Theory of the lattice Boltzmann method:From the Boltzmann equationto the lattice Boltzmann equationXiaoyi He 1,2,*and Li-Shi Luo 2,3,†1Center for Nonlinear Studies,MS-B258,Los Alamos National Laboratory,Los Alamos,New Mexico 875452Complex Systems Group T-13,MS-B213,Theoretical Division,Los Alamos National Laboratory,Los Alamos,New Mexico 875453ICASE,MS 403,NASA Langley Research Center,6North Dryden Street,Building 1298,Hampton,Virginia 23681-0001͑Received 29April 1997;revised manuscript received 26August 1997͒In this paper,the lattice Boltzmann equation is directly derived from the Boltzmann equation.It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation.Various approxi-mations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail.A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is dem-onstrated explicitly.The lattice Boltzmann models derived include the two-dimensional 6-bit,7-bit,and 9-bit,and three-dimensional 27-bit models.͓S1063-651X ͑97͒12512-0͔PACS number ͑s ͒:47.10.ϩg,47.11.ϩj,05.20.DdI.INTRODUCTIONIn the last few years,we have witnessed a rapid develop-ment of the method known as the lattice Boltzmann equation ͑LBE ͓͒1–6͔.Although only in its infancy,the LBE method has demonstrated its ability to simulate hydrodynamic sys-tems ͓1–5͔,magnetohydrodynamic systems ͓7͔,multiphase and multicomponent fluids ͓8͔including suspensions ͓9͔and emulsions ͓10͔,chemical-reactive flows ͓11͔,and multi-component flow through porous media ͓12͔.Together with modern computers of massively parallel processors,the LBE method has become a powerful computational method for studying various complex systems.The obvious advantages of the LBE method are the parallelism of the method,the simplicity of the programming,and the capability of incor-porating model interactions.However,a rigorous framework of the LBE method is still lacking in spite of the great inter-est in the method.Historically,the models of the lattice Boltzmann equation directly evolve from the models of the lattice-gas automata ͑LGA ͓͒13͔.While the LGA models are Boolean ones,the LBE models are indeed the floating-number counterpart of the corresponding LGA models—a particle in the LGA model ͑represented by a Boolean number ͒is replaced by the single-particle distribution function ͑represented by a real number ͒.Even though the LBE models appear to be rather different from their LGA counterparts because various approximations,such as the linearization of the collision op-erator ͓2͔and the Bhatnagar-Gross-Krook ͑BGK ͓͒15–17͔approximation ͓3͔,have been applied,the theoretical frame-work of the LBE method nevertheless rests upon the Chapman-Enskog analysis of the LGA method ͓14͔.Al-though the connection between the LBE models and the con-tinuous Boltzmann equation are discussed in various places ͓18,19͔,so far there exists no rigorous result in this direction.The lack of a thorough understanding of the LBE method has some immediate implications.For instance,the LBE method has not been very successful in simulating thermohydrody-namic systems ͓19–22͔,nor has the LBE method been able to be implemented on arbitrary mesh grids ͓5,23͔,even though considerable effort has been applied in these direc-tions.As we have shown recently ͓24–26͔,substantial progress can be made in the aforementioned areas once a better understanding of the LBE method is attained.Further-more,through the derivation we can directly show the con-nection between the LBE method and other newly developed gas kinetic methods ͓27–33͔.In this paper we will show that the lattice Boltzmann equation can be directly derived from the continuous Boltz-mann equation discretized in some special manner in both time and phase space.Our analysis shows that theoretically the lattice Boltzmann equation is independent of the lattice-gas automata.The lattice Boltzmann equation is a finite-difference form of the continuous Boltzmann equation.We provide a detailed account of an a priori derivation of the lattice Boltzmann equation from its continuous counter-part—the continuous Boltzmann equation.A general proce-dure to derive lattice Boltzmann models from the continuous Boltzmann equation is established.A number of LBE models in both two-and three-dimensional ͑2D,3D ͒space are de-rived to illustrate the procedure.The kinetic-model equation used in this paper is the BGK equation with a single relax-ation time ͓15–17͔.Although the BGK equation has its in-herent limitations and shortcomings,such as fixed Prandtl number,the equation can be generalized to remedy the short-comings ͓34͔.Therefore,it is sufficient to use the BGK equation for the purpose of studying hydrodynamics of simple fluids.This paper is organized as follows.In Sec.II we discuss the discretization of time for the Boltzmann BGK equation.In Sec.III we discuss the low Mach number ͑small velocity ͒expansion and the discretization of the phase space.In Sec.IV we derive the lattice Boltzmann 6-bit,7-bit,and 9-bit models in two-dimensional space and the 27-bit model in three-dimensional space.Section V concludes the paper.*Electronic address:xyh@†Electronic address:luo@PHYSICAL REVIEW E DECEMBER 1997VOLUME 56,NUMBER 6561063-651X/97/56͑6͒/6811͑7͒/$10.006811©1997The American Physical SocietyII.DISCRETIZATION OF THE BOLTZMANN EQUATIONIn the following analysis,we shall use the Boltzmann equation with the BGK,or single-relaxation-time,approxi-mation ͓15–17͔:ץfץt ϩ•ٌf ϭϪ1͑f Ϫg ͒,͑1͒where f ϵf (x ,,t )is the single-particle distribution function,is the microscopic velocity,is the relaxation time due to collision,and g is the Boltzmann-Maxwellian distribution function:g ϵ͑2RT ͒D /2exp ͩϪ͑Ϫu ͒22RT ͪ,͑2͒where R is the ideal gas constant,D is the dimension of the space,and ,u ,and T are the macroscopic density of mass,velocity,and temperature,respectively.The macroscopic variables,,u ,and T are the ͑microscopic velocity ͒mo-ments of the distribution function,f :ϭ͵f d ϭ͵gd ,͑3a ͒u ϭ͵f d ϭ͵gd ,͑3b ͒ϭ12͵͑Ϫu ͒2f d ϭ12͵͑Ϫu ͒2gd .͑3c ͒The energy can also be written in terms of temperature T :ϭD 02RT ϭD 02N A k BT ,͑4͒where D 0,N A ,and k B are the number of degrees of freedom of a particle,Avogadro’s number,and the Boltzmann con-stant,respectively.In Eqs.͑3͒,an assumption of Chapman-Enskog ͓16͔has been applied:͵h ͑͒f ͑x ,,t ͒d ϭ͵h ͑͒g ͑x ,,t ͒d ,͑5͒where h ()is a linear combination of collisional invariants ͑conserved quantities ͒h ͑͒ϭA ϩB •ϩC •.͑6͒In the above equation,A and C are arbitrary constants,and B is an arbitrary constant vector.A.Discretization of timeEquation ͑1͒can be formally rewritten in the form of an ordinary differential equation:d f dt ϩ1f ϭ1g ,͑7͒whered dt ϵץץtϩ•“is the time derivative along the characteristic line .The above equation can be formally integrated over a time step of ␦t :f ͑x ϩ␦t ,,t ϩ␦t ͒ϭ1e Ϫ␦t /͵␦te t Ј/g ͑x ϩt Ј,,t ϩt Ј͒dt Јϩe Ϫ␦t /f ͑x ,,t ͒.͑8͒Assuming that ␦t is small enough and g is smooth enoughlocally,the following approximation can be made:g ͑x ϩt Ј,,t ϩt Ј͒ϭͩ1Ϫt Ј␦tͪg ͑x ,,t ͒ϩt Ј␦tg ͑x ϩ␦t ,,t ϩ␦t ͒ϩO ͑␦t 2͒,0рt Јр␦t .͑9͒The leading terms neglected in the above approximation are of the order of O (␦t 2).With this approximation,Eq.͑8͒be-comesf ͑x ϩ␦t ,,t ϩ␦t ͒Ϫf ͑x ,,t ͒ϭ͑e Ϫ␦t /Ϫ1͓͒f ͑x ,,t ͒Ϫg ͑x ,,t ͔͒ϩͩ1ϩ␦t͑e Ϫ␦t /Ϫ1͒ͪϫ͓g ͑x ϩ␦t ,,t ϩ␦t ͒Ϫg ͑x ,,t ͔͒.͑10͒If we expand e Ϫ␦t /in its Taylor expansion and,further,neglect the terms of order O (␦t 2)or smaller on the right-hand side of Eq.͑10͒,then Eq.͑10͒becomes f ͑x ϩ␦t ,,t ϩ␦t ͒Ϫf ͑x ,,t ͒ϭϪ1͓f ͑x ,,t ͒Ϫg ͑x ,,t ͔͒,͑11͒where ϵ/␦t is the dimensionless relaxation time ͑in the unit of ␦t ͒.Therefore,Eq.͑11͒is accurate to the first order in ␦t .Equation ͑11͒is the evolution equation of the distribu-tion function f with discrete time.Although g is written as an explicit function of t ,the time dependence of g lies solely in the hydrodynamic variables ,u ,and T ͑the Chapman-Enskog ansatz ͓16͔͒,that is,g (x ,,t )ϭg (x ,;,u ,T ).Therefore,one must first compute ,u ,and T before constructing the equilibrium distribution function,g .Thus,the calculation of ,u ,and T becomes one of the most crucial steps in discretizing the Boltzmann equa-tion.B.Calculation of the hydrodynamic momentsIn order to numerically evaluate the hydrodynamic mo-ments of Eq.͑3͒,appropriate discretization in momentumspace must be accomplished.With appropriate discretiza-tion,integration in momentum space ͑with weight function g ͒can be approximated by quadrature up to a certain degree of accuracy,that is,681256XIAOYI HE AND LI-SHI LUO͵͑͒g͑x,,t͒dϭ͚␣W␣͑␣͒g͑x,␣,t͒,͑12͒where͑͒is a polynomial of,W␣is the weight coefficient of the quadrature,and␣is the discrete velocity set or the abscissas of the quadrature.Accordingly,the hydrodynamic moments of Eqs.͑3͒can be computed byϭ͚␣f␣ϭ͚␣g␣,͑13a͒uϭ͚␣␣f␣ϭ͚␣␣g␣,͑13b͒ϭ12͚␣͑␣Ϫu͒2f␣ϭ12͚␣͑␣Ϫu͒2g␣,͑13c͒wheref␣ϵf␣͑x,t͒ϵW␣f͑x,␣,t͒,͑14a͒g␣ϵg␣͑x,t͒ϵW␣g͑x,␣,t͒.͑14b͒It should also be noted that f␣͑or g␣)has the unit of f d͑or gd).III.DERIVATION OF THE LATTICE BOLTZMANN EQUATION AND ITS EQUILIBRIUM DISTRIBUTIONFUNCTIONThe lattice Boltzmann equation has the following ingredi-ents:͑1͒an evolution equation,in the form of Eq.͑11͒with discretized time and phase space of which configuration space is of a lattice structure and momentum space is re-duced to a small set of discrete momenta;͑2͒conservation constraints in the form of Eq.͑13͒;͑3͒a proper equilibrium distribution function which leads to the Navier-Stokes equa-tions.In what follows,the low Mach number expansion is first applied to the Boltzmann-Maxwellian distribution func-tion.Then quadrature to approximate the integration in the momentum spaceis discussed in detail for a variety of LBE models in both2D and3D space.Through the quadra-ture,the lattice structure,and the equilibrium distribution function of the LBE are constructed.A.Low-Mach-number approximationIn the lattice Boltzmann equation,the equilibrium distri-bution function is obtained by a truncated small velocity ex-pansion͑or low-Mach-number approximation͓͒14͔.The same can be done here:gϭ2RT D/2exp͑Ϫ2/2RT͒exp͕͑•u͒/RTϪu2/2RT͖ϭ͑2RT͒D/2exp͑Ϫ2/2RT͒ϫͭ1ϩ͑•u͒RTϩ͑•u͒22͑RT͒2Ϫu22RTͮϩO͑u3͒.͑15͒With the equilibrium distribution function g of the aboveform,Eq.͑11͒bears a strong resemblance to the latticeBoltzmann equation,and what remains to be accomplished isthe discretization of phase space.For convenience,the fol-lowing notation for the equilibrium distribution function withtruncated small velocity expansion shall be used in what fol-lows:f͑eq͒ϭ͑2RT͒D/2exp͑Ϫ2/2RT͒ϫͭ1ϩ͑•u͒RTϩ͑•u͒22͑RT͒2Ϫu22RTͮ.͑16͒Although f(eq)only retains the terms up to O(u2),it is trivialto maintain high-order terms of u in the above expansion ifnecessary.B.Discretization of phase spaceThere are two considerations in the discretization of phasespace.First of all,the discretization of momentum space iscoupled to that of configuration space such that a latticestructure is obtained.This is a special characteristic of thelattice Boltzmann equation.Second,the quadrature must beaccurate enough so that not only the conservation constraintsof Eq.͑13͒are preserved exactly,but also the necessary sym-metries required by the Navier-Stokes equations are retained.In deriving the Navier-Stokes equations from the Boltz-mann equation via the Chapman-Enskog analysis͓16,17͔,thefirst two order approximations of the distribution func-tion͑i.e.,f(eq)and f(1)͒must be considered.Therefore,giventhe equilibrium distribution function,f(eq)of Eq.͑16͒,thequadrature used to evaluate the hydrodynamic moments mustbe able to compute the following moments with respect tof(eq)exactly::1,i,ij,͑17a͒u:i,ij,ijk,͑17b͒T:ij,ijk,ijkl,͑17c͒wherei is the component ofin Cartesian coordinates.͑Wehave assumed that the particle only has the linear degree offreedom,i.e.,D0ϭD.͒Thus,to obtain the Navier-Stokesequations,one must be able to evaluate the moments of1,,...,6with respect to weight function exp(Ϫ2/2RT)ex-actly,owing to the small u expansion of g.For isothermalmodels,which are discussed in what follows,the momentsthat are to be evaluated are1,,...,5.Calculating the hydrodynamic moments of f(eq)is equiva-lent to evaluating the following integral in general:Iϭ͵͑͒f͑eq͒dϭ͑2RT͒D/2͵͑͒exp͑Ϫ2/2RT͒ϫͭ1ϩ͑•u͒RTϩ͑•u͒22͑RT͒2Ϫu22RTͮd,͑18͒where͑͒is a polynomial of.The above integral has thefollowing structure:566813THEORY OF THE LATTICE BOLTZMANN METHOD:...͵e Ϫx 2͑x ͒dx ,which can be calculated numerically with Gaussian-type quadrature ͓35͔.Our objective is to use quadrature to evalu-ate the hydrodynamic moments ͓,u ,and T ,given by Eq.͑3͔͒.Given proper discretization of phase space,we can evaluate the above integral with desirable accuracy.In the meantime,the lattice Boltzmann equation with appropriate equilibrium distribution function can also be derived.TTICE BOLTZMANN MODELS IN TWO-AND THREE-DIMENSIONAL SPACE A.Two-dimensional 6-bit and 7-bit triangular lattice modelThe 7-bit model is constructed on a two-dimensional (D ϭ2)triangular lattice space.The triangular lattice has the necessary rotational symmetry required by the hydrodynam-ics ͓14͔.The polar coordinates of space,͑,͒,are used here.For the sake of simplicity but without losing generality,assumingm ,n ͑͒ϭ͑ͱ2RT ͒m ϩn m ϩncos m sin n,͑19͒where ϭ/ͱthe integral in Eq.͑18͒becomesI ϭ͵m ,n ͑͒f͑eq ͒d ϭ͑ͱ2RT ͒m ϩn͵02͵ϱe Ϫ2m ϩn cos m sin n ϫͭ1ϩ2͑eˆ•u ͒ͱ2RTϩ2͑eˆ•u ͒2RT Ϫu 22RT ͮd d ,͑20͒where eˆϭ(cos ,sin ).To obtain the 7-bit lattice Boltzmann equation on the triangular lattice space,the angular variable must be discretized evenly into six sections in the interval ͓0,2͒,that is,␣ϭ(␣Ϫ1)/3,for ␣ϭ͕1,2,...,6͖.With the discretization of ,we have͵2cos m sin n d ϭͭ3͚␣ϭ16cos m ␣sin n ␣,͑m ϩn ͒even 0,͑m ϩn ͒odd͑21͒for (mϩn )рing the above result,we obtainI ϭΆ3͑ͱ2RT ͒m ϩn͚␣ϭ16cos m ␣sin n ␣ͭͩ1Ϫu 22RT ͪI m ϩn ϩ͑eˆ␣•u ͒2RTI m ϩn ϩ2ͮ,͑m ϩn ͒even3͑ͱ2RT ͒m ϩn͚␣ϭ16cos m ␣sin n ␣2͑eˆ␣•u ͒ͱ2RT I m ϩn ϩ1,͑m ϩn ͒odd,͑22͒where eˆ␣ϭ(cos ␣,sin ␣),and I m ϭ͵ϩϱ͑e Ϫ2͒m d ͑23͒is the m th moment with respect to the weight functione Ϫ2.Since the 7-bit model only has two speeds ͑i.e.,n ϭ2͒and one of them is fixed at 0,it is clear that the abscissas of the quadrature to evaluate I m should be 0ϭ0and 1ϭ␥Ϫ1,where ␥is a positive parameter to be adjusted later.The Radau-Gauss formula ͓35͔is a natural choice to evaluate the integral I m :I m ϭ00m ϩ͚j ϭ1nj j m .For the 7-bit model,n ϭ1,and the integrals needed to be evaluated are I 0,I 2,and I 4.͓I 1and I 3do not play any role because of the symmetry of the integral I ,as shown in Eq.͑22͒.͔Then,we have the following three equations:I 0ϭ0ϩ1ϭ1/2,͑24a ͒I 2ϭ1␥Ϫ2ϭ1/2,͑24b ͒I 4ϭ1␥Ϫ4ϭ1,͑24c ͒of which,the solutions are0ϭ1/4,͑25a ͒1ϭ1/4,͑25b ͒␥ϭ1/&.͑25c ͒Therefore,we haveI m ϭ14͑0m ϩ1m͒,m ϭ0,2,4.͑26͒Note that the above quadrature for I m is exact for m ϭ0,2,and 4.Consequently,the following equality is expected to be exact for (m ϩn )р5:681456XIAOYI HE AND LI-SHI LUOIϭ12͑ͱ2RT͒mϩn͚␣ϭ16cos m␣sin n␣ͭͩ1Ϫu22RTͪϫ͑0mϩnϩ1mϩn͒ϩ2͑eˆ␣•u͒ͱ2RT͑0mϩnϩ1ϩ1mϩnϩ1͒ϩ͑eˆ␣•u͒2RT͑0mϩnϩ2ϩ1mϩnϩ2͒ͮϭ2m,n͑0͒ͭ1Ϫu22RTͮϩ12͚␣ϭ16m,n͑␣͒ϫͭ1ϩ͑␣•u͒RTϩ͑␣•u͒22͑RT͒2Ϫu22RTͮ,whereʈ0ʈϭͱ2RT0ϭ0and␣ϭͱ2RT1eˆ␣ϭ2ͱRT eˆ␣.It becomes obvious that the equilibrium distribution function for the7-bit model isf␣͑eq͒ϭw␣ͭ1ϩ4͑e␣•u͒c2ϩ8͑e␣•u͒2c4Ϫ2u2c2ͮ,͑27͒where␣͕0,1,2,...,6͖,cϭ␦x␦t͑28͒is‘‘the speed of light’’in the system͑which is usually set to be unity in the literature͒,e␣ϭͭ͑0,0͒,␣ϭ0͑cos␣,sin␣͒c,␣ϭ͑␣Ϫ1͒/3,␣ϭ1,2,...,6͑29͒andw␣ϭͭ1/2,␣ϭ01/12,␣ϭ1,2, (6)͑30͒Note that the substitution of RTϭc s2ϭc2/4,where c s is the sound speed of the system,has been made in Eq.͑27͒,and RTϭc s2ϭc2/4is equivalent toʈ␣ʈϭc for␣ 0.The coef-ficient W␣defined in Eq.͑14͒is explicitly given byW␣ϭ͑2RT͒D/2e␣2/2RT w␣.͑31͒Similarly,the equilibrium distribution function for the6-bit LBE model,which is a degenerate case of the7-bit model,can be obtained easily by solving two equations for one abscissa and the weight coefficient of the Hermite-Gauss formula͓35͔with modification for the integral on half real axis͓36͔.The solutions are1ϭ12and␥ϭ1.Then,we havef␣͑eq͒ϭ6ͭ1ϩ2͑e␣•u͒c2ϩ4͑e␣•u͒2c4Ϫu2c2ͮ,͑32͒where␣͕1,2,...,6͖.The sound speed of the6-bit LBE model is c sϭc/&,and w␣ϭ16.B.Two-dimensional9-bit square lattice modelTo recover the9-bit LBE model on square lattice space, the Cartesian coordinate system is used and,accordingly,͑͒can be set tom,n͑͒ϭx my n,wherex andy are the x and y components of.The integral of the moments,defined by Eq.͑18͒,becomesIϭ͵m,n͑͒f͑eq͒dϭ͑ͱ2RT͒mϩnͭͩ1Ϫu22RTͪI m I n ϩ2͑u x I mϩ1I nϩu y I m I nϩ1͒ͱ2RTϩu x2I mϩ2I nϩ2u x u y I mϩ1I nϩ1ϩu y2I m I nϩ2RTͮ,͑33͒whereI mϭ͵ϪϱϩϱeϪ2m d,ϭ/ͱ2RTis the m th order moment of the weight function eϪ2on the real axis.Naturally,the third-order Hermite formula͓35͔is the optimal choice to evaluate I m for the purpose of deriving the9-bit LBE model:I mϭ͚jϭ13jj m.The three abscissas of the quadrature are1ϭϪͱ3/2,2ϭ0,3ϭͱ3/2,͑34͒and the corresponding weight coefficients are1ϭͱ/6,2ϭ2ͱ/3,3ϭͱ/6.͑35͒Then,the integral of the moment in Eq.͑33͒becomesIϭ͚i,jϭ13ij͑i,j͒ͭ1ϩ͑i,j•u͒RTϩ͑i,j•u͒22͑RT͒2Ϫu22RTͮ,͑36͒wherei,jϭ(i,j)ϭͱ2RT(i,j).Obviously,we can identify the equilibrium distribution function withf i,j͑eq͒ϭijͭ1ϩ͑i,j•u͒RTϩ͑i,j•u͒22͑RT͒2Ϫu22RTͮ.͑37͒Employing the notations of566815THEORY OF THE LATTICE BOLTZMANN METHOD:...e ␣ϭͭ͑0,0͒,␣ϭ0͑cos ␣,sin ␣͒c ,␣ϭ͑␣Ϫ1͒/2,␣ϭ1,2,3,4&͑cos ␣,sin ␣͒c ,␣ϭ͑␣Ϫ5͒/2ϩ/4,␣ϭ5,6,7,8͑38͒andw ␣ϭi jϭͭ4/9,i ϭj ϭ2,␣ϭ01/9,i ϭ1,j ϭ2,...,␣ϭ1,2,3,41/36,i ϭj ϭ1,...,␣ϭ5,6,7,8,͑39͒and with the substitution of RT ϭc s 2ϭc 2/3͑or ʈͱ2RT 1ʈϭͱ3RT ϭc ͒,we obtain the equilibrium distribution function of the 9-bit LBE model:f ␣͑eq ͒ϭw ␣ͭ1ϩ3͑e ␣•u ͒c 2ϩ9͑e ␣•u ͒22c 4Ϫ3u 22c2ͮ.͑40͒C.Three-dimensional 27-bit square lattice modelThe 27-bit LBE model in 3D square lattice space is a straightforward extension of the 2D 9-bit model.The abscis-sas i ,as well as the corresponding weight coefficients i ,of the quadrature to evaluate the moments remain the same.Therefore,I ϭ3/2͚i ,j ,k ϭ13i j k ͑i ,j ,k ͒ϫͭ1ϩ͑i ,j ,k •u ͒RT ϩ͑i ,j ,k •u ͒22͑RT ͒2Ϫu 22RTͮ,͑41͒where i ,j ,k ϭ(i ,j ,k )ϭͱ2RT (i ,j ,k ).With the fol-lowing notations similar to the 9-bitmodel,e ␣ϭΆ͑0,0,0͒,␣ϭ0͑Ϯ1,0,0͒c ,͑0,Ϯ1,0͒c ,͑0,0,Ϯ1͒c ,␣ϭ1,2,...,6͑Ϯ1,Ϯ1,0͒c ,͑Ϯ1,0,Ϯ1͒c ,͑0,Ϯ1,Ϯ1͒c ,␣ϭ7,8,...,18͑Ϯ1,Ϯ1,Ϯ1͒c ,␣ϭ19,20,...,26͑42͒and w ␣ϭi j k3/2ϭΆ8/27,i ϭj ϭk ϭ2,␣ϭ02/27,i ϭj ϭ2,k ϭ1,...,␣ϭ1,2,...,61/54,i ϭj ϭ1,k ϭ2,...,␣ϭ7,8,...,181/216,i ϭj ϭk ϭ1,...,␣ϭ19,20, (26)͑43͒we obtain the equilibrium distribution function of the 27-bit LBE model:f ␣͑eq ͒ϭw ␣ͭ1ϩ3͑e ␣•u ͒c 2ϩ9͑e ␣•u ͒22c 4Ϫ3u 22c2ͮ.͑44͒Note that the f ␣(eq)of the 27-bit model is exactly the same as that of the 9-bit model except for the values of the coeffi-cients w ␣.Also,the sound speed of the 27-bit model is identical to that of the 9-bit model because in both modelsRT ϭc s 2ϭc 2/3.V.CONCLUSIONRecently,a number of gas kinetic methods have been de-veloped for solving Navier-Stokes equations of simple andcomplex fluids ͓27–33͔.All these methods are based uponthe solution of the BGK equation ͑1͒.In particular,these gas kinetic methods are basically a finite-volume solution of Eq.͑8͒and they do not require discretization of the velocity space .While the lattice Boltzmann equation is in the in-compressible limit due to the small-Mach-number expansion,the gas kinetic methods are particularly suitable for shock capture.In conclusion,we have derived lattice Boltzmann models from the Boltzmann BGK equation,which is completely in-dependent of lattice-gas automata.The derivation directly connects the lattice Boltzmann equation to the Boltzmann equation;thus,the framework of the lattice Boltzmann equa-tion can rest on that of the Boltzmann equation and the rig-orous results of the Boltzmann equation can be extended to the lattice Boltzmann equation via this explicit 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