Progress in the Understanding of the Fluctuating Lattice Boltzmann Equation

Transcription

1 Progress n the Understandng of the Fluctuatng Lattce Boltzmann Equaton Burkhard Dünweg a, Ulf D. Schller a, Anthony J. C. Ladd b a Max Planck Insttute for Polymer Research, Ackermannweg 10, D Manz, Germany b Chemcal Engneerng Dept., Unversty of Florda, Ganesvlle, FL , USA Abstract We gve a bref account of the development of methods to nclude thermal fluctuatons nto lattce Boltzmann algorthms. Emphass s put on our recent work (Phys. Rev. E 76, (2007)) whch provdes a clear understandng n terms of statstcal mechancs. Key words: Lattce Boltzmann, thermal fluctuatons, Langevn equaton, Monte Carlo, detaled balance PACS: Qr, s The lattce Boltzmann (LB) equaton has, n the last few decades, emerged as a powerful tool to solve flud dynamcs problems numercally [1, 2]. The algorthm s a fully dscretzed verson of the Boltzmann equaton, known from the knetc theory of gases. Space r s dscretzed n terms of a regular (usually smple-cubc) lattce wth spacng b, tme t n terms of a tme step h, and velocty space n terms of a small set of veloctes c that are chosen such that c h s a vector whch connects two nearby lattce stes. For example, the popular D3Q19 model [3] employs nneteen veloctes, correspondng to zero and the sx nearest and twelve next-nearest neghbors on a smple-cubc lattce. The central quanttes on whch the algorthm operates are the populatons n ( r, t), representng the mass densty correspondng to velocty c, such that the total mass densty ρ( r, t) at the ste r at tme t s gven by ρ( r, t) = n ( r, t). (1) Preprnt submtted to Computer Physcs Communcatons January 7, 2009

2 Smlarly, the momentum densty s obtaned as the frst velocty moment, j( r, t) = n ( r, t) c, (2) and the hydrodynamc flow velocty s gven by u( r, t) = j( r, t) ρ( r, t). (3) The algorthm s then descrbed by the lattce Boltzmann equaton n ( r + c h, t + h) = n ( r, t) = n ( r, t) + ({n ( r, t)}). (4) The collson operator modfes the populatons on the ste ({n } denotes the set of all populatons on the ste), such that mass and momentum are conserved. Energy conservaton s not taken nto account, snce we are here nterested n an sothermal verson, where the temperature nstead of the energy s fxed (formally, ths corresponds to a system wth nfnte heat conductvty). The conservaton equatons therefore read = c = 0. (5) Ths results n a set of post-collsonal populatons n, whch are then propagated to the neghborng stes. In most applcatons, t s assumed that s a determnstc varable,. e. that t can be calculated n a unque fashon from the populatons n ( r, t). Ths s very much n sprt of the orgnal contnuum Boltzmann equaton, and applcable to many practcal problems of flud flow. However, for soft-matter applcatons, where one s nterested n Brownan moton of suspended partcles, or smlar phenomena, ths s not suffcent. Rather, one must take nto account that here both the lattce spacng b and the tme step h are so small that on these scales thermal fluctuatons are szeable and cannot be vewed as just averaged out. Indeed, assumng that the underlyng physcal model s an deal gas, one can see ths rather easly by startng from the equaton of state k B T = m p c 2 s, (6) where k B s Boltzmann s constant, T s the absolute temperature, m p s the mass of a gas partcle, and c s s the sothermal speed of sound (c 2 s = 2

3 p/ρ, where p s the thermodynamc pressure). Usually, c s s chosen as an adjustable parameter, pcked n such a way that even n nonequlbrum stuatons lke shear flow the typcal flow velocty u s small compared to c s. Ths s the condton of low Mach number flow, whch s needed because of the restrcted velocty space (note that c s s of the order of the c ). Furthermore, the physcs of the problem usually dctates the values of k B T and ρ for example, we may assume that we study water at room temperature. Equaton 6 then allows us to determne the mass of a gas partcle, m p, whch, n turn, determnes the number of partcles on a lattce ste (assumng a smple-cubc lattce n three dmensons), N p = ρb3 m p. (7) If ths number s very large, fluctuatons wll strongly average out,. e. one can consder the sngle lattce ste as a thermodynamc system. Ths s the case for typcal engneerng applcatons. However, f N p s comparable to unty, as t s the case for many soft-matter applcatons, then fluctuatons are mportant, and must be taken nto account n the algorthm. Snce the system s an deal gas, N p s a random varable whose probablty dstrbuton s Posson. For such a dstrbuton, the varance s dentcal to the mean,. e. the relatve mportance of fluctuatons s gven by Bo = ( N 2 p Np 2) 1/2 N p = N p 1/2 = ( mp ) 1/2 ( ) 1/2 kb T = (8) ρb 3 ρb 3 c 2 s (we coned the word Boltzmann number for ths parameter). We thus see that the degree of fluctuatons s controlled by the degree of coarse-granng, through the lattce spacng b. It s also useful to ntroduce the parameter µ = m p b 3 = k BT, (9) b 3 c 2 s whch may be called the thermal mass densty. The queston of how to actually mplement these fluctuatons n the collson operator has found dfferent answers durng the last ffteen years, wth ncreasng level of refnement and understandng. In what follows, we wsh to brefly outlne these developments. Snce all the materal has been publshed prevously, we would lke to be bref, and refer the nterested reader 3

4 to the orgnal papers [4, 5, 6, 7] as well as to a recent revew [8], n whch all the techncal detals have been worked out and explaned n depth. The frst mplementaton of a fluctuatng lattce Boltzmann equaton was by Ladd [4, 5]. He started from the well-understood determnstc verson (Bo = 0), and added a stochastc term to the collson operator, wth the requrement that ths s consstent, on the macroscopc scale, wth fluctuatng hydrodynamcs, as gven by Landau and Lfshtz [9]. Let us frst dscuss the determnstc verson n some more detal. It s based upon a lnearzed collson operator, = j L j (n j n eq j ), (10) where the matrx L j contans constant elements, and s mplctly gven va a dagonal representaton (see below), whle n eq s the lattce analog to a velocty-dependent Maxwell-Boltzmann dstrbuton: ( ) n eq (ρ, u) = a c ρ 1 + u c + ( u c ) 2 u2. (11) c 2 s 2c 4 s 2c 2 s Here c s s the speed of sound, and the weghts a c > 0 are normalzed such that ac = 1. Ths notaton has been chosen n order to emphasze that, for symmetry reasons, the weghts only depend on the absolute values of the speeds c, but not on ther drecton. Furthermore, the weghts are adjusted n such a way that n eq satsfes the propertes n eq = ρ, (12) n eq c = j, (13) n eq c c = ρc 2 s 1 +ρ u u = Π eq. (14) For D3Q19, ths mples a c = 1/3 for the rest populaton, a c = 1/18 for the nearest neghbors, and a c = 1/36 for the next-nearest neghbors. Furthermore c 2 s = (1/3)(b 2 /h 2 ). L j s mplemented as follows: Frst, one transforms to so-called modes,. e. lnear combnatons of the n whch are adapted to the symmetry of the problem. The frst ten modes have a drect hydrodynamc nterpretaton: 4

5 Mode 0: Mass densty ρ = n. Modes 1-3: Momentum densty j α = n c α ; here α denotes a Cartesan ndex. Modes 4-9: Stresses Π αβ = n c α c β, whch are convenently decomposed nto trace and traceless part: Π αβ = Π αβ δ αβπ γγ ; here we use the Ensten summaton conventon. The addtonal modes (so-called knetc or ghost modes) do not have a drect relaton to hydrodynamcs. In the D3Q19 model, there are nne such modes, whch are explctly lsted n Ref. [8]. After havng calculated the (pre-collsonal) modes, one leaves the conserved modes unchanged, whle the other modes are lnearly relaxed towards ther local equlbrum value. The stresses are changed from pre- to post-collsonal values accordng to where we use the notaton n neq Π neq αβ = γ Πneq s αβ, (15) Π αα neq = γ b Π neq αα, = n n eq. The knetc modes are defned n such a way that ther equlbrum part s zero, and the acton of L j on them s, n the smplest verson, just a projecton, such that the post-collsonal knetc modes vansh. A Chapman-Enskog analyss shows that ths procedure yelds the Naver- Stokes equatons of hydrodynamcs n the lmt of large length and tme scales, wth shear and bulk vscostes that are unquely determned by the values of γ s and γ b, respectvely. Lnear stablty requres γ s < 1, γ b < 1, correspondng to postve values of the vscostes. Ths determnstc procedure was modfed by Ladd [4, 5] by just changng Eq. 15 to Π neq αβ = γ Πneq s αβ + R αβ, (16) Π αα neq = γ b Π neq αα + R αα, wth sutably chosen random stresses R αβ, whle the treatment of the knetc modes was left unchanged. The ratonale behnd ths procedure was that the knetc modes do not contrbute to hydrodynamcs, and the goal was to smulate the fluctuatons correctly on the hydrodynamc scale. On ths 5

6 scale, however, the fluctuatng stresses ˆR αβ that appear n the Naver-Stokes equaton (dfferent from R αβ that appears n Eq. 16) satsfy the relatons [9] ˆRαβ = 0, (17) ˆRαβ ( r, t) ˆR γδ ( r, t ) = 2k B T η αβγδ δ ( r r ) δ (t t ) 2k BT b 3 h η αβγδ δ r r δ tt, where η αβγδ s the sotropc fourth-rank vscosty tensor, parameterzed by shear and bulk vscosty, or the relaxaton parameters γ s and γ b. In the last step, we have dscretzed the delta functons by the lattce parameter b and the tme step h, as t s approprate for a lattce smulaton. One mght expect that the LB noses are just gven by R αβ = ˆR αβ. However, ths turns out not to be correct [4, 5]. Rather, the correct fluctuatng LB stresses are obtaned by a sutable modfcaton of the ampltude. For the shear stresses one has, for example, R 2 xy = (1 γs ) 2 ˆR2 xy. (18) The same modfcaton factor occurs for all other shear stresses, too, whle the correspondng factor for the bulk stresses s (1 γ b ) 2. The reason has been explaned n detal n Refs. [4, 5]; essentally the renormalzaton of the ampltude comes from the fact that Eq. 17 descrbes the physcs on a more coarse-graned tme scale than Eq. 16 the delta correlaton n tme s n LB replaced by an exponental decay. However, the tme ntegral of the correlaton functons must be the same n order to obtan the same macroscopc vscostes. Adhkar et al. [6] then generalzed ths procedure by not only thermalzng the stresses, but also the knetc modes, whch were treated n a rather smlar fashon to Eq. 16. The argument was that the relaxaton of knetc modes ntroduces an addtonal dsspatve mechansm nto the system, whch should be balanced by a compensatng Langevn nose. A projecton should be vewed as the lmt of such a relaxaton, wth relaxaton parameter γ 0, such that the fluctuaton-dsspaton relaton should hold n ths case, too. Whle ths argument makes ntutve sense, and led to a substantally mproved representaton of the fluctuatons at short length scales [6], the theoretcal foundaton of ths procedure remaned somewhat obscure (at least to the present authors). 6

7 In a recent publcaton [7] we have been able to resolve these questons by developng a frst-prncples theory of thermal fluctuatons n LB models. The startng pont was the observaton that for a dscrete system the concept of a fluctuaton-dsspaton theorem should rather be replaced by the concept of detaled balance as t apples to Monte Carlo smulatons [10]. In order to be able to check whether an update rule satsfes or volates the detaled-balance condton, we therefore explctly constructed the probablty densty for the random varables n on a ste n thermal equlbrum. Takng advantage of the underlyng pcture of a gas of partcles, we frst transform from the n to varables ν, the number of partcles on the ste whch have velocty c (cf. Eqs. 7 and 9): ν = n µ. (19) In terms of these varables, the probablty densty (except for normalzaton, whch s unmportant for our purposes) s wrtten as ( ) ( ) ( ) ν ν P ({ν }) ν! exp ( ν ) δ µν ρ δ µ c ν j. (20) The underlyng pcture s that of a velocty bn n thermal contact wth a huge reservor of partcles, resultng n a Posson dstrbuton of the varable ν. Ths dstrbuton s characterzed by ts mean value ν, whch, for reasons of consstency wth the determnstc verson, should be proportonal to the weght a c (see Eq. 11). Normalzaton requres ν = ac ρ µ. (21) Equaton 20 then results from assumng that all the velocty bns on the ste are statstcally ndependent, except for the constrants of conserved mass and momentum, whch are taken nto account by the delta functons, n close analogy to the statstcal descrpton of the mcrocanoncal ensemble [11]. The further development s somewhat techncal but straghtforward and shall be sketched only brefly. We use Strlng s formula and transform back to the n to wrte the factor n front of the delta functons as exp(s), where the entropy S has a Boltzmann-lke form. Maxmzng P s equvalent to maxmzng S under the constrants of gven values for ρ and j, and the 7

8 soluton of ths problem, up to second order n u, s just Eq. 11, as s wellknown from prevous studes of the entropc lattce Boltzmann approach [12]. Fluctuatons around the most probable populatons are descrbed by, whch, wthn a saddle-pont approxmaton, obey a Gaussan dstrbuton, whose varance s, wthn a u 0 approxmaton, gven by µρa c. Normalzng the fluctuatons to unt varance, followed by an orthonormal transformaton to normalzed modes ˆm neq k, yelds a very smple form for the probablty dstrbuton, ( ) P ({ ˆm neq k }) exp 1 ˆm neq 2 k, (22) 2 n neq where modes = 0,..., 3 do not occur due to mass and momentum conservaton. These modes are updated accordng to the rule ˆm neq k k>3 = γ k ˆm neq k + ϕ k r k, (23) wth adjustable parameters γ k, ϕ k, and normalzed, ndependent Gaussan random numbers r k. It s then straghtforward to show [7, 8] that detaled balance holds exactly for ϕ k = ( 1 γ 2 k) 1/2, (24) whch turns out to be dentcal to the prescrpton of Adhkar et al. [6]. Ths shows that the stochastc analog of projectng out the knetc modes s to sample them from scratch, and explans the non-trval prefactor n the fluctuatng stresses n a straghtforward way. Furthermore [7, 8], one may apply the Chapman-Enskog procedure to the stochastc verson of the algorthm. Ths shows n a partcularly concse way that the behavor n the hydrodynamc lmt s gven by Landau-Lfshtz fluctuatng hydrodynamcs [9], and that the detals of the dynamcs of the knetc modes are ndeed mmateral for the behavor n that lmt, as already antcpated n Refs. [4, 5]. For practcal smulatons, however, one should prefer the more recent verson whch does satsfy detaled balance on the local scale as well. We beleve that ths s really an mprovement that outweghs the computatonal costs, whch are unfortunately not completely neglgble. Whle smple LB algorthms have so few operatons per collson step that they are typcally lmted by the bandwth of memory access n the streamng step [13], ths does not seem to be true here, where the generaton of random numbers 8

9 L stresses only full thermalzaton Table 1: Performance of the stochastc D3Q19 algorthm, usng an mplementaton on a 64-bt AMD Athlon processor wth 2.2 GHz CPU speed and 512 kb cache sze. The program s part of the Manz ESPResSo [14] package. Smulatons were run on smple-cubc lattces of sze L 3 for 10 5 lattce sweeps, and Gaussan random numbers were generated by the Box-Muller [15] method. Performance data are gven n MLUPS (mllon lattce-ste updates per second). combned wth the lnear transformaton to mode space and back contrbutes notceably. In practce, one may say that the addtonal thermalzaton of the knetc modes wll slow down the algorthm by roughly 20%... 40% at least ths s what we observed for our D3Q19 mplementaton, see Table 1. For large lattces the memory bottlenecks become more mportant than for small ones; for ths reason, the smulatons become systematcally slower, whle the performance dfference between stresses-only vs. full thermalzaton becomes less pronounced. So far, only the case of an sothermal deal gas has been thoroughly understood. For the future, t s hoped that the present theoretcal approach wll also help develop an mproved understandng of systems wth non-trval equatons of state, and systems where thermal conducton and energy conservaton are taken nto account. References [1] Succ, S., The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, Oxford Unversty Press, Oxford, [2] Benz, R., Succ, S., and Vergassola, M., Phys. Rep. 222 (1992) 145. [3] Qan, Y. H., D Humeres, D., and Lallemand, P., Europhys. Lett. 17 (1992) 479. [4] Ladd, A. J. C., J. Flud Mech. 271 (1994) 285. [5] Ladd, A. J. C., J. Flud Mech. 271 (1994)

10 [6] Adhkar, R., Stratford, K., Cates, M. E., and Wagner, A. J., Europhys. Lett. 71 (2005) 473. [7] Dünweg, B., Schller, U. D., and Ladd, A. J. C., Phys. Rev. E 76 (2007) [8] Dünweg, B. and Ladd, A. J. C., Adv. Polym. Sc. 221 (2009) 89. [9] Landau, L. D. and Lfshtz, E. M., Flud Mechancs, Addson-Wesley, Readng, [10] Landau, D. P. and Bnder, K., A Gude to Monte Carlo Smulatons n Statstcal Physcs, Cambrdge Unversty Press, Cambrdge, [11] Landau, L. D. and Lfshtz, E. M., Statstcal Physcs, Addson-Wesley, Readng, [12] Karln, I. V., Ferrante, A., and (1999) 182. Öttnger, H. C., Europhys. Lett. 47 [13] Wellen, G., Zeser, T., Hager, G., and Donath, S., Computers & Fluds 35 (2006) 910. [14] Lmbach, H.-J., Arnold, A., Mann, B. A., and Holm, C., Comput. Phys. Commun. 174 (2006) 704. [15] Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterlng, W. T., Numercal Recpes, Cambrdge Unversty Press, Cambrdge,