular automata are two dimensional cellular automata which generalize lattice versions of the lorentz gas and certain biochemistry models of artificial life we show that rotators and time reversibility play a special role in the creation of closed orbits in cyclic cellular automata we also prove that almost every orbit is closed periodic and the absence of diffusion for the flipping rotator model also known as the ant

the efficiency and consistency of approximations to the jackknife variance estimators abstract the problem considered is the computation reduction for general delete d jackknife variance estimators the delete d jackknife estimator was proved consistent shao and wu 1986 and in this article its mean squared error is shown to have order o n 2 where n is the sample size these properties are not shared by the traditional delete 1 jackknife in some situations use of the delete d jackknife however requires n d recomputations of a point estimate which increases rapidly as n and d increase using techniques from survey sampling a shortcut can be taken with evaluated only m times m n d the efficiency and consistency of the resulting jackknife sampling hybrid variance estimators jsve s are studied if m is chosen so that n m 0 the increase in mean squared error by using the jsve is relatively negligible for the consistency of jsve m is sufficient hence the jsve with m n can also be used to alleviate the computational burden for the delete 1 jackknife in th

algebraic spatial correlations in lattice gas automata violating detailed balance in this paper we discuss the existence of generic long range correlations in spatially homogeneous and stable equilibrium states of closed lattice gas automata whose stochastic collision rules violate the symmetry conditions of detailed balance and in addition satisfy local conservation laws such correlations occur even though the collision rules are strictly local and invariant under all symmetries of the lattice first a phenomenological langevin equation approach is discussed next we present a theoretical analysis on the basis of an approximate microscopic ring kinetic theory this theory is used to calculate the amplitude ofr tails in the spatial correlations and the result is compared with computer simulations

a berry esseen bound for finite population student s statistic a general and precise berry esseen bound is proved for the studentized mean based on n random observations drawn without replacement from a finite population the bound yields the optimal rate o n 1 2 under minimal conditions if the erdos renyi condition holds this bound implies the asymptotic normality of student s statistic and the self normalized sum

an edgeworth expansion for u statistics based on samples from finite populations suppose that u is a u statistic of degree 2 based on a simple random sample of size n selected without replacement from a finite population of n elements a bound for the difference between the distribution function of a standardized version of u and its single term edgeworth expansion is given we apply these results to obtain an edgeworth equation for the variance estimator in a finite population some simulation results are reported in this case

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

the edgeworth expansion for u statistics of degree two an edgeworth expansion with remainder o n 1 is established for a u statistic with a kernel h of degree 2 the assumptions involved appear to be very mild in particular the common distribution of the summands h x i x j is not assumed to be smooth

maximum entropy formalism for a dense gas abstract the maximum entropy formalism is developed to implement a complete description of a dense gas in non equilibrium situations two variational schemes are discussed in this context the kinetic equations describing the system are consistently closed with the n particle distribution function obtained via this formalism the equilibrium state obtained on this basis is consistent with the description of the theory of liquids in the equilibrium state

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

relation between the lattice boltzmann equation and the navier stokes equations abstract it is shown that the lattice gas boltzmann equation may be rewritten in a form which brings out its close relation with the navier stokes equations various consequences are pointed out

a rank statistics approach to the consistency of a general bootstrap a general notion of a bootstrapped mean constructed by exchangeably weighting sample points is introduced consistency of this generalized bootstrapped mean which includes proposals of efron and rubin among others is proved by classical linear rank statistics theory the consistency of generalized bootstrapped empirical and quantile processes is also established

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

theory of the lattice boltzmann method two fluid model for binary mixtures a two fluid lattice boltzmann model for binary mixtures is developed the model is derived formally from kinetic theory by discretizing two fluid boltzmann equations in which mutual collisions and self collisions are treated independently in the resulting lattice boltzmann model viscosity and diffusion coefficients can be varied independently by a suitable choice of mutual and self collision relaxation time scales further the proposed model can simulate miscible and immiscible fluids by changing the sign of the mutual collision term this is in contrast to most existing single fluid lattice boltzmann models that employ a single relaxation time scale and hence are restricted to unity prandtl and schmidt numbers the extension of binary mixing model to multiscalar mixing is quite straightforward

a knudsen layer theory for lattice gases abstract a knudsen layer theory is presented for lattice gases with arbitrary boundary conditions analytical results are obtained for two special orientations these exhibit anisotropic knudsen layers provided suitable conditions are satisfied however the standard boundary conditions used in previous simulations are shown to be isotropic the bulk steady state extending everywhere in the gas this theory allows a more accurate localization of the obstacle with respect to the lattice nodes these results are in good agreement with the numerical simulations

error bound in a central limit theorem of double indexed permutation statistics an error bound in the normal approximation to the distribution of the double indexed permutation statistics is derived the derivation is based on stein s method and on an extension of a combinatorial method of bolthausen the result can be applied to obtain the convergence rate of order n 1 2 for some rank related statistics such as kendall s tau spearman s rho and the mann whitney wilcoxon statistic its applications to graph related nonparametric statistics of multivariate observations are also mentioned

general h theorem for hard spheres the maximum entropy formalism is used to investigate the growth of entropy h theorem for an isolated system of hard spheres in an external potential under general boundary geometry assuming that only correlations of a finite number of particles are controlled and the rest maximizes entropy we obtain an h theorem for such a system the limiting cases such as the modified enskog equation and linear kinetic theory are discussed

local variational description for a dense gas abstract we present the study of a dense gas at a hydrodynamical level of description we start from the entropy functional obtained at a mesoscopic regime l romero salazae et al physica a 237 1997 150 mef1 and we apply the maximum entropy formalism where the restrictions are now the hydrodynamic variables and the density fluctuations an entropy functional is obtained both in the local and the equilibrium inhomogeneous situation

asymptotic expansions based on smooth functions in the central limit theorem stein s method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables together with lyapounov estimates of the error in the approximation

on the asymptotic properties of the jackknife histogram we study the asymptotic normality of the jackknife histogram for one sample mean it holds if and only if r the number of observations retained and d n r the number of observations deleted both diverge to infinity the best convergence rate n 1 2 is obtained when r o n and d o n for u statistics of degree 2 and nonlinear statistics admitting a certain expansion similar results are obtained under conditions on r and d a second order approximation based on the edgeworth expansion is discussed

the closure hypothesis in the maximum entropy formalism abstract closure hypotheses in kinetic theory constitute a point of comparison between different approaches to describe the behavior of a macroscopic system the maximum entropy formalism mef i gives us a way to close the bbgky hierarchy equations and its discussion is the main objective of this paper

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

h theorem and generalized semi detailed balance condition for lattice gas systems for lattice gas systems obeying fermi dirac statistics an h theorem can be proved with a more general condition that the semi detailed balance condition this new condition allows more flexible transition rates among states so that it has broader applicability for various lattice gas models including those which have multiple phase properties

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

local h theorem for a kinetic variational theory a localh theorem is derived for a recently proposed extension of enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core

an edgeworth expansion for symmetric finite population statistics let t be a symmetric statistic based on sample of size n drawn without replacement from a finite population of size n where n n assuming that the linear part of hoeffding s decomposition of t is nondegenerate we construct a one term edgeworth expansion for the distribution function of t and prove the validity of the expansion with the remainder o 1 n as n where n min n n n

an edgeworth expansion for symmetric statistics we consider asymptotically normal statistics which are symmetric functions of n i i d random variables for these statistics we prove the validity of an edgeworth expansion with remainder o n 1 under cramer s condition on the linear part of the statistic and moment assumptions for all parts of the statistic by means of a counterexample we show that it is generally not possible to obtain an edgeworth expansion with remainder o n 1 without imposing additional assumptions on the structure of the nonlinear part of the statistic

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

fisher information and bounds to the entropy increase with reference to systems obeying conservation of flow for which a probability density function may be used in their description it is shown that the interesting relationship established by nikolov and frieden nf between entropy increase and fisher information phys rev e 49 4815 1994 holds only in special cases we will here i exhibit a counterexample to the hypothetical nf relationship and ii derive a valid fisher s information related upper bound to entropy increases

viscous drag by cellular automata a simple method to compute the drag coefficient of two dimensional bodies with arbitrary shapes is presented the procedure is based on cellular automata as an extreme idealization of the molecular dynamics of a viscous fluid we verify the algorithm by examples and obtain results in quantitative agreement with experiments even when eddies behind the obstacle are formed

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

entropy balance equation for a dense gas abstract this work is concerned with the description of the time evolution for the entropy functional in a dense fluid out of equilibrium we consider the fluid as a collection of n particles interacting via the superposition of pairwise additive short range intermolecular potentials such an interaction is assumed to be central and the particles do not have internal structure to reach this goal we use the maximum entropy formalism in which the gibbs entropy is maximized under the restrictions imposed by the normalization of the n particle distribution function and the knowledge of the one and two particle distribution functions this procedure leads to an expression for the entropy functional in terms of such distribution functions on the other hand the evolution equations allow for the construction of a balance equation for the entropy functional in the system an analysis of the entropy flux and entropy source is given for a particular expression in the lagrange multipliers coming from the maximization procedure such an analysis shows the connection of the entropy production with the conversion rate of kinetic and potential energy between particles interesting conclusions about the temperature are reached in this case

hoeffding decompositions and urn sequences let x x1 x2 be a nondeterministic infinite exchangeable sequence with values in 0 1 we show that x is hoeffding decomposable if and only if x is either an i i d sequence or a polya sequence this completes the results established in peccati ann probab 32 2004 1796 1829 the proof uses several combinatorial implications of the correspondence between hoeffding decomposability and weak independence our results must be compared with previous characterizations of i i d and polya sequences given by hill lane and sudderth ann probab 15 1987 1586 1592 and diaconis and ylvisaker ann statist 7 1979 269 281 the final section contains a partial characterization of hoeffding decomposable sequences with values in a set with more than two elements

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

a fluctuation formula for the nongalilean factor in lattice gas automata a lattice gas automaton lacks galilei invariance and equilibria of systems moving with a finite speed mod u mod are not simply related by a galilei transformation to the equilibrium distribution in the rest frame in the hydrodynamic description of low speed equilibria in lattice gas automata a factor g p appears in the nonlinear convective term delta g p puu of the navier stokes equation that differs from unity due to lack of galilei invariance for this non galilean factor an expression in terms of fluctuating quantities is derived a grand ensemble where the total momentum is fluctuating around a zero average the formula is valid as long as there exists a unique equilibrium state consequently the results can also be used for a direct simulation of g p in lattice gas models where the explicit form of the equilibrium distribution is not known such as in models that violated semi detailed balance

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

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

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

entropy production bound in a dense gas abstract this paper concerns the behavior of the entropy production in a dense gas a bound for this quantity is found in terms of the energy conversion rate between particles and fisher s information integrals describing the system

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

theory of the lattice boltzmann equation symmetry properties of discrete velocity sets the lattice boltzmann equation replaces continuous particle velocity space by a finite set the velocity distribution function then varies over a finite dimensional vector space instead of over an infinite dimensional function space the number of linearly independent moments of the distribution function in a lattice boltzmann model cannot exceed the number of velocities finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory given a finite velocity set it is important to know which moments are free of these dependences elementary group theory is applied to the solution of this problem it is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments the construction of some standard two and three dimensional models is reviewed from this viewpoint and procedures for constructing higher dimensional models are suggested

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field

a short proof of motoo s combinatorial central limit theorem using stein s method a new proof of motoo s combinatorial central limit theorem see motoo 1957 is given using a method of stein 1972 and a combinatorial method of bolthausen 1984 this proof is shorter than motoo s and other wellknown proofs see e g hajek 1961