Statistical mechanics of the fluctuating lattice Boltzmann equation

PHYSICAL REVIEW E 76, 036704 2007Statistical mechanics of the fluctuating lattice Boltzmann equationBurkhard Dünweg and Ulf D. SchillerMax Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, GermanyAnthony J. C. LaddMax Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany and Chemical Engineering Department,University of Florida, Gainesville, Florida 32611-6005, USAReceived 11 July 2007; published 12 September 2007We propose a derivation of the fluctuating lattice Boltzmann equation that is consistent with both equilibriumstatistical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gasmodel, with each velocity direction occupied by many particles. We show that the most probable state of thismodel corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuationsabout this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rulesare described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivationof discrete Langevin equations for the fluctuating modes. It is shown that all nonconserved modes should bethermalized, as first pointed out by Adhikari et al. Europhys. Lett. 71, 473 2005; any other choice violatesthe condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuatinghydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamicallyconsistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful indeveloping new algorithms for thermal and multiphase flows.DOI: 10.1103/PhysRevE.76.036704PACS numbers: 47.11.Qr, 47.57.sI. INTRODUCTIONThe collisions conserve mass and momentum, henceur,t = j r,t/r,t. 4 stress tensorLattice Boltzmann LB methods 1,2 have become apopular tool for simulating hydrodynamics, particularly in i = i c i =0.i icomplex geometries. The underlying model is a regular lattice5of sites r, combined with a small set of velocity vectorsc i , which, within one time step h, connect a given site withThe algorithm thus satisfies important requirements for simulatinghydrodynamic flows—mass and momentum conserva-some of its neighbors. The set of velocities is chosen to becompatible with the symmetry of the lattice. The basic dynamicalvariables are real-valued populations n i ; in thetion, and locality—but lacks Galilean invariance due to thefinite number of velocities. Full rotational symmetry is alsopresent paper, we will consider n i as the mass density associatedwith the velocity c i . The LB algorithm is then de-lost, but by a suitable choice of velocity set, isotropic momentumtransport can be recovered on sufficiently large hydrodynamiclength scales. Nevertheless, the finite number ofscribed by the update rulevelocities always confines the method to flows with smalln i r + c i h,t + h = n i r,t = n i r,t + i n i r,t, 1Mach number u/c s ≪1. The speed of sound c s is of orderb/h, where b is the lattice spacing, or of order c i .Most of the LB literature deals with deterministic collisionrules, with i describing a linear relaxation of the dis-where n i denotes the complete set of populations. Then i r,t at each site are first rearranged in a “collision” step, tribution n i toward the local equilibrium 3,4:described by i , and then propagated along their respectivelinks. The hydrodynamic fields, mass densityn eq i ,u = a c i1+ u · c i2+ u · c i 24− u2cr,t = n i r,t,2s 2c s 2c2, 6siwhere a c i0 is the weight associated with the speed c i . Theand momentum densityviscosity of the LB fluid is determined by the choice of relaxationrates.However, to simulate Brownian motion of suspended particles,thermal fluctuations must be included. At the hydro-j r,t = n i r,tc i3idynamic level, this means adding uncorrelated noise to thefluid stress tensor 5. In Refs. 6–8 an analogous fluctuatingLB model was introduced by making i a stochasticare moments of the discrete velocity distribution n i r,t,while the fluid velocity is given byvariable, but in such a way that the noise was only applied tothe modes linear combinations of n i related to the viscous1539-3755/2007/763/03670410036704-1©2007 The American Physical Society

DÜNWEG, SCHILLER, AND LADDneq = n neq i c i c i ; ihere and denote Cartesian components and n i neq =n i−n i eq is the nonequilibrium distribution. Although this procedureis correct in the hydrodynamic limit 7,9, it providespoor thermalization on smaller length scales, as was firstobserved by Adhikari et al. 10. They introduced a thermalizationprocedure which applies to all nonconserved modes,with significantly improved numerical behavior at shortscales 10. The procedure was derived by considering afluctuating LB model, making explicit use of the transformationbetween the populations n i and the modes 11.The purpose of the present paper is to rederive the stochasticupdating rule of Ref. 10 from a generalized latticegasmodel. The difference in our formulation lies in the introductionof an ensemble of population densities at eachgrid point, so that a fluctuating LB simulation is a singlerealization of this ensemble. There follows naturally a probabilitydistribution, Pn i , for the set of populations n i ata position r and time t. The equilibrium distribution at asingle site can be derived by maximizing P subject to theconstraints of fixed mass and momentum densities, and j .This distribution agrees with the standard equilibrium distributionfor LB models Eq. 6 up to terms of order u 2 .Asimilar procedure has been followed in deriving H-theoremsfor LB models 12–14, but these papers were not concernedwith fluctuations.A coarse-graining of the microscopic collision operatorleads to a Langevin description for the nonconserved degreesof freedom. However, these stochastic collisions may also beviewed as a Monte Carlo process 15, satisfying the principleof detailed balance governed by Pn i . The procedureof Refs. 7,9 can be shown to violate detailed balance, whilethe improved version of Ref. 10 satisfies it.In summary, our goal is to reconnect the lattice Boltzmannequation with its lattice gas origins, and thus to establisha firm statistical mechanical foundation for stochasticLB simulations, in addition to the usual connection to fluctuatinghydrodynamics 7,10. We believe this provides acomparable theoretical framework to that already availablefor other stochastic simulation methods, such as dissipativeparticle dynamics 16 and stochastic rotation dynamics 17.This formulation also offers the possibility for future modificationsand generalizations, for example, to thermal flows18, models with nonideal equations of state 19,20, ormulticomponent mixtures 21.The paper is organized as follows. In Sec. II we describethe underlying lattice-gas model, derive the probability distributionPn i , and show that the most probable value forn i is equivalent to Eq. 6. In Sec. III we consider smallfluctuations around the equilibrium distribution. We showthey are approximately Gaussian distributed, with the levelof thermal fluctuations governed by the degree of coarsegraining: a given amount of mass on a lattice site can bedistributed between many particles, in which case the fluctuationsare small, or between few, in which case they arelarge. In this way we can adjust the level of fluctuationswhile keeping the temperature fixed. In Sec. IV we construct7PHYSICAL REVIEW E 76, 036704 2007a stochastic collision operator such that detailed balance issatisfied. From this, we derive the random stresses at an individualsite. In Sec. V we apply the Chapman-Enskog procedure9 to the algorithm in order to find the behavior onthe hydrodynamic scale; the deterministic and stochasticterms are here treated on an equal basis 22. We then findthat, on the macroscopic scale, the procedure yields exactlythe stress correlations given by Landau and Lifshitz 5. SectionVI discusses how to choose parameters for a coupledparticle-fluid system. Section VII summarizes our conclusions.II. SINGLE-SITE PROBABILITY DISTRIBUTIONHistorically, the lattice-Boltzmann model 3,23 was developedfrom earlier work on lattice-gas LG models24,25, in which each velocity direction was occupied by atmost one particle. Here, we imagine a generalized lattice-gasmodel GLG where each velocity direction can be occupiedby many particles. Each particle has the same mass, but differentvelocity directions may have different mean populations,even in a fluid at rest. The microscopic state of thesystem at any given site is specified by a set of integers i giving the occupancies of each direction. Then the update ofthe GLG is analogous to the standard LG or LB models, butwith i an integer as opposed to a Boolean or real variable: i r + c i h,t + h = i r,t = i r,t + ˜ i i r,t,where ˜ i operates on i to compute the change in population i − i . While collisions may be both deterministic andmicroscopically reversible, we shall assume only that thecollision operator satisfies detailed balance.Without considering the collision rules in detail, we constructan equilibrium distribution from the following thoughtexperiment. Consider a velocity direction, i, at a particularsite, r. Particles are drawn randomly from a large reservoirand assigned to the site r with velocity c i ; the number ofparticles in the reservoir is assumed to be much larger thanthe number of particles selected, i r. Under these circumstances i r follows a Poisson distribution,P i = ¯i i i ! e−¯i,9with a mean number of particles ¯i, and a variance 2 i − i 2 = ¯i.10Let m p be the mass of a particle and =m p /b d , where b isthe lattice spacing and d is the spatial dimension. Then n i= i , andn 2 i − n i 2 = n i .11The fluctuations in mass density at a site are controlled bythe mass of an LB particle: small m p means that the mass isdistributed onto many particles, and therefore fluctuationsare small. For fixed m p , and therefore the level of fluctuationsbecomes large as b decreases. This is natural, sincea fine spatial resolution means fewer particles per cell, andlarger fluctuations relative to the mean.8036704-2

STATISTICAL MECHANICS OF THE FLUCTUATING …If we now imagine sampling each velocity with an independentreservoir, but taking only those sets of populationswhich produce specific values for the total mass and momentum,the probability density for the occupation numbers isexcept for normalizationP i i i ¯i i ! e−¯ii i − i c i − j .i12Using Stirling’s approximation for i ≫1, we can write thedistribution in terms of the entropy associated with the occupationnumbers,S i =− i ln i − i − i ln ¯i + ¯i, 13iand the constraints:P i expS i i i − i c i − j .i14The equilibrium distribution, eq i , can be found by maximizingS, treating i as a continuous variable, and taking intoaccount the mass and momentum constraints via Lagrangemultipliers, and , j respectively:S i+ + j · c i =0, i − =0,i1516 i c i − j =0.17iIt should be noted that this procedure is closely related to thedetermination of an entropy function for the LB equation13. Equation 15 can be solved to give the equilibriumpopulations in terms of the Lagrange multipliers, i eq = ¯i exp + j · c i,18which are then determined from the constraints, Eqs. 16and 17, substituting i eq for i .The mean populations in the absence of constraints, ¯i,can be expressed in terms of the mean number of particles ata site,¯i = ¯a c i,19where ¯ = i ¯i. The symmetry of the lattice constrains theweights, a c i, to be dependent on the speed of the particle, butnot its direction. Thus for a lattice with cubic symmetry, a c i =1,20i a c ic i =0,i21PHYSICAL REVIEW E 76, 036704 2007 a c ic i c i = 2 ,i22 a c ic i c i c i =0,23iwhere is the Kronecker delta, and 2 is a constant withunits b/h 2 .A solution of the nonlinear equations 16–18 requiresan iterative numerical procedure, but it is more practical toseek an approximate expression for the equilibrium distributionin the limit that j ·c i is small 14. To second order in , j the mass and momentum constraints yield: 2 2¯e j 1+ = , 242¯e 2 j = u .25Inserting these results into Eq. 18, we find the equilibriumdistribution can be written in the form of Eq. 6,n eq i = a c i1+ u · c i+ u · c i 22− u226 2 2 22 2.For the sake of completeness, we now briefly mention theprocedure 4,9 to determine the weights a c i such that the LBmodel is consistent with hydrodynamics. This requires thatthe second moment of the equilibrium distribution, eq = n eq i c i c i ,27ishould equal the Euler stress p +u u , with the pressuregiven by the ideal gas equation of state, p=k B T/m p , wherek B is Boltzmann’s constant and T is the absolute temperature.For an isothermal gas of particles of mass m p , k B T=m p c 2 s ,and therefore the equation of state is also given by p=c 2 s ,with c s the speed of sound.To evaluate eq we require the fourth moment of a c i,which from cubic symmetry must be of the form a c ic i c i c i c i = 4 + 4 + + ,i28where is unity if all four indexes are the same and zerootherwise; 4 and 4 have units of b/h 4 . Consistency betweenEq. 27 and the Euler stress requires that 2 = c s 2 = k B T/m p , 4 = 2 2 ,2930 4 =0.31These conditions, together with the normalization condition, i a c i=1, determine the weights uniquely for a model withthree different speeds. For example, for the D3Q19 model4 19 velocities on a three-dimensional simple cubic lattice,a 0 =1/3 for the stationary particles, a 1 =1/18 for the036704-3

DÜNWEG, SCHILLER, AND LADDsix nearest-neighbor directions, and a 2 =1/36 for the 12next-nearest-neighbor directions: the sound speed is then c s2=1/3b/h 2 . In the D2Q9 model 4 nine velocities on atwo-dimensional square lattice the weights are a 0 =4/9, a 1=1/9, and a 2 =1/36; the sound speed is again c s 2 =1/3b/h 2 .III. SINGLE-SITE FLUCTUATIONSWe now consider the distribution of small fluctuations inthe mass densities associated with each velocity direction,n i neq =n i −n i eq . Using the results of the Appendix to incorporatethe constraints, and converting from fluctuations in i tofluctuations in n i ,Pn i neq exp− in i neq 22n ieqin neq i c i n neq i .i32The variance in the fluctuations depends on direction, but,since n neq i is already a small quantity in comparison with n eq i ,we will approximate the variance by the low-velocity limit,lim n eq i = a c i.33u→0The velocity dependence of the fluctuations in the GLGmodel is a consequence of the broken Galilean invariance,which is only entirely restored in the limit u→0. However,the approximation in Eq. 33 makes no difference to themacroscopic dynamics of the fluctuating LB model Eqs.82–85, since the stress fluctuations are already secondorderin the Chapman-Enskog expansion.We now introduce normalized fluctuations x i , via the definitionn neq i = ac i x i ,and transform Eq. 32 to the simplified expressionx i2i ac i x i ac i c i x i .i34Px i exp− 1 2 i35Equations 6, 34, and 35 define the statistics of our fluctuatingLB model. Equation 35 is entirely consistent withthe proposed GLG model, within the approximation expressedin Eq. 33.The LB collision operator can be conveniently representedin terms of modes, which are linear combinations ofthe mass densities, n i 11, with basis vectors constructedfrom orthogonal polynomials in the velocity set c i . There ismore than one possible choice for these basis vectors 26,and we use the “weighted” set 10,26, for which only thehydrodynamic modes mass density, momentum density, andstress have a projection on the equilibrium distribution. Wethen write the nonequilibrium distribution as an orthonormaltransformation of the scaled variables, x i :m k = ê ki x i ,i36x i = ê ki m k ,37kwhere m k is the amplitude of the kth mode, and the basisvectors satisfy the orthonormality conditions ê ki ê li = kl .38iIt should be noted that the basis vectors ê ki are different fromthe e ki defined in Ref. 26, since there the transformationwas for unscaled variables, n i , rather than the scaled variables,x i , used here. The essential physics of the transformationis, however, unchanged; the present expressions are justa reparametrization. The basis vectors ê ki are related to theweighted basis vectors used in Ref. 26:ê ki = ac iw ke ki ,39where w k is the length of the kth basis vector,w k = a c ie 2 ki . 40iThe hydrodynamic modes, mass density, momentum density,and stress, can be written in a model-independent form.Explicitly,ê 0i = ac i41for the mass mode, andê i = ac i2c c i,s =1, ...,d 42for the momentum modes. Note that in our formalism m 0 andm =1,...,d are zero.In addition to the conserved modes, there are dd+1/2viscous modes: one bulk mode, d−1 shear modes involvingdiagonal elements of the c i c i tensor, and dd−1/2 offdiagonalelements. The bulk stress mode is given by2d c 2 i − dc 2 s , 43where orthogonality to the mass mode is assured by Schmidtorthogonalization. There is a shear mode of the formê d+1,i = 1 c s2 ac iê d+2,i = 1 c s2 ac i2dd −1 dc 2 ix − c 2 i , 44and d−2 shear modes of the form d2 a c iê d+3,i =22c c 2 iy − c 2 iz , 45stogether with additional modes formed by cyclic permutationsof the Cartesian indexes. The dd−1/2 off-diagonalshear stresses are of the form a c iê 2d+1,i =2c c ixc iystogether with cyclic permutations.PHYSICAL REVIEW E 76, 036704 200746036704-4

STATISTICAL MECHANICS OF THE FLUCTUATING …TABLE I. Basis vectors of the D2Q9 model. Each row correspondsto a different basis vector, with the actual polynomial in ĉ ishown in the second column; the components of ĉ i =c i h/b arenormalized to unity. The orthonormal basis vectors ê ki can be obtainedfrom the table using Eq. 39, ê ki = ac i /w k e ki : the normalizingfactor for each basis vector is in the third column.k e ki w k0 1 11 ĉ ix 1/32 ĉ iy 1/33 3ĉ 2 i −2 44 2ĉ 2ix −ĉ i 4/95 ĉ ix ĉ iy 1/96 3ĉ 2 i −4ĉ ix 2/37 3ĉ 2 i −4ĉ iy 2/38 9ĉ 4 i −15ĉ 2 i +2 16All these vectors are mutually orthogonal. Further orthogonalvectors, whose span are the kinetic or “ghost”10 modes, may be constructed in terms of higher-orderpolynomials of c i 11; these are model dependent. Completesets of basis vectors 26 for the D2Q9 and D3Q19 LB models4 are given in Tables I and II, respectively.TABLE II. Basis vectors of the D3Q19 model. Each row correspondsto a different basis vector, with the actual polynomial in ĉ ishown in the second column; the components of ĉ i =c i h/b arenormalized to unity. The orthonormal basis vectors ê ki can be obtainedfrom the table using Eq. 39, ê ki = ac i /w k e ki : the normalizingfactor for each basis vector is in the third column.k e ki w k0 1 11 ĉ ix 1/32 ĉ iy 1/33 ĉ iz 1/34 ĉ 2 i −1 2/35 3ĉ 2ix −ĉ i 4/36 ĉ 2iy −ĉ iz4/97 ĉ ix ĉ iy 1/98 ĉ iy ĉ iz 1/99 ĉ iz ĉ ix 1/910 3ĉ 2 i −5ĉ ix 2/311 3ĉ 2 i −5ĉ iy 2/312 3ĉ 2 i −5ĉ iz 2/313 ĉ iy −ĉ iz ĉ ix 2/914 ĉ iz −ĉ ix ĉ iy 2/915 ĉ ix −ĉ iy ĉ iz 2/916 3ĉ 4 i −6ĉ 2 i +1 217 2ĉ 2 i −33ĉ ix −ĉ 2 i 4/318 2ĉ 2 i −3ĉ iy −ĉ iz 4/9PHYSICAL REVIEW E 76, 036704 2007Equation 35 can be rewritten using Eqs. 37 and 38 togive the nonequilibrium probability distribution of the modesm k ,Pm k exp− 1 2 k47There is no contribution to P from the conserved modes.m k2 m i exp− 1 mid 2 k2.kdIV. STOCHASTIC COLLISIONS AS A MONTE CARLOPROCESSIn this section we construct a stochastic collision operator,viewed as a Monte Carlo process, and consider the localdynamics at the level of a single lattice site. In the nextsection Sec. V we will consider the global dynamics,through a Chapman-Enskog expansion. A deterministic collisionoperator at the microscopic level is quite complicatedto construct, even for the simplest three-dimensional LGmodels 27, and cannot be easily extended to the largernumber of particles in Eq. 8. Collision rules are mucheasier to construct at the Boltzmann level 3; the stochasticupdate from precollision to postcollision populations, n i→n i , is facilitated by making the transition between modes,m k →m k , since each degree of freedom is then independent.Denoting a transition probability between the pre- and postcollisionstates of a particular mode m by m→m , thecondition of detailed balance, governed by the distribution inEq. 47, readsm → m m → m = exp− m2 /2exp− m 2 /2 . 48A simulation at the hydrodynamic level does not need tosatisfy this condition, and typically does not, but it is essentialfor a proper thermal equilibrium of the LB fluid.There are many possible realizations of Eq. 48: onewell-known example is the Metropolis method, involving atrial move followed by a stochastic acceptance or rejectionstep to enforce detailed balance. Here we consider the linearrelaxation model typically used in LB simulations, balancedby Gaussian noise:m = m + r,49where is related to an eigenvalue of the linearized collisionoperator, =1+ see Eq. 8 of Ref. 26, and r is a Gaussianrandom variable with zero mean and unit variance. Thedissipation parameter is restricted by the linear stabilitylimit, 1, with the case 0 corresponding to “overrelaxation”.Equation 49 has the technical advantage of beingrejection-free, and the conceptual advantage of enablingan analytic calculation to be made at the Chapman-Enskoglevel see Sec. V.The parameter must be adjusted to satisfy detailed balance,Eq. 48, using the relation Eq. 49 r= −1 m −m. Since the transition probability for m→m is identicalto the probability for generating the value of r that gives m from m,036704-5

DÜNWEG, SCHILLER, AND LADDm → m = 2 2 −1/2 exp− m − m 2 /2 2 . 50There is a similar expression for the reverse transition, m →m, with m and m interchanged. From Eq. 48, we thenfind that detailed balance is satisfied for = 1− 2 1/2 .51Thus the case =1 corresponds to a conserved mode, while=0 corresponds to m being entirely random, with nomemory of its previous value.Each mode, m k , in the LB model is assigned its own relaxationrate k , subject to the constraints of symmetry andconservation laws; the conserved modes kd require that k =1. For the bulk stress we choose a value b , and for thed+2d−1/2 shear stresses a single value s . In Refs.7,9,10 the kinetic modes were updated with k =0, but it ispossible to achieve more accurate boundary conditions witha proper tuning of the kinetic eigenvalues 26,28. Equation51 ensures that detailed balance is satisfied for all choicesof k . A purely deterministic LB model is obtained by setting k =0 for all modes; physically, this corresponds to the limitof m p →0, or ¯i→.The original derivation of the fluctuating LB model 7,9is obtained by setting k = k =0 for all the kinetic modes, butchoosing the variance of the stresses according to Eq. 51.The kinetic modes are projected out at every time step bythis collision rule, and m k →m k =0=1. However, there isno route back to the precollisional state, m k =0→m k =0,and detailed balance Eq. 48 is clearly violated. Nevertheless,this model still yields the correct fluctuating hydrodynamicsin the limit of large length scales 9, as is shown bythe analysis in Sec. V. Treating all the nonconserved modeson an equal basis 10 satisfies detailed balance on all scales,and is entirely equivalent to Eqs. 48–51.As a general rule, proper thermalization requires as manyrandom variables as there are degrees of freedom in the systemnot counting the conserved variables. However, in thespecial case where k =0 for all kinetic modes, the deterministicLB model can be propagated forward in time from themass, momentum, and stress at each lattice site. Thus it istempting to conclude that in this instance only the stressmodes should be thermalized 6–8, since the n i play therole of auxiliary variables. However, this is incorrect; if both k =0 and k =0 for any mode, it is impossible to reconstructthe reverse trajectory and the system will not reach thermalequilibrium. The number of degrees of freedom is only welldefined,therefore, when all k 0. Situations where some k =0 should be treated as a limit of finite k , and continuitytells us that the number of random variables should be thesame.The update rule in Eq. 49, with 0, is an exact solutionof a continuous Langevin equation 29,30,ddt m =−m + 52with t=0 and tt=2t−t. Integrating Eq.52 from t=0 to t=h i.e., one LB time step gives Eq. 49,with =exp−h. The standard first-order Euler approximationto Eq. 52 corresponds to =1−h, and is only validfor small h. By contrast Eq. 49 does not impose any restrictionon the time step.For the Chapman-Enskog analysis in Sec. V, we will needthe collisional update of the nonequilibrium stress tensor, neq = i n neq i c i c i ; the equilibrium part of the stress is unchangedby the collision process. We first decompose neqinto a multiple of the unit tensor bulk stress, and a tracelesspart shear stresses, denoted by an overbar: neq = ¯ neq + 1 d neq ,53where we have used the Einstein summation convention forthe Cartesian components. The change in the nonequilibriumstress tensor at a lattice site, due to collisions, can be determinedfrom Eqs. 49 and 51,¯ neq = s ¯ neq + R¯ ,54 neq = b neq + R .55The variables R are Gaussian random variables with zeromean; in addition R¯ is traceless. The covariance matrixR R is determined by the variances of the stochasticstress modes. The calculation can be simplified by observingthat the matrix is a fourth rank tensor and is therefore isotropicby the symmetries of the LB model,R R = R 1 + + R 2 . 56The unknown constants, R 1 and R 2 , can be determined fromspecial cases. For example, in the D3Q19 model defined inTable II, neq xy = 2 cs m 7 ,57and therefore, from Eqs. 49 and 54,R 2 xy = c 4 s 1− 2 s = R 1 , 58where the final equality follows from Eq. 56. Similarly, neq yy − neq zz =2 2 cs m 6 ,59and thereforeR yy − R zz 2 =4c 4 s 1− 2 s =4R 1 . 60This result is consistent with Eq. 56, which demonstratesthat the fluctuating stresses are indeed isotropic. Finally, thefluctuations in the trace, R , are related to b :R R =6c 4 s 1− 2 b =6R 1 +9R 2 . 61The general expression for the covariance in the randomstresses isR R c s4PHYSICAL REVIEW E 76, 036704 2007= 1− 2 s + + 2 d 2 s − 2 b .62This covariance matrix is different from the global fluctuationsin stress, which are superposed onto the hydrodynamicmodes Sec. V.036704-6

STATISTICAL MECHANICS OF THE FLUCTUATING …V. CHAPMAN-ENSKOG EXPANSIONIn order to determine the behavior on hydrodynamiclength and time scales, we apply the Chapman-Enskogmethod to the stochastic dynamics of the fluctuating LBmodel. We modify the derivation of Ref. 9 to include thermalfluctuations: for an alternative procedure, see Ref. 31.Here, the expansion parameter is used to separate the latticescale, r, from the hydrodynamic scale, r 1 =r. Thus = 1 , with the notation =/r .Since the collision operator is local in space and time, thenonequilibrium distribution is also taken to be of order :n i neq =n i 1 , with n i 1 of order unity in the expansion,n i = n i eq + n i 1 .63We use the usual multiple time scale expansion 32, t= t1 + 2 t2 , to separate the convective t 1 and diffusive t 2 relaxation processes. The left-hand side of Eq. 1 is expandedabout r,t in a Taylor series with respect to h: to firstorder in , t1 + c i 1 n i eq = h −1 i .64Multiplying this equation by one of the basis vectors andsumming over all the directions, we obtain the equations forthe dynamics of the fluctuating LB model on the t 1 timescale, t1 in i eq e ki + 1 in eq i c i e ki = h −1 i e ki .i65Note that we use the e ki basis vectors here, in conjunctionwith n i eq and n i neq , not the normalized basis vectors ê ki ,which are for the x i .When applied to the conserved degrees of freedom, kd, Eq. 65 leads to the inviscid fluid equations: t1 + 1 j =0, t1 j + 1 c s 2 + u u =0.6667The equilibrium distribution contains polynomials in c i up tosecond order, and is thus automatically orthogonal to thekinetic modes, which are made up of third-order and fourthorderpolynomials in c i . Since the equilibrium distributionhas no projection on the kinetic modes, the time-derivative inEq. 65 vanishes identically for kd 2 +3d/2. However,the gradient term in Eq. 65 includes an additional factor ofc i : thus third-order polynomials survive, making small equilibriumcontributions of order u 2 to the dynamics.At order 2 , the Boltzmann equation is t2 n i eq + t1 n i neq + c i 1 n i neq + h 2 t 1 t1 + c i 1 n ieq+ h 2 1 t1 + c i 1 n i eq c i =0, 71where the terms have been grouped to suggest the most expedientmeans of calculation. Since only the hydrodynamicmodes survive to the t 2 time scale, we consider just themodes up to k=d. It follows immediately from Eq. 71 andthe conservations laws Eqs. 66 and 67 that the fluid isincompressible on the t 2 time scale, t2 =0.72The momentum equation can be written as t2 j + 1 neq + 1 2 − =0, 73where we can use Eq. 69 to substitute the velocity gradientsfor − . This is the usual lattice correction to the viscousmomentum flux 9. The kinetic modes make no contributionto the hydrodynamic variables, and j , at longtimes.The nonequilibrium stress can be calculated by combiningthe stress update rule, Eqs. 54 and 55, with Eq. 69. Forexample, from Eq. 54,and from Eq. 69PHYSICAL REVIEW E 76, 036704 2007 xy − eq xy = s xy − eq xy + R xy ,74Similarly, for the stress modes, dkd 2 +3d/2, we find: t1 c s 2 + u u + c s 2 1 j + 1 j + 1 j Eliminating xy xy − xy = hc 2 s 1 x u y + 1 y u x .from these two equations,75= h −1 − . 68Evaluating the time derivatives in Eq. 68 gives a simplifiedexpression for the nonequilibrium stress, apart from smallterms of order u 3 9, − = hc 2 s 1 u + 1 u . 69Thus, on the t 1 time scale, the viscous stresses fluctuatearound a mean value that is slaved to the velocity gradient, 1 u + 1 u .The kinetic modes fluctuate around zero on the t 1 timescale, with at most a small correction of order u 2 :m k − m k = Ou 2 .701− s neq xy + hc 2 s 1 x u y + 1 y u x = R xy . 76In the general case, we again decompose the stress into itstrace and traceless parts, neq =− hc 2s1− s 1 u + 1 u − 2 d 1 u − hc 2s1− b 2 d 1 u − Q ,77where the random stress tensor on the macroscopic level is1 1 1Q =− R¯ −1− s 1− b d R . 78Equation 73 can now be rewritten in terms of the viscousand fluctuating stresses036704-7

DÜNWEG, SCHILLER, AND LADD t2 j = 1 Q + 1 1 u + 1 u + − 2 d 1 u .79The deterministic part of the stress tensor has the desiredNewtonian form 5, with the usual expressions 9 for theshear viscosity and bulk viscosity :2 = hc s 1+ s,2 1− s280 = hc s 1+ b.81d 1− bCombining the momentum transport on the t 1 and t 2 timescales we obtain the equations of fluctuating hydrodynamics5, t u + u u + c s 2 t + u =0,= Q + u + u + − 2 d u ,8283with random stresses Q . These are Gaussian variables withzero mean and a covariance matrix that can be calculatedfrom the analogous result on the microscopic level, Eq. 62:Q Q = 2m 2pc sb d + + − 2 .h d84This is the discrete analog of the covariance matrix of thefluctuating stresses given by Landau and Lifshitz 5. Thedelta functions in space and time that appear in the continuumtheory are here converted into factors b −d and h −1 .Thus the stress fluctuations depend on the discretization ofspace and time. Equation 84 can be made consistent withthe amplitude of fluctuating stresses in Ref. 5 by choosingk B T = m p c 2 s .85This is exactly the relation expected from the equation ofstate of an isothermal, ideal gas. In other words, our resultsare simultaneously consistent with macroscopic thermodynamicsand fluctuating hydrodynamics.VI. CHOICE OF PARAMETERSThe fluctuating LB model has been used to simulate arange of soft-matter physics, such as colloidal suspensions6 and polymer solutions 33,34. In such cases it is necessaryto match the LB parameters to the mass density, temperature,and viscosity of the molecular system. In additionthere are two parameters that control the accuracy of the LBsimulation without affecting the physics being simulated;namely the grid spacing, b, and the time step, h. The gridspacing must be related to the characteristic length scale ofPHYSICAL REVIEW E 76, 036704 2007the physical system. For example, in coupling the LB fluid tosoft matter, like polymer chains, colloidal particles, or membranes,the length would be the size of the object. For flow incomplex geometries, it would be the channel width, while forsimulations of turbulent flow, it would be the Kolmogorovlength. This length scale, plus the desired spatial resolution,fixes the lattice spacing b in absolute units. Choosing a suitabletime step then automatically sets the speed of soundc s =ĉ s b/h, where ĉ s is a dimensionless property of the LBmodel; for example, in the D2Q9 and D3Q19 models ĉ s= 1/3. Typically, the sound speed will be unrealisticallysmall for a dense liquid; however, this is not crucial since theLB method only runs in flow regimes where density fluctuationsare negligible.Once the length and time scales have been set, we canmatch the shear and bulk viscosities to the molecular system.Equations 80 and 81 suggest using b and h to computenondimensional viscosities from the reference values,ˆ =ˆ =2hc = hs b 2 2ĉ ,s2hc = hs b 2 2ĉ .sThe parameters s and b are then set by ˆ and ˆ: s =2ˆ −12ˆ +1 ,868788dˆ −1 b = . 89dˆ +1Small time steps therefore imply that the LB simulation isrun in the over-relaxation regime. The relaxation rates of thekinetic modes can be chosen for convenience k =0 or toimprove the accuracy of the boundary conditions 26,28.The remaining LB parameter is the particle mass, m p ,which must be fixed, for a given b and h, so that the fluctuationsin the LB fluid are consistent with the temperature, Eq.85. The parameter =m p /b d determines the variance in thefluctuations Eq. 32, = k BTh 2ĉ 2 s b d+2 ,90from which we see that too fine a grid or too large a time stepwill cause an unacceptably high noise level. A stable simulationwill require that the time step scales as hb d/2+1 orb 5/2 in three dimensions, which is slightly more stringentthan the usual diffusive scaling, hb 2 .VII. CONCLUSIONSFor models of the D3Q19 type, our analysis has shownthat a fluctuating LB equation can be developed from statisticalmechanical considerations. We have shown that thefluctuations are governed by the degree of coarse graining,and that the relevant parameter is the mass of the LB particle,036704-8

STATISTICAL MECHANICS OF THE FLUCTUATING …m p , which, for a given temperature, is determined by thediscretization of space, b, and time, h. The temperature appearingin the equation of state is identical to that whichcontrols the fluctuations, as it should be.The beauty of the present approach is that one only needsto take care that the statistical properties are correct at the LBlevel. The correct fluctuation-dissipation theorem at theNavier-Stokes level is then an automatic consequence of themicroscopic physics. We have introduced the principle ofdetailed balance into the LB model, which is the microscopiccounterpart of the fluctuation-dissipation theorem used inprevious work 7,9,10. We have demonstrated that all nonconservedmodes must be thermalized 10 in order to satisfydetailed balance; early implementations of the fluctuating LBmodel 7,9 did not preserve detailed balance. On the otherhand, all these methods have been shown to be correct in thehydrodynamic limit. Only the stress fluctuations survive tolong times, and fluctuations in the kinetic modes becomeasymptotically irrelevant. Nevertheless, practical simulationsrarely probe the asymptotic limit, and then a procedurewhich is statistically correct on all length scales is clearlypreferable.ACKNOWLEDGMENTSWe thank R. Adhikari, M. E. Cates, and A. J. Wagner forvery stimulating discussions on the subject. U.D.S. thanksthe Volkswagen Foundation for support within the frameworkof the program “New conceptual approaches to modelingand simulation of complex systems.” A.J.C.L. thanks theAlexander von Humboldt Foundation for supporting his stayat the Max Planck Institute for Polymer Research.APPENDIX: CONSTRAINED DISTRIBUTIONSLet us consider a constrained probability distributionP i of the following general form:P i expS i j i ij − q j ,iA1where S is a function of i , and ij and q j are constants.The constraints can be eliminated by making use of the Fourierrepresentation of the delta function, x=2 −1 expikxdk:whereP i dk j expŜ i ,k j , A2jŜ i ,k j = S i + ijk j i ij − q j .iA3Now, let i 0 , k j 0 denote the saddle point of Ŝ, which can befound by solving the system of equations:Ŝ=0⇔ S i i+ ijk j ij =0,A4Ŝ=0⇔k j i ij − q j =0. A5iThe solution 0 isatisfies the constraints in Eq. A1 and isidentical to the one obtained by maximizing S, taking intoaccount the constraints via Lagrange multipliers, ik j .The second-order Taylor expansion of Ŝ around the saddlepoint isŜ i ,k j = Ŝ i 0 ,k j 0 + ilwhere we have introduced the abbreviations il = 1 2 2 il i l + i ij i k j ,ijS, i l0 i i = i − i 0 ,k j = k j − k j 0 .A6A7A8A9The probability distribution for i is then approximatelyGaussian.The expansion of Ŝ is now inserted into Eq. A2. Ignoringthe constant term, which can be absorbed in the normalizationof P, and transforming to the new variables k j ,wefind ij i P i jPHYSICAL REVIEW E 76, 036704 2007 dk j exp ik jiexp il i l .ilReintroducing delta functions, we obtain the final resultP i exp il i l iljA10 ij i . A11iAssuming the coefficients ij form a negative-definite matrixotherwise the Gaussian approximation would not makesense, the saddle point is a maximum in P.1 S. Succi, The Lattice Boltzmann Equation for Fluid Dynamicsand Beyond Oxford University Press, New York, 2001.2 R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 1451992.3 F. Higuera, S. Succi, and R. Benzi, Europhys. Lett. 9, 3451989.4 Y. H. Qian, D. d’Humières, and P. Lallemand, Europhys. Lett.17, 479 1992.036704-9

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