w covariance of the weyl wigner groenewold moyal quantization the differential structure of operator bases used in various forms of the weyl wigner groenewold moyal wwgm quantization is analyzed and a derivative based approach alternative to the conventional integral based one is developed thus the fundamental quantum relations follow in a simpler and unified manner an explicit formula for the ordered products of the heisenberg weyl algebra is obtained the w covariance of the wwgm quantization in its most general form is established it is shown that the group action of w that is realized in the classical phase space induces on bases operators in the corresponding hilbert space a similarity transformation generated by the corresponding quantum w which provides a projective representation of the former w explicit expressions for the algebra generators in the classical phase space and in the hilbert space are given it is made manifest that this w covariance of the wwgm quantization is a genuine property of the operator bases

second order boltzmann schemes for compressible euler equations in one and two space dimensions a class of second order numerical schemes for the compressible euler equations is described and their l 1 stability i e rho geqq 0 t geqq 0 is proved following van leer s approach the solution rho u sqrt t here is represented as piecewise linear functions the necessity of a slope limitation appears naturally in the derivation of the schemes but it can be less strict than the slope reconstructions usually used these schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space the upwinding is obtained through a very generalized notion of characteristics the kinetic one

realizable high order finite volume schemes for quadrature based moment methods dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however direct numerical solution of kinetic equations is often infeasible because of the large number of independent variables an alternative is to reformulate the problem in terms of the moments of the velocity distribution recently a quadrature based moment method was derived for approximating solutions to kinetic equations the success of the new method is based on a moment inversion algorithm that is used to calculate non negative weights and abscissas from the moments the moment inversion algorithm does not work if the moments are non realizable which might lead to negative weights it has been recently shown 14 that realizability is guaranteed only with the 1st order finite volume scheme that has an inherent problem of excessive numerical diffusion the use of high order finite volume schemes may lead to non realizable moments in the present work realizability of the finite volume schemes in both space and time is discussed for the 1st time a generalized idea for developing realizable high order finite volume schemes for quadrature based moment methods is presented these finite volume schemes give remarkable improvement in the solutions for a certain class of problems it is also shown that the standard runge kutta time integration schemes do not guarantee realizability however realizability can be guaranteed if strong stability preserving ssp runge kutta schemes are used numerical results are presented on both cartesian and triangular meshes

growth rates and fractional dimensions in cellular automata abstract this paper concerns some different ways by which a fractional dimension may be associated to a cellular automaton whose transition rule is linear modulus p the growth rate dimensions are shown to equal the kolmogorov dimensions suggestions are made as to the relevance for dendritic crystal growth

a class of quadrature based moment closure methods with application to the vlasov poisson fokker planck system in the high field limit quadrature based moment closure methods are a class of approximations that replace high dimensional kinetic descriptions with lower dimensional fluid models in this work we investigate some of the properties of a sub class of these methods based on bi delta bi gaussian and bi b spline representations we develop a high order discontinuous galerkin dg scheme to solve the resulting fluid systems finally via this high order dg scheme and strang operator splitting to handle the collision term we simulate the fluid closure models in the context of the vlasov poisson fokker planck system in the high field limit we demonstrate numerically that the proposed scheme is asymptotic preserving in the high field limit

radiation transport modeling using extended quadrature method of moments abstract the radiative transfer equation describes the propagation of radiation through a material medium while it provides a highly accurate description of the radiation field the large phase space on which the equation is defined makes it numerically challenging as a consequence significant effort has gone into the development of accurate approximation methods recently an extended quadrature method of moments eqmom has been developed to solve univariate population balance equations which also have a large phase space and thus face similar computational challenges the distinct advantage of the eqmom approach over other moment methods is that it generates moment equations that are consistent with a positive phase space density and has a moment inversion algorithm that is fast and efficient the goal of the current paper is to present the eqmom method in the context of radiation transport to discuss advantages and disadvantages and to demonstrate its performance on a set of standard one dimensional benchmark problems that encompass optically thin thick and transition regimes special attention is given in the implementation to the issue of realizability that is consistency with a positive phase space density numerical results in one dimension are promising and lay the foundation for extending the same framework to multiple dimensions

multispeed models in off lattice boltzmann simulations the lattice boltzmann method is a highly promising approach to the simulation of complex flows here we realize recently proposed multispeed lattice boltzmann models s chikatamarla et al phys rev lett 97 190601 2006 by exploiting the flexibility offered by off lattice boltzmann methods the approach is based on the general characteristic based algorithm for off lattice boltzmann simulations that preserves all appealing properties of the standard lattice boltzmann method while extending the method to unstructured grids we show that the use of multispeed models indeed gives rise to major improvements in accuracy the suggested approach thus renders truly large scale off lattice boltzmann computations practical

eulerian multi fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays in this paper we present a new eulerian multi fluid modeling for dense sprays of evaporating liquid droplets which is able to describe droplet coalescence and size polydispersion as well as the associated size conditioned dynamics it is an uncommon feature of eulerian spray models which are required in a number of non stationary simulations because of the optimization capability of a solver coupling a eulerian description for both phases the chosen framework is the one of laminar flows or the one of direct numerical simulations since no turbulence models are included in the present study the model is based on a rigorous derivation from the kinetic level of description p d f equation and can be considered as a major extension of the original sectional method introduced by tambour et al we obtain a set of conservation equations for each fluid a statistical average of all the droplets in given size intervals associated to a discretization of the size phase space the coalescence phenomenon appears as quadratic source terms the coefficients of which the collisional integrals can be pre calculated from a given droplet size discretization and do not depend on space nor time we validate this eulerian model by performing several comparisons for both stationary and non stationary cases to a classical lagrangian model which involves a stochastic algorithm in order to treat the coalescence phenomenon the chosen configuration is a self similar 2d axisymmetrical decelerating nozzle with evaporating sprays having various size distributions ranging from smooth ones up to dirac delta functions through discontinuous ones we show that the eulerian model if the discretization in the size phase space is fine enough the problem is then 3d unstationary 2d in space and 1d in size is able to reproduce very accurately the non stationary coupling of evaporation dynamics and coalescence moreover it can still reproduce the global features of the behavior of the spray with a coarse size discretization which is a nice feature compared to lagrangian approaches the computational efficiency of both approaches are then compared and the eulerian model is provided to be a good candidate for more complex and realistic configurations

gaussian process regression for maximum entropy distribution abstract maximum entropy distributions offer an attractive family of probability densities suitable for moment closure problems yet finding the lagrange multipliers which parametrize these distributions turns out to be a computational bottleneck for practical closure settings motivated by recent success of gaussian processes we investigate the suitability of gaussian priors to approximate the lagrange multipliers as a map of a given set of moments examining various kernel functions the hyperparameters are optimized by maximizing the log likelihood the performance of the devised data driven maximum entropy closure is studied for couple of test cases including relaxation of non equilibrium distributions governed by bhatnagar gross krook and boltzmann kinetic equations

conditional hyperbolic quadrature method of moments for kinetic equations the conditional quadrature method of moments cqmom was introduced by yuan and fox j comput phys 230 22 8216 8246 2011 to reconstruct a velocity distribution function vdf from a finite set of its integer moments the reconstructed vdf takes the form of a sum of weighted dirac delta functions in velocity phase space and provides a closure for the spatial flux term in the corresponding kinetic equation the cqmom closure for the flux leads to a weakly hyperbolic system of moment equations in subsequent work chalons et al proceed ctr sum prog 2010 347 358 2010 the dirac delta functions were replaced by gaussian distributions which make the moment system hyperbolic but at the added cost of dealing with continuous distributions here a hyperbolic version of cqmom is proposed that uses weighted dirac delta functions while the moment set employed for multi gaussian and conditional hyqmom chyqmom are equivalent the latter is able to access all of moment space whereas the former cannot e g arbitrary values of the fourth order velocity moment in 1 d phase space with two nodes by making use of the properties of chyqmom in 2 d phase space it is possible to control a symmetrical subset of the optimal moments fox indust engng chem res 48 21 9686 9696 2009 furthermore the moment sets for 2 d problems are smaller for chyqmom than in the original cqmom thanks to a judicious choice of the velocity abscissas in phase space

the algebra of weyl symmetrised polynomials and its quantum extension the algebra of weyl symmetrised polynomials in powers of hamiltonian operatorsp andq which satisfy canonical commutation relations is constructed this algebra is shown to encompass all recent infinite dimensional algebras acting on two dimensional phase space in particular the moyal bracket algebra and the poisson bracket algebra of which the moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra we propose the introduction of a second deformation by the replacement of the heisenberg algebra forp q with aq deformed commutator and construct algebras ofq symmetrised polynomials

accuracy of higher order lattice boltzmann methods for microscale flows with finite knudsen numbers accuracy of the lattice boltzmann lb method for microscale flows with finite knudsen numbers is investigated we employ up to the eleventh order gauss hermite quadrature for the lattice velocities and diffuse scattering boundary condition for fluid wall interactions detailed comparisons with the direct simulation monte carlo dsmc method and the linearized boltzmann equation are made for planar couette and poiseuille flows all higher order lb methods considered here give improved results as compared with the standard lb method the accuracy of the lb hierarchy however does not monotonically increase with the order of the gauss hermite quadrature at moderate and large knudsen numbers the results also show the sensitivity to a quadrature chosen even when the gauss hermite quadratures have the same order of formal accuracy among the schemes investigated here d2q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes the higher order lb methods predict the knudsen layer up to kn o 0 1 the knudsen layer however rapidly disappears when the knudsen number approaches unity due to a finite number of the lattice velocities while it is still present for kn o 1 in the boltzmann equation it is also found that the higher order lb methods adopted here do not capture the asymptotic behavior of the boltzmann equation at large knudsen numbers

higher order mixed moment approximations for the fokker planck equation in one space dimension we study mixed moment models full zeroth moment half higher moments for a fokker planck equation in one space dimension mixed moment minimum entropy models are known to overcome the zero net flux problem of full moment minimum entropy m n models a realizability theory for these mixed moments of arbitrary order is derived as well as a new closure which we refer to as kershaw closure they provide nonnegative distribution functions combined with an analytical closure numerical tests are performed with standard first order finite volume schemes and compared with a finite difference fokker planck scheme

hamiltonian closures for fluid models with four moments by dimensional analysis fluid reductions of the vlasov ampere equations that preserve the hamiltonian structure of the parent kinetic model are investigated hamiltonian closures using the first four moments of the vlasov distribution are obtained and all closures provided by a dimensional analysis procedure for satisfying the jacobi identity are found two hamiltonian models emerge for which the explicit closures are given along with their poisson brackets and casimir invariants

a connection between operator orderings and representations of the lie algebra mathfrak sl _2 we investigate special classes of polynomials in the quantum mechanical position and momentum operators arising from various operator orderings in particular from the so called orderings generalizing well known operator orderings in quantum mechanics such as the weyl ordering the normal ordering etc viewing orderings as maps from the polynomial algebra on the phase space to the weyl algebra generated by the quantum mechanical position and momentum operators we formulate conditions under which these maps intertwine certain naturally defined actions of the lie algebra these conditions arise via certain regularities in coefficients defining the orderings which can nicely be described in terms of some combinatorial objects called here inverted pascal diagrams at the end we establish a connection between radial elements in the weyl algebra and certain polynomials of the number operator expressible in terms of the hypergeometric function this is related to another representation of realized in terms of difference operators

the moyal lie theory of phase space quantum mechanics a lie algebraic approach to the unitary transformations in weyl quantization is discussed this approach being formally equivalent to the quantization is an extension of the classical poisson lie theory and can be used as an efficient tool in the quantum phase space transformation theory

a quadrature based moment method for dilute fluid particle flows gas particle and other dispersed phase flows can be described by a kinetic equation containing terms for spatial transport acceleration and particle processes such as evaporation or collisions in principle the kinetic description is valid from the dilute non collisional to the dense limit however its numerical solution in multi dimensional systems is intractable due to the large number of independent variables as an alternative lagrangian methods discretize the density function into parcels that are simulated using monte carlo methods while quite accurate as in any statistical approach lagrangian methods require a relatively large number of parcels to control statistical noise and thus are computationally expensive a less costly alternative is to solve eulerian transport equations for selected moments of the kinetic equation however it is well known that in the dilute limit eulerian methods have great difficulty to describe correctly the moments as predicted by a lagrangian method here a two node quadrature based eulerian moment closure is developed and tested for the kinetic equation it is shown that the method can successfully handle highly non equilibrium flows e g impinging particle jets jet crossing particle rebound off walls finite stokes number flows that heretofore could not be treated accurately with the eulerian approach

efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes this paper presents efficient second order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows for compressible unsteady flows a time dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables as a result a compact scheme on the unstructured meshes is developed for incompressible steady flows a continuous second order gas kinetic bgk type scheme is presented for which the time dependent gas distribution function with a continuous particle velocity is used on unstructured meshes the efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells therefore even with the stencil of a first order scheme a high resolution method is constructed numerical examples are presented which are compared with the benchmark solutions and the experimental measurements

a quadrature based third order moment method for dilute gas particle flows dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows away from the equilibrium maxwellian limit in this work a quadrature based third order moment closure is derived that can be applied to gas particle flows at any knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the weights and abscissas a robust inversion procedure is proposed for moments up to third order and tested for three example applications riemann shock problem impinging jets and vertical channel flow extension of the moment inversion algorithm to fifth or higher order is possible but left to future work the spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient diffusion model is not used to close the fluxes because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the boltzmann equation the mass momentum and energy are conserved for arbitrary knudsen number including the euler limit while developed here for dilute gas particle flows quadrature based moment methods can in principle be applied to any application that can be modeled by a kinetic equation e g thermal and non isothermal flows currently treated using lattice boltzmann methods and examples are given from the literature

numerical simulation of spray coalescence in an eulerian framework direct quadrature method of moments and multi fluid method the scope of the present study is eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation the fundamental mathematical description is the williams spray equation governing the joint number density function f v u x t of droplet volume and velocity eulerian multi fluid models have already been rigorously derived from this equation in laurent et al f laurent m massot p villedieu eulerian multi fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays j comput phys 194 2004 505 543 the first key feature of the paper is the application of direct quadrature method of moments dqmom introduced by marchisio and fox d l marchisio r o fox solution of population balance equations using the direct quadrature method of moments j aerosol sci 36 2005 43 73 to the williams spray equation both the multi fluid method and dqmom yield systems of eulerian conservation equations with complicated interaction terms representing coalescence in order to focus on the difficulties associated with treating size dependent coalescence and to avoid numerical uncertainty issues associated with two way coupling only one way coupling between the droplets and a given gas velocity field is considered in order to validate and compare these approaches the chosen configuration is a self similar 2d axisymmetrical decelerating nozzle with sprays having various size distributions ranging from smooth ones up to dirac delta functions the second key feature of the paper is a thorough comparison of the two approaches for various test cases to a reference solution obtained through a classical stochastic lagrangian solver both eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the lagrangian solver the third key point of the study is a detailed description of the limitations associated with each method thus giving criteria for their use as well as for their respective efficiency

conditional quadrature method of moments for kinetic equations kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution moment methods are an important class of approximate models derived from kinetic equations but require closure to truncate the moment set in quadrature based moment methods qbmm closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments e g spatial fluxes can be related to the finite moment set in this work a novel moment inversion algorithm based on 1 d adaptive quadrature of conditional velocity moments is introduced and shown to always yield realizable distribution functions i e non negative quadrature weights this conditional quadrature method of moments cqmom can be used to compute exact n point quadratures for multi valued solutions also known as the multi variate truncated moment problem and provides optimal approximations of continuous distributions in order to control numerical errors arising in volume averaging and spatial transport an adaptive 1 d quadrature algorithm is formulated for use with cqmom the use of adaptive cqmom in the context of qbmm for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing i e collision less systems elastic and inelastic particle particle collisions and external forces i e fluid drag

higher order quadrature based moment methods for kinetic equations kinetic equations containing terms for spatial transport body forces and particle particle collisions occur in many applications e g rarefied gases dilute granular gases fluid particle flows the direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows sufficiently far away from the maxwellian limit in previous work a quadrature based third order moment closure was derived for approximating solutions to the kinetic equation for arbitrary knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the non negative weights and velocity abscissas here a robust inversion procedure is proposed for three component velocity moments up to ninth order by reconstructing the velocity distribution function the spatial fluxes in the moment equations are treated using a kinetic based finite volume solver because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the kinetic equation the mass momentum and energy are conserved for arbitrary knudsen and mach numbers the computational algorithm is tested for the riemann shock problem and for increasing knudsen numbers i e larger deviations from the maxwellian limit the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes

a solution algorithm for fluid particle flows across all flow regimes abstract many fluid particle flows occurring in nature and in technological applications exhibit large variations in the local particle volume fraction for example in circulating fluidized beds there are regions where the particles are close packed as well as very dilute regions where particle particle collisions are rare thus in order to simulate such fluid particle systems it is necessary to design a flow solver that can accurately treat all flow regimes occurring simultaneously in the same flow domain in this work a solution algorithm is proposed for this purpose the algorithm is based on splitting the free transport flux solver dynamically and locally in the flow in close packed to moderately dense regions a hydrodynamic solver is employed while in dilute to very dilute regions a kinetic based finite volume solver is used in conjunction with quadrature based moment methods to illustrate the accuracy and robustness of the proposed solution algorithm it is implemented in openfoam for particle velocity moments up to second order and applied to simulate gravity driven gas particle flows exhibiting cluster induced turbulence by varying the average particle volume fraction in the flow domain it is demonstrated that the flow solver can handle seamlessly all flow regimes present in fluid particle flows

quantum canonical invariance a moyal approach the quasiclassical generalised moyal quantisation procedure is applied to the problem of quantum canonical invariance simple new formulae for the moyal bracket corresponding to various correspondence rules are given and are used to calculate the additional potential terms arising from a point canonical transformation made on a free hamiltonian the status of canonical transformations in other equivalent quantum phase space formulations is also reviewed

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

realizable high order finite volume schemes for quadrature based moment methods applied to diffusion population balance equations population balance equations with advection and diffusion terms can be solved using quadrature based moment methods recently high order realizable finite volume schemes with appropriate realizability criteria have been derived for the advection term however hitherto no work has been reported with respect to realizability problems for the diffusion term the current work focuses on developing high order realizable finite volume schemes for diffusion the pitfalls of existing finite volume schemes for the diffusion term based on the reconstruction of moments are discussed and it is shown that realizability can be guaranteed only with the 2 nd order scheme and that the realizability criterion for the 2 nd order scheme is the same as the stability criterion however realizability of moments cannot be guaranteed when higher order moment based reconstruction schemes are used to overcome this problem realizable high order finite volume schemes based on the reconstruction of weights and abscissas are proposed and suitable realizability criteria are derived the realizable schemes can achieve higher than 2 nd order accuracy for problems with smoothly varying abscissas in the worst case scenario of highly nonlinear abscissas the realizable schemes are 2 nd order accurate but have lower error magnitudes compared to existing schemes the results obtained using the realizable high order schemes are shown to be consistent with those obtained using the 2 nd order moment based reconstruction scheme

a robust two node 13 moment quadrature method of moments for dilute particle flows including wall bouncing for flows where the particle number density is low and the stokes number is relatively high as found when sand or ice is ingested into aircraft gas turbine engines streams of particles can cross each other s path or bounce from a solid surface without being influenced by inter particle collisions the aim of this work is to develop an eulerian method to simulate these types of flow to this end a two node quadrature based moment method using 13 moments is proposed in the proposed algorithm thirteen moments of particle velocity including cross moments of second order are used to determine the weights and abscissas of the two nodes and to set up the association between the velocity components in each node previous quadrature method of moments qmom algorithms either use more than two nodes leading to increased computational expense or are shown here to give incorrect results under some circumstances this method gives the computational efficiency advantages of only needing two particle phase velocity fields whilst ensuring that a correct combination of weights and abscissas is returned for any arbitrary combination of particle trajectories without the need for any further assumptions particle crossing and wall bouncing with arbitrary combinations of angles are demonstrated using the method in a two dimensional scheme the ability of the scheme to include the presence of drag from a carrier phase is also demonstrated as is bouncing off surfaces with inelastic collisions the method is also applied to the taylor green vortex flow test case and is found to give results superior to the existing two node qmom method and is in good agreement with results from lagrangian modelling of this case

numerical approximations of pressureless and isothermal gas dynamics we study several schemes of first or second order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations the pressureless gas system is weakly hyperbolic giving rise to the formation of density concentrations known as delta shocks for the isothermal gas system the infinite speed of expansion into vacuum leads to zero timestep in the godunov method based on exact riemann solver the schemes we consider are able to bypass these difficulties they are proved to satisfy positiveness of density and discrete entropy inequalities to capture the delta shocks and to treat data with vacuum

a comparative study of the lbe and gks methods for 2d near incompressible laminar flows we compare the lattice boltzmann equation lbe and the gas kinetic scheme gks applied to 2d incompressible laminar flows although both methods are derived from the boltzmann equation thus share a common kinetic origin numerically they are rather different the lbe is a finite difference method while the gks is a finite volume one in addition the lbe is valid for near incompressible flows with low mach number restriction ma 0 3 while the gks is valid for fully compressible flows in this study we use the generalized lattice boltzmann equation glbe with multiple relaxation time mrt collision model which overcomes all the apparent defects in the popular lattice bgk equation we use both the lbe and gks methods to simulate the flow past a square block symmetrically placed in a 2d channel with the reynolds number re between 10 and 300 the lbe and gks results are validated against the well resolved results obtained using finite volume method our results show that both the lbe and gks yield quantitatively similar results for laminar flow simulations and agree well with existing ones provided that sufficient grid resolution is given for 2d problems the lbe is about 10 and 3 times faster than the gks for steady and unsteady flow calculations respectively while the gks uses less memory we also observe that the gks method is much more robust and stable for under resolved cases due to its upwinding nature and interpolations used in calculating fluxes

entropic quadrature for moment approximations of the boltzmann bgk equation abstract using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the boltzmann equation to model rarefied gases one attractive example is the maximum entropy closure which however is computationally barely affordable in this paper we will combine the maximum entropy approach with the quadrature method of moments qmom two methods which have in some sense diametric properties thereby introducing the entropic quadrature eq closure eq calculates a sparse quadrature based reconstruction of the unknown velocity distribution like the qmom where the physical meaningful maximum entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments we will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions