singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

large deviations of the lyapunov exponent and localization for the 1d anderson model the proof of anderson localization for the 1d anderson model with arbitrary e g bernoulli disorder originally given by carmona klein martinelli in 1987 is based in part on the multi scale analysis later in the 90s it was realized that for one dimensional models with positive lyapunov exponents some parts of multi scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle leading to nonperturbative proofs for 1d quasiperiodic models in this paper we present a short proof along these lines for the anderson model to prove dynamical localization we also develop a uniform version of craig simon s bound that works in high generality and may be of independent interest

evaluation of the external force term in the discrete boltzmann equation a representation of the forcing term in the boltzmann equation based on a hermite expansion of the boltzmann distribution function in velocity phase space is derived based on this description of the forcing term a systematic comparison of previous lattice boltzmann models describing a nonideal gas behavior is given

two edge connected spanning subgraphs and polyhedra this paper studies the problem of finding a two edge connected spanning subgraph of minimum weight this problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks we discuss the polytope associated with the solutions to this problem we show that when the graph is series parallel the polytope is completely described by the trivial constraints and the so called cut constraints we also give some classes of facet defining inequalities of this polytope when the graph is general

regularity of the invariant measure and of the density of states in the one dimensional anderson model abstract let h 1 2 v on l2 z where v x x z are i i d r v s with common probability distribution let h t e itvd v let n e be the integrated density of states and let e be the invariant measure at energy e it is proven 1 1 let h be d times differentiable on 0 with bounded derivatives such that h j t 0 as t for j 0 1 d then i n e is a cd function ii if d 3 e is absolutely continuous for all e r and d e dx is of class c d 1 2 1 jointly in e and x 2 2 let h be differentiable on 0 with h j t o e b t for j 0 1 and some b 0 then d e dx is jointly analytic in e and x

lattice boltzmann scheme for mixture modeling analysis of the continuum diffusion regimes recovering maxwell stefan model and incompressible navier stokes equations a finite difference lattice boltzmann scheme for homogeneous mixture modeling which recovers maxwell stefan diffusion model in the continuum limit without the restriction of the mixture averaged diffusion approximation was recently proposed p asinari phys rev e 77 056706 2008 the theoretical basis is the bhatnagar gross krook type kinetic model for gas mixtures p andries k aoki and b perthame j stat phys 106 993 2002 in the present paper the recovered macroscopic equations in the continuum limit are systematically investigated by varying the ratio between the characteristic diffusion speed and the characteristic barycentric speed it comes out that the diffusion speed must be at least one order of magnitude in terms of knudsen number smaller than the barycentric speed in order to recover the navier stokes equations for mixtures in the incompressible limit some further numerical tests are also reported in particular 1 the solvent and dilute test cases are considered because they are limiting cases in which the maxwell stefan model reduces automatically to fickian cases moreover 2 some tests based on the stefan diffusion tube are reported for proving the complete capabilities of the proposed scheme in solving maxwell stefan diffusion problems the proposed scheme agrees well with the expected theoretical results

random magnetic fields on line graphs we study the spectral and transport properties of schrodinger operators on line graphs with random magnetic fields we show that it has a pure point spectrum with exponentially decaying eigenfunctions on spectral edges whereas there appears an eigenvalue with infinite multiplicity due to the structure of line graphs we compute the electrical conductivity which is zero on spectral edges but is nonzero and finite on the isolated eigenvalue mentioned above some related problems are also discussed

localization in the ground state of the one dimensional x y model with a random transverse field we consider the ground state of the quantum spin model h j sigma langle i j rangle sigma x i sigma x j sigma y i sigma y j sigma i h i sigma z i in one dimension where hi iez are independent identically distributed random variables by means of a jordan wigner transformation the model is mapped into a free fermi gas in the presence of a random external potential we then use exponential localization of the one particle states to prove exponential decay for the spin spin correlation functions

thermodynamic foundations of kinetic theory and lattice boltzmann models for multiphase flows this paper demonstrates that thermodynamically consistent lattice boltzmann models for single component multiphase flows can be derived from a kinetic equation using both enskog s theory for dense fluids and mean field theory for long range molecular interaction the lattice boltzmann models derived this way satisfy the correct mass momentum and energy conservation equations all the thermodynamic variables in these lbm models are consistent the strengths and weaknesses of previous lattice boltzmann multiphase models are analyzed

pressure tensor calculation in a class of nonideal gas lattice boltzmann models in nonideal gas lattice boltzmann lb models obtaining the correct form of the pressure tensor is essential in determining many of the statistical mechanical properties such as the surface tension and the density profile here we outline a general approach for calculating the pressure tensor in lb models with interactions beyond nearest neighbors the statistical mechanical properties calculated from such a pressure tensor are shown to agree very well with those measured from numerical experiments comparisons with alternative theories are also made

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

the traveling salesman problem in graphs with some excluded minors given a graph and a length function defined on its edge set the traveling salesman problem can be described as the problem of finding a family of edges an edge may be chosen several times which forms a spanning eulerian subgraph of minimum length in this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints this characterization is given in terms of excluded minors a constructive characterization is also given which uses a small number of basic graphs

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

smoothness of the integrated density of states on strips smoothness of the integrated density of states k e of random schrodinger operators i i d and non i i d cases on a discrete strip lattice is investigated it is proven that k e is c if only the potentials on the top surface of the strip have distributions with compactly supported densities in some fractional sobolev space the c result for the case of the anderson model i e all potentials having a distribution with compactly supported density in some sobolev space is also recovered

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

polyhedral results for two connected networks with bounded rings we study the polyhedron associated with a network design problem which consists in determining at minimum cost a two connected network such that the shortest cycle to which each edge belongs a ring does not exceed a given length k we present here a new formulation of the problem and derive facet results for different classes of valid inequalities we study the separation problems associated to these inequalities and their integration in a branch and cut algorithm and provide extensive computational results

a mean field approximation for the maximum lyapunov exponent of bernoulli trials of matrices the author considers the infinite products of d d random matrices xi which are taken according to a bernoulli distribution i e xi a with probability 1 p and xi b with probability the maximum lyapunov exponent is estimated by selecting a suitable periodic sequence one need only then compute the eigenvalues of one particular matrix which is obtained as a finite product of matrices a and b he also discusses the cases in which the method fails because of fluctuation effects

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

1d quasiperiodic operators latent symmetries a large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofsl 2 z action with equal densities of states moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum some applications of these results are presented

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

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

critical energies in random palindrome models we investigate the occurrence of critical energies where the lyapunov exponent vanishes in random schrodinger operators when the potentials have some local order which we call random palindrome models we give necessary and sufficient conditions for the presence of such critical energies the commutativity of finite word elliptic transfer matrices finally we perform some numerical calculations of the lyapunov exponents showing their behavior near the critical energies and the respective time evolution of an initially localized wave packet obtaining the exponent ruling the algebraic growth of the second momentum we also consider special random palindrome models with one letter bounded gap property the transport effects of such long range order are showed numerically

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

dynamical localization for discrete anderson dirac operators we establish dynamical localization for random dirac operators on the d dimensional lattice with d in left 1 2 3 right in the three usual regimes large disorder band edge and 1d these operators are discrete versions of the continuous dirac operators and consist in the sum of a discrete free dirac operator with a random potential the potential is a diagonal matrix formed by different scalar potentials which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support we prove the exponential decay of fractional moments of the green function for such models in each of the above regimes i e j throughout the spectrum at larger disorder jj for energies near the band edges at arbitrary disorder and jjj in dimension one for all energies in the spectrum and arbitrary disorder dynamical localization in theses regimes follows from the fractional moments method the result in the one dimensional regime contrast with one that was previously obtained for 1d dirac model with bernoulli potential

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

facets for polyhedra arising in the design of communication networks with low connectivity constraints this paper addresses the important practical problem of designing survivable fiber optic communication networks this problem can be formulated as a minimum cost network design problem with certain low connectivity constraints previous work presented structural properties of optimal solutions and heuristic methods for obtaining near optimal network designs some facet inducing inequalities for the convex hull of the solutions to this problem are given a companion paper describes computational results on real world telephone network design problems with a cutting plane method based on this work these computational results are summarized in the last section of this paper

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

zariski closure of subgroups of the symplectic group and lyapunov exponents of the schrodinger operator on the strip we consider the schrodinger equation with a random potential y n 1 q n ei y n y n 1 0 whereqn is a sequence of independent identically distributed random symmetricm m matrices with real valued elements yn m n e is the real parameter andi is the identity matrix we show that if the smallest jordan algebra of matrices containing the support of the distribution of matricesqn coincides with jordan algebra of all real valued symmetric matrices then for all but maybe a finite number of values ofe all the lyapunov exponents of our schrodinger equation are different and thus the spectrum of the corresponding schrodinger operator is pure point

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

localization of the continuous anderson hamiltonian in 1 d we study the bottom of the spectrum of the anderson hamiltonian mathcal h l partial x 2 xi on 0 l driven by a white noise xi and endowed with either dirichlet or neumann boundary conditions we show that as l rightarrow infty the point process of the appropriately shifted and rescaled eigenvalues converges to a poisson point process on mathbf r with intensity e x dx and that the appropriately rescaled eigenfunctions converge to dirac masses located at independent and uniformly distributed points furthermore we show that the shape of each eigenfunction recentered around its maximum and properly rescaled is given by the inverse of a hyperbolic cosine we also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate and we obtain very precise information on the zeros and local maxima of these eigenfunctions finally we show that the eigenvalues eigenfunctions in the dirichlet and neumann cases are very close to each other and converge to the same limits

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

prost a parabolic reconstruction of surface tension for the volume of fluid method volume of fluid vof methods are popular for the direct numerical simulation of time dependent viscous incompressible flow of multiple liquids as in any numerical method however it has its weaknesses namely for flows in which the capillary force is the dominant physical mechanism the lack of convergence with spatial refinement or convergence to a solution that is slightly different from the exact solution has been documented in the literature a well known limiting case for this is the existence of spurious currents for the simulation of a spherical drop with zero initial velocity these currents are present in all previous versions of vof algorithms in this paper we develop an accurate representation of the body force due to surface tension which effectively eliminates spurious currents we call this algorithm prost parabolic reconstruction of surface tension there are several components to this procedure including the new body force algorithm improvements in the projection method for the navier stokes solver and a higher order interface advection scheme the curvature to the interface is calculated from an optimal fit for a quadratic approximation to the interface over groups of cells

critical extreme points of the 2 edge connected spanning subgraph polytope in this paper we study the extreme points of the polytope p g the linear relaxation of the 2 edge connected spanning subgraph polytope of a graph g we introduce a partial ordering on the extreme points of p g and give necessary conditions for a non integer extreme point of p g to be minimal with respect to that ordering we show that if is a non integer minimal extreme point of p g then g and can be reduced by means of some reduction operations to a graph g and an extreme point of p g where g and satisfy some simple properties as a consequence we obtain a characterization of the perfectly 2 edge connected graphs the graphs for which the polytope p g is integral

spurious velocity from the cutoff and magnification equation in free energy based lbm for two phase flow with a large density ratio abstract the spurious velocity resulting from the cutoff and magnification equation in the free energy based one component two phase flow lattice boltzmann method lbm for a large density ratio proposed by inamuro et al 34 is investigated in inamuro s model the phase interface is tracked by the order parameter which is the dimensionless density of a two phase flow system with a small density ratio the velocity field is calculated based on the density profile of a two phase system with a large density ratio which is derived from the order parameter profile through a sine type of cutoff and magnification equation this velocity field is then used to drive the movement of a phase interface described by the order parameter the implementation of the cutoff and magnification equation causes the density profiles in the two systems to differ from each other such that force balances across the interfaces break from the ideal state and lead to spurious velocity three measures 1 the linear cutoff and magnification equation which equalizes the density profiles in the interface region 2 a new scheme which enhances the smoothness of the density profile around the cutoff points in the large density ratio system and 3 a special method that increases the interface thickness of the large density ratio system are adopted to reduce the magnitude of the spurious velocity from the profile difference respectively we show that all of these measures can successfully reduce the magnitude of the spurious velocity to some degree and the scheme that enhances the smoothness of the density profile around the cutoff points reduces the spurious velocity of the original scheme by one order of magnitude

generalized lattice boltzmann method with multirange pseudopotential the physical behavior of a class of mesoscopic models for multiphase flows is analyzed in details near interfaces in particular an extended pseudopotential method is developed which permits to tune the equation of state and surface tension independently of each other the spurious velocity contributions of this extended model are shown to vanish in the limit of high grid refinement and or high order isotropy higher order schemes to implement self consistent forcings are rigorously computed for 2d and 3d models the extended scenario developed in this work clarifies the theoretical foundations of the shan chen methodology for the lattice boltzmann method and enhances its applicability and flexibility to the simulation of multiphase flows to density ratios up to o 100

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

lower transport bounds for one dimensional continuum schrodinger operators we prove quantum dynamical lower bounds for one dimensional continuum schrodinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states this general result is applied to a number of models including the bernoulli anderson model with a constant single site potential

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

localization for discrete one dimensional random word models we consider schrodinger operators in l2 z whose potentials are obtained by randomly concatenating words from an underlying set w according to some probability measure on w our assumptions allow us to consider models with local correlations such as the random dimer model or more generally random polymer models we prove spectral localization and away from a finite set of exceptional energies dynamical localization for such models these results are obtained by employing scattering theoretic methods together with furstenberg s theorem to verify the necessary input to perform a multiscale analysis

quantum particle in a hierarchical potential with tunnelling over arbitrarily large scales the authors introduce a notion of hierarchical potential by iterating a basic elementary geometric construction over an increasing sequence of length scales they then show that a quantum particle in such a potential exhibits a subdiffusive behaviour characterised by r2 or c ln t beta if the initial state is a normalised wavepacket superposition of states of sufficiently low energy

two edge connected subgraphs with bounded rings polyhedral results and branch and cut we consider the network design problem which consists in determining at minimum cost a 2 edge connected network such that the shortest cycle a ring to which each edge belongs does not exceed a given length k we identify a class of inequalities called cycle inequalities valid for the problem and show that these inequalities together with the so called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables we then study the polytope associated with that problem and describe further classes of valid inequalities we give necessary and sufficient conditions for these inequalities to be facet defining we study the separation problem associated with these inequalities in particular we show that the cycle inequalities can be separated in polynomial time when k 4 we develop a branch and cut algorithm based on these results and present extensive computational results

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

thermodynamic consistency of liquid gas lattice boltzmann simulations lattice boltzmann simulations have been very successful in simulating liquid gas and other multiphase fluid systems however the underlying second order analysis of the equation of motion has long been known to be insufficient to consistently derive the fourth order terms that are necessary to represent an extended interface these same terms are also responsible for thermodynamic consistency i e to obtain a true equilibrium solution with both a constant chemical potential and a constant pressure in this article we present an equilibrium analysis of nonideal lattice boltzmann methods of sufficient order to identify those higher order terms that lead to a lack of thermodynamic consistency we then introduce a thermodynamically consistent forcing method

making almost commuting matrices commute suppose two hermitian matrices a b almost commute vert a b vert leq delta are they close to a commuting pair of hermitian matrices a b with vert a a vert vert b b vert leq epsilon a theorem of h lin 3 shows that this is uniformly true in that for every epsilon 0 there exists a 0 independent of the size n of the matrices for which almost commuting implies being close to a commuting pair however this theorem does not specify how depends on epsilon we give uniform bounds relating and epsilon the proof is constructive giving an explicit algorithm to construct a and b we provide tighter bounds in the case of block tridiagonal and tridiagonal matrices within the context of quantum measurement this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously finally we comment briefly on the case of approximately measuring three or more approximately commuting operators using povms positive operator valued measures instead of projective measurements

dynamical localization of quantum walks in random environments the dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom the coin states is considered the discrete time unitary dynamics is determined by the repeated action of a coin operator in u 2 on the internal degrees of freedom followed by a one step shift to the right or left conditioned on the state of the coin for a fixed coin operator the dynamics is known to be ballistic

positive lyapunov exponents and a large deviation theorem for continuum anderson models briefly abstract in this short note we prove positivity of the lyapunov exponent for 1d continuum anderson models by leveraging some classical tools from inverse spectral theory the argument is much simpler than the existing proof due to damanik sims stolz and it covers a wider variety of random models along the way we note that a large deviation theorem holds uniformly on compacts

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

solving the two connected network with bounded meshes problem we study the problem of designing at minimum cost a two connected network such that the shortest cycle to which each edge belongs a mesh does not exceed a given lengthk this problem arises in the design of fiber optic based backbone telecommunication networks a branch and cut approach to this problem is presented for which we introduce several families of valid inequalities and discuss the corresponding separation algorithms because the size of the problems solvable to optimality by this approach is too small we also develop some heuristics the computational performances of these exact and approximate methods are then thoroughly assessed both on randomly generated instances as well as instances suggested by real applications

a rigorous replica trick approach to anderson localization in one dimension leth v onl2 wherev x x are i i d r v s and letg l x y e i x hl e i 1 y whereh l denotes the operatorh restricted to l l 1 l with dirichlet boundary conditions we use a supersymmetric replica trick to prove that e g l 0 x e i eta 2 leqq k eta 2 exp m log eta sigma x for somem 0 0 k uniformly inl ande this estimate together with the usual necessary estimate on the density of states implies zero conductivity and gives exponential localization by the frohlich martinelli scoppola and spencer method

send and split method for minimum concave cost network flows many problems from inventory production and capacity planning and from network design exhibit scale economies and can be formulated in terms of finding minimum additive concave cost nonnegative network flows we reduce the problem to an equivalent one in which the arc flow costs are nonnegative and give a dynamic programming method called the send and split method to solve it the main work of the method entails repeatedly solving set splitting and minimum cost chain problems in uncapacitated networks with n nodes a arcs and d 1 demand nodes i e nodes with nonzero exogenous demand the algorithm requires up to n2 13d s2d operations additions and comparisons where s n log2n 3a is the number of operations required to solve a minimum cost chain problem with nonnegative arc costs on the augmented graph formed by appending a node and an arc thereto from each node in the graph if also the network is k planar i e the graph is planar with all demand nodes lying on the boundary of k faces the method requires at most n2 kd3k sd2k operations the algorithm can be applied to capacitated networks because they can be reduced to equivalent uncapacitated ones these results unify significantly generalize e g to cyclic problems and sometimes improve upon e g for tandem facilities known polynomial time dynamic programming algorithms for wagner and whitin s wagner h m whitin t m 1958 dynamic version of the economic lot size model management sci 5 89 96 dynamic economic order quantity problem zangwill s zangwill w i 1969 a backlogging model and a multi echelon model of a dynamic economic lot size production system a network approach management sci 15 506 527 generalization to tandem facilities and veinott s veinott jr a f 1969 minimum concave cost solution of leontief substitution models of multi facility inventory systems oper res 17 262 291 increasing capacity warehousing problem the networks for the finite period versions of these problems are each 1 planar the method improves upon zangwill s zangwill w i 1968 minimum concave cost flows in certain networks management sci 14 429 450 related oand running time dynamic programming method for finding minimum additive concave cost non negative flows in circuitless single source networks we also implement the method to solve in polynomial time the d 1 demand node and k planar versions of the minimum cost forest and steiner problems in graphs the running time for the d 1 demand node steiner problem in graphs is comparable to that of dreyfus and wagner s dreyfus s e wagner r a 1971 the steiner problem in graphs networks1 195 207 method

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

unified framework for a side by side comparison of different multicomponent algorithms lattice boltzmann vs phase field model lattice boltzmann models lbm and phase field models pfm are two of the most widespread approaches for the numerical study of multicomponent fluid systems both methods have been successfully employed by several authors but despite their popularity still remains unclear how to properly compare them and how they perform on the same problem here we present a unified framework for the direct one to one comparison of the multicomponent lbm against the pfm we provide analytical guidelines on how to compare the shan chen sc lattice boltzmann model for non ideal multicomponent fluids with a corresponding free energy fe lattice boltzmann model then in order to properly compare the lbm vs the pfm we propose a new formulation for the free energy of the cahn hilliard navier stokes equations finally the lbm model is numerically compared with the corresponding phase field model solved by means of a pseudo spectral algorithm this work constitute a first attempt to set the basis for a quantitative comparison between different algorithms for multicomponent fluids we limit our scope to the few of the most common variants of the two most widespread methodologies namely the lattice boltzmann model sc and fe variants and the phase field model

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

multiphase lattice boltzmann flux solver for incompressible multiphase flows with large density ratio a multiphase lattice boltzmann flux solver mlbfs is proposed in this paper for incompressible multiphase flows with low and large density ratios in the solver the flow variables at cell centers are given from the solution of macroscopic governing differential equations navier stokes equations recovered by multiphase lattice boltzmann lb model by the finite volume method at each cell interface the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice boltzmann equation lbe the forcing terms in the governing equations are directly treated by the finite volume discretization the phase interfaces are captured by solving the phase field cahn hilliard equation with a fifth order upwind scheme unlike the conventional multiphase lb models which restrict their applications on uniform grids with fixed time step the mlbfs has the capability and advantage to simulate multiphase flows on non uniform grids the proposed solver is validated by several benchmark problems such as two phase co current flow taylor couette flow in an annulus rayleigh taylor instability and droplet splashing on a thin film at density ratio of 1000 with reynolds numbers ranging from 20 to 1000 numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high reynolds number

disordered quantum walks in one lattice dimension we study a spin 12 particle moving on a one dimensional lattice subject to disorder induced by a random space dependent quantum coin the discrete time evolution is given by a family of random unitary quantum walk operators where the shift operation is assumed to be deterministic each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices we derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization put differently the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance our findings imply that this effect takes place if the coin is chosen at random from the haar measure or some measure continuous with respect to it but also for a class of discrete probability measures which support consists of two coins one of them being the hadamard coin

local wegner and lifshitz tails estimates for the density of states for continuous random schrodinger operators we introduce and prove local wegner estimates for continuous generalized anderson hamiltonians where the single site random variables are independent but not necessarily identically distributed in particular we get wegner estimates with a constant that goes to zero as we approach the bottom of the spectrum as an application we show that the differentiated density of states exhibits the same lifshitz tails upper bound as the integrated density of states

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

random composition of two rational maps singularity of the invariant measure we study the invariant measure of a markov chain obtained by randomly composing two rational maps related to the anderson model with a bernoulli potential for a certain range of the parameters we show that the invariant measure is singular continuous in certain cases the support turns out to be a cantor set with a multifractal structure

from the continuous to the lattice boltzmann equation the discretization problem and thermal models the velocity discretization is a critical step in deriving the lattice boltzmann lbe from the continuous boltzmann equation this problem is considered in the present paper following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the lbe with respect to the continuous boltzmann equation and with the lattice structure considering n to be the order of the polynomial approximation to the maxwell boltzmann equilibrium distribution it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the hermite polynomial tensors in the hilbert space generated by the functions that map the velocity space onto the real numbers space as a consequence it is shown that for each order n of approximation the even parity velocity tensors are isotropic up to rank 2n in the discrete space the norm and the orthogonality restrictions lead to space filling lattices with increased dimensionality when compared with presently known lattices this problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space two dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed

localization near the edge for the anderson bernoulli model on the two dimensional lattice we consider a hamiltonian given by the laplacian plus a bernoulli potential on the two dimensional lattice we prove that for energies sufficiently close to the edge of the spectrum the resolvent on a large square is likely to decay exponentially this implies almost sure anderson localization for energies sufficiently close to the edge of the spectrum our proof follows the program of bourgain kenig using a new unique continuation result inspired by a liouville theorem of buhovsky logunov malinnikova sodin

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies

lattice boltzmann model for binary mixtures the lattice boltzmann equation lbe 1 6 is emerging as an effective computational method based on fundamental physics for simulating complex flows such as multiphase 7 10 and multiple component flows 11 13 flows through porous media cf ref 5 and particulate suspensions in fluid flows e g ref 14 recently important strides have been made on the theoretical front establishing from fundamental principles the physical legitimacy and mathematical rigor of the lbe method most importantly it has been proved that the lattice boltzmann equation can be derived from the boltzmann equation a priori 2 4 9 10 i t should be pointed out that the boltzmann equation bridges the gap between the microscopic dynamics and the macroscopic hydrodynamics indeed the navier stokes equations can be rigorously derived from the boltzmann equation via the chapman enskog analysis the second important theoretical result is the demonstration that the lattice boltzmann equation is indeed equivalent to an explicit finite difference scheme of the navier stokes equations 15 these theoretical developments have completely and comprehensively resolved all doubts surrounding the early lattice gas automata 16 and lattice boltzmann models the present day lattice boltzmann equation is a viable alternative to the continuum methods for simulating fluid flows much of the rigorous work with lattice boltzmann methods so far has been restricted to simple single phase single component fluids recently the lbe model for single component multiphase fluids has been derived from the enskog equation 9 10 a rigorous mathematical development of the lattice boltzmann method for multicomponent fluids is still in its infancy and such is the object of the present work in many practical flows involving pollutant dispersion chemical processing and combustor mixing and reaction mass and momentum transport in multispecies fluids plays an important role for these applications the continuumbased models can be difficult to compute due to various reasons such as complexity of flow geometry and phase change moreover it is difficult to construct the continuum based models from first principles therefore for these flows there is a growing interest in using the lattice boltzmann equation 11 13 in this paper we develop a unified approach for developing the lattice boltzmann models for multicomponent fluids within the framework of kinetic theory this work is a part of our continuing effort to set the lattice boltzmann equation on a rigorous foundation 2 3 9 10 specifically we will derive a conservative discretized version of the continuum boltzmann equation for fluid mixtures we shall present a model that is capable of simulating either a miscible or immiscible binary mixture the lattice boltzmann equation considered here can be extended to a mixture of three or more species the kinetic theory of gas mixtures encompasses a significant amount of literature e g refs 17 27 in a manner similar to the derivation of the boltzmann equation for a pure system of single species one can derive n simultaneous equations for a system of n species by reducing the appropriate liouville equation for the sake of simplicity without loss of generality we shall only discuss the boltzmann equations for a binary system t f a 1jiif a 1aaiij f a 5q aa 1q ab 1