Relativistic Lattice Boltzmann Model with Improved Dissipation

Transcription

1 Relatvstc Lattce Boltzmann Model wth Improved Dsspaton M. Mendoza, 1, I. Karln, 2, S. Succ, 3, 1, 4, and H. J. Herrmann 1 TH Zürch, Computatonal Physcs for ngneerng Materals, Insttute for Buldng Materals, Schafmattstrasse 6, HIF, CH-8093 Zürch (Swtzerland 2 TH Zürch, Department of Mechancal and Process ngneerng, Sonneggstrasse 3, ML K 20, CH-8092 Zürch (Swtzerland 3 Isttuto per le Applcazon del Calcolo C.N.R., Va de Taurn, , Rome (Italy, and Freburg Insttute for Advanced Studes, Albertstrasse, 19, D-79104, Freburg, (Germany 4 Departamento de Físca, Unversdade Federal do Ceará, Campus do Pc, Fortaleza, Ceará, (Brazl (Dated: January 15, 2013 We develop a relatvstc lattce Boltzmann (LB model, provdng a more accurate descrpton of dsspatve phenomena n relatvstc hydrodynamcs than prevously avalable wth exstng LB schemes. The procedure apples to the ultra-relatvstc regme, n whch the knetc energy (temperature far exceeds the rest mass energy, although the extenson to massve partcles and/or low temperatures s conceptually straghtforward. In order to mprove the descrpton of dsspatve effects, the Maxwell-Jüttner dstrbuton s expanded n a bass of orthonormal polynomals, so as to correctly recover the thrd order moment of the dstrbuton functon. In addton, a tme dlataton s also appled, n order to preserve the compatblty of the scheme wth a cartesan cubc lattce. To the purpose of comparng the present LB model wth prevous ones, the tme transformaton s also appled to a lattce model whch recovers terms up to second order, namely up to energymomentum tensor. The approach s valdated through quanttatve comparson between the second and thrd order schemes wth BAMPS (the soluton of the full relatvstc Boltzmann equaton, for moderately hgh vscosty and veloctes, and also wth prevous LB models n the lterature. xcellent agreement wth BAMPS and more accurate results than prevous relatvstc lattce Boltzmann models are reported. PACS numbers: j, k, Sf Keywords: I. INTRODUCTION Relatvstc hydrodynamcs and knetc theory play a major role n many forefronts of modern physcs, from large-scale applcatons n astrophyscs and cosmology, to mcroscale electron flows n graphene [1 3], all the way down to quark-gluon plasmas [4 6]. Due to ther strong non-lnearty and, for the case of knetc theory, hgh dmensonalty as well, the correspondng equatons are extremely challengng even for the most powerful numercal methods, let alone analytcs. Recently, a promsng approach, based on a mnmal form of relatvstc Boltzmann equaton, whose dynamcs takes place n a fully dscrete phase-space and tme lattce, known as relatvstc lattce Boltzmann (RLB, has been proposed by Mendoza et al. [7 9]. To date, the RLB has been appled to the smulaton of weakly and moderately relatvstc flud dynamcs, wth Lorentz factors of γ 1.4, where γ = 1/ 1 v 2 /c 2, c beng the speed of lght and v the speed of the flud. Ths model reproduces correctly shock waves n quark-gluon plasmas, showng excellent lectronc address: mmendoza@ethz.ch lectronc address: karln@lav.mavt.ethz.ch lectronc address: succ@ac.cnr.t lectronc address: hjherrmann@ethz.ch agreement wth the soluton of the full Boltzmann equaton as obtaned by Bouras et al. usng BAMPS (Boltzmann Approach Mult-Parton Scatterng [10, 11]. The RLB makes use of two dstrbuton functons, the frst one modelng the conservaton of number of partcles, and the second one, the momentum-energy conservaton equaton. The model was constructed by matchng the frst and second order moments of the dscrete-velocty dstrbuton functon to those of the contnuum equlbrum dstrbuton of a relatvstc gas In a subsequent work, Hupp et al.[12] mproved the scheme by extendng the equlbrum dstrbuton functon for the number of partcles, n such a way as to nclude second order terms n the velocty of the flud, thereby tamng numercal nstabltes for hgher pressure gradents and veloctes. However, the model was not able to reproduce the rght velocty and pressure profles for the Remann problem n quark-gluon plasmas, for the case of large values of the rato between the shear vscosty and entropy densty, η/s 0.5, at moderate flud speeds (v/c 0.6. In order to set up a theoretcal background for the lattce verson of the relatvstc Boltzmann equaton, Romatschke et al. [13] developed a scheme for an ultrarelatvstc gas based on the expanson on orthogonal polynomals of the Maxwell-Jüttner dstrbuton [14] and, by followng a Gauss-type quadrature procedure, the dscrete verson of the dstrbuton and the weghtng func-

2 2 tons was calculated. Ths procedure was smlar to the one used for the non-relatvstc lattce Boltzmann model [15, 16]. Ths relatvstc model showed very good agreement wth theoretcal data, although t was not compatble wth a lattce, thereby requrng lnear nterpolaton n the free-streamng step. Ths mples the loss of some key propertes of the standard lattce Boltzmann method, such as negatve numercal dffuson and exact streamng. Very recently, L et al. [17] notced that the equaton of conservaton for the number of partcles, recovered by the RLB model [7, 8], exhbts ncorrect dffusve effects. Therefore, they proposed an mproved verson of RLB, usng a mult-relaxaton tme collson operator n the Boltzmann equaton, showng that ths fxes the ssue wth the equaton for the conservaton of the number of partcles. The generalzed collson operator allows to tune ndependently the bulk and shear vscostes, yeldng results for the Remann problem closer to BAMPS [10] when the bulk vscosty s decreased. However, the thrd order moment of the equlbrum dstrbuton stll does not match ts contnuum counterpart and therefore the model stll has problems to reproduce hgh η/s 0.5, for moderately hgh veloctes, β = v/c = 0.6. Thus, whle surely provdng an mprovement on the orgnal RLB model, the work [17] dd not succeed n reproducng the vanshng bulk vscosty, whch s pertnent to the ultra-relatvstc gas, whle allowng the bulk vscosty to vary ndependently on the shear vscosty. Note that n the much more studed case of the lattce Boltzmann models for the non-relatvstc fluds, the queston of the choce of the lattce wth hgher-order symmetry requrements has only recently been solved, n the framework of the entropy-complant constrcton [18, 19]. However, the lattces (space-fllng dscretevelocty sets found n that case are talored to reproduce the moments of the non-relatvstc Maxwell-Boltzmann dstrbuton, and do not seem to be drectly transferable to the present case of the relatvstc (Maxwell-Jüttner equlbrum dstrbuton, whch has farly dfferent symmetres as compared to the non-relatvstc Maxwell- Boltzmann dstrbuton. Therefore, the extenson of the prevous LB models has to be consdered anew. In ths paper, we develop a new lattce Boltzmann model capable of reproducng the thrd order moment of the contnuum equlbrum dstrbuton, and stll realzable on a cubc lattce. The model s based on a sngle dstrbuton functon and satsfes conservaton of both number of partcles and momentum-energy equatons. The model s based on the sngle relaxaton tme collson operator proposed by Anderson and Wttng [14, 20] whch s more approprate for the ultra-relatvstc regme than the Marle model used n the prevous works, Thus, the proposed model offers sgnfcant mprovement on prevous relatvstc lattce Boltzmann models n two respects: ( It captures the symmetry of the hgher-order equlbrum moments suffcently to reproduce the dsspatve relatvstc hydrodynamcs at the level of the Grad approxmaton to the relatvstc Boltzmann equaton; ( It represents a genune lattce Boltzmann dscretzaton of space and tme, wth no need of any nterpolaton scheme, thereby avodng the otherwse ubqutous spurous dsspaton. The new lattce Boltzmann model s shown to reproduce wth very good accuracy the results of the shock-waves n quark-gluon plasmas, for moderately hgh veloctes and hgh ratos η/s. The paper s organzed as follows: n Sec. II we descrbe n detal the model and the way t s constructed; n Sec. III, we mplement smulatons of the Remann problem n order to valdate our model and compare t wth BAMPS and prevous relatvstc lattce Boltzmann models; fnally, n Sec. IV, we dscuss the results and future work. II. MODL DSCRIPTION A. Symmetres of the relatvstc Boltzmann equaton To buld our model, we start from the relatvstc Boltzmann equaton for the probablty dstrbuton functon f: p µ µ f = p µu µ c 2 τ (f, (1 where the local equlbrum s gven by the Maxwell- Jüttner equlbrum dstrbuton [14], = A exp( p µ U µ /k B T, (2 In the above, A s a normalzaton constant, c the speed of lght, and k B the Boltzmann constant. The 4-momentum vectors are denoted by p µ = (/c, p, and the macroscopc 4-velocty by U µ = (c, uγ(u, wth u the threedmensonal velocty of the flud. Note that we have used the Anderson-Wttng collson operator[20] (rhs of q. (1, makng our model compatble wth the ultrarelatvstc regme. Hereafter, we wll use natural unts, c = k B = 1, and work n the ultrarelatvstc regme, ξ mc 2 /k B T 1. Accordng to a standard procedure [13, 15, 16], we frst expand the Maxwell-Jüttner dstrbuton n the rest frame, = A exp( p 0 /T, n an orthogonal bass. Snce n the ultrarelatvstc regme, p 0 /T = p2 /T 2 + m 2 /T 2 p/t, beng p = p 2, we can wrte the equlbrum dstrbuton n sphercal coordnates, ge p0/t d3 p π 2π p 0 = gpe p/t dp sn(θdθdφ, (3 where g s an arbtrary functon of momentum. Followng Romatschke [13], we can expand the dstrbuton n each coordnate separately, and subsequently, by usng a Gauss quadrature, calculate the dscrete values of the 4-momentum vectors. Thus, the dscrete equlbrum ds-

3 3 trbuton can be wrtten as, l =,j,k a jk (U µ P (θ l R j (p l F k (φ l, (4 where the coeffcents a jk (U µ are the projectons of the dstrbuton on the polynomals P (θ l R j (p l F k (φ l, and the dscrete 4-momenta are denoted by p µ l = (p l, p l cos(φ l sn(θ l, p l sn(φ l sn(θ l, p l cos(θ l. Consequently, the dscrete form of the Boltzmann equaton takes the form, f l (x µ +p µ l /p0 l δt, t+δt f l (x µ, t = p lµu µ δt τp 0 (f l l. l (5 However, note that, n the streamng process on the rghthand-sde of q.(5, the dstrbuton moves at velocty p µ l /p0 l, whch mples that the nformaton travels (n a sngle tme step from each cell center to a poston that belongs to the surface of a sphere of radus c δt = 1. Furthermore, to represent correctly the thrd order moment of the equlbrum dstrbuton, P αβλ = l l p α l p β l pλ l, (6 the number of ponts needed on the surface of the unt sphere exceeds 6 and 12, whch correspond to the frst neghbors for a cubc and hexagonal closed packed (HCP lattces, respectvely. Ths mples that, n general, the 4-vectors p µ /p 0 cannot be embedded nto a regular lattce, and therefore, an nterpolaton algorthm has to be used. By dong ths, we are losng one of the most mportant features of lattce Boltzmann models, whch s the exact streamng. Thus, wthn ths sphercal coordnate representaton, the streamng process cannot take place on a regular lattce. B. Moment projecton of the equlbrum In ths work, we shall use a dfferent approach to the quadrature representaton. We frst expand the equlbrum dstrbuton at rest, w(p 0 = = A exp( p 0 by usng Cartesan coordnates, unlke the sphercal coordnate system used n Ref. [13], and choose the 4- momentum vectors such that they belong to the lattce (from now on and wthout loss of generalty, we wll use the notaton p 0 /T p 0. Ths procedure also avods extra terms n the product, P (θ l R j (p l F k (φ l for the sphercal case, whch are not necessary f we only need to recover correctly the frst three moments of the equlbrum dstrbuton. Ths consderably smplfes the dscrete equlbrum dstrbuton. By performng a Gram-Schmdt procedure wth the weght w(p 0, we construct a set of orthonormal polynomals. The orthonormal polynomals n cartesan coordnates up to thrd order, herefrom denoted by J k, where the ndex k runs from 0 to 29, are shown n Table I. Order Polynomal J k k 0th 1 0 1st 2nd 3rd p (p 0 6p 0 +6 (p 0 4p z 2 2, px 2, py 2, pz 2 1, 2, 3, 4, (p0 4p x 2, (p0 4p y , p02 +p x2 +2p y2 4 2 p x p z 2, py p z 2 2, px p y 2 2 2, p02 3p x (p0 6 2 p 0 2, ((p0 10p 0 +20p x (p0 6 ( p 02 3p x2, 5px3 3p 02 p x ((p 0 10p 0 +20p y 4 5 (p0 6(p 02 p x2 2p y2 8 3 p y ( 3p 02 +3p x2 +4p y (p 0 6p x p z (p 0 6p y p z 4 3, (p0 6p x p y 4 3, pz (p 02 5p x2 24 5, px p y p z 4 3, px ( p 02 +p x2 +2p y2 8 3, ((p0 10p 0 +20p z 4 5, (p0 6p y p z 4 3, pz ( p 02 +p x2 +4p y , 6, 7 8, 9, 10 11, 12, 13 14, 15 16, 17 18, 19, 20 21, 22 23, 24 25, 26, 27 28, 29 TABL I: Polynomals J k that are orthonormal on the weght functon w(p 0 n Cartesan coordnates (x, y, z. Note that n ths Table, the 4-momentum has the notaton p µ = (p 0, p x, p y, p z. Snce these polynomals are orthonormal, there are no normalzaton factors, and the Maxwell-Jüttner dstrbuton can be approxmated, up to thrd order n the momentum space, by an expanson as smple as 29 w(p 0 a k (T, U µ J k (p µ, (7 k=0 where the projectons a k are calculated by, a k = J k (p µ d3 p p 0. (8 Snce the Anderson-Wttng model s only compatble wth the Landau-Lfshtz decomposton [14, 20], we must calculate the energy densty of the flud by solvng the egenvalue problem, T αβ U β = ɛu α, (9 ɛ beng the energy densty of the flud, and T αβ = fp α p β d3 p p 0, (10 the momentum-energy tensor. For the partcle densty, we use the relaton, n = U α fp α d3 p p 0, (11 and, by usng the equaton of state, ɛ = 3nT, we can calculate the temperature of the flud. C. Dscrete-velocty representaton of the quadratures Note that the above dervaton usng Cartesan coordnates stll refers to the contnuous 4-momenta. In order

4 4 to dscretze the above moment projecton of the equlbrum dstrbuton, we must choose a set of 4-momentum vectors that satsfes the very same orthonormalty condtons, namely: w(p 0 J l (p µ J k (p µ d3 p p 0 = w J l (p µ J k(p µ = δ lk, (12 whle, at the same tme, p µ /p 0 corresponds to lattce ponts. Here, we choose to work wth a cubc lattce, although the procedure descrbed here also apples to other ones, e.g. HCP lattce. Snce, due to ts nature, p µ /p 0 leads to velocty vectors whch belong to a sphere of radus c n the space components, usng the procedure n Ref. [13] wll generally result n off-ste lattce ponts. For ths reason, we opt for another quadrature based on ths orthonormalty condton, and mpose that the dstrbuton functon at rest frame should satsfy the moments of the equlbrum dstrbuton, up to 6-th order. Ths s made to ensure that the 5-th order moment of the equlbrum dstrbuton s recovered (at least at very low flud veloctes, whch, n the context of the Grad theory for the Anderson-Wttng model [14], s a requrement for the correct calculaton of the transport coeffcents, namely the shear and bulk vscostes and thermal conductvty. The condton for the 6-th order moment, s to choose from the multple lattce solutons, the one that presents the hghest symmetry to model the Maxwell-Jüttner dstrbuton. In order to use general features of classcal lattce Boltzmann models, lke bounce-back boundary condtons to mpose zero velocty on sold walls, we wll also requre that the weghts w correspondng to the dscrete 4-momentum vectors p k have the same values as the ones correspondng to p k (latn ndces run over spatal components. In order to generate on-ste lattce ponts, let us frst analyse the relatvstc Boltzmann equaton, whch can be wrtten as, p 0 t f + p a a f = p µu µ (f, (13 τ and n the ultrarelatvstc regme, p 0 ( t f + v a a f = p µu µ (f. (14 τ where v a are the components of the mcroscopc velocty. These mcroscopc veloctes have the same magntude but, n general, dfferent drectons. Dvdng both sdes of q.(16 by p 0, we obtan t f + v a a f = p µu µ τp 0 (f. (15 In other words, n the ultra-relatvstc regme, the relatvstc Boltzmann equaton can be cast nto a form where the tme dervatve and the propagaton term become the same as n the non-relatvstc case, at the prce of an addtonal dependence on p 0 n the relaxaton term. FIG. 1: Drectons of the velocty vectors ϑ to recover up to the thrd order moment of the Maxwell-Jüttner dstrbuton. The radus of the sphere s R = 41. The ponts represent lattce stes belongng to the sphere surface. However, snce ths newly acqured dependence remans local, we shall be able to fnd a dscrete-velocty quadrature whch also allows for a lattce Boltzmann-type dscretzaton n tme and space wthout any nterpolaton. Indeed, n a cubc cell of length δx = 1 there are only 6 neghbors, whch are not suffcent to satsfy the orthogonalty condtons and the thrd order moment of the equlbrum dstrbuton. However, by multplyng ths equaton by a constant R at both sdes, and performng a tme transformaton (dlataton, δt Rδt and τ Rτ, we obtan t f + ϑ a a f = p µu µ τ p 0 (f, (16 where we have defned ϑ a = Rv a. Due to ths transformaton, the 4-momentum vectors are reconstructed through the relaton p µ = p 0 (1, ϑ/r, (17 At ths stage, we can choose the radus of the sphere such that the lattce ponts that belong to the surface of the sphere and the cubc lattce exhbt enough symmetres to satsfy both condtons. Ths s equvalent to solvng the Dophantne equaton, n 2 x + n 2 y + n 2 z = R 2, (18 where n x, n y, and n z are nteger numbers, beng ϑ = (n x, n y, n z. Thus, we can determne the components of the dscrete verson of the veloctes ϑ whch are needed for the streamng term n the Boltzmann equaton, lhs of

5 5 q. (16. However, on the rhs of ths equaton, and for the calculaton of the dscrete 4-momentum vectors va q. (17, we also need to know the dscrete values of p 0. The 4-vector p µ s needed to compute the orthonormalty condtons gven by q. (12 and the moments of the equlbrum dstrbuton. Due to the fact that p 0 s the magntude of the 4- momentum, p 0 = p µ p µ, n dmensonal spacetme, t s natural to assume that ts dscrete values can be calculated by usng the weght functon n sphercal coordnates, w(p = 4πAp 2 exp( p, where the angular components have been ntegrated out, and usng the zeros of ts respectve orthonormal polynomal of fourth order (ths s because we are nterested n an expanson up to thrd order, so we need one more order to calculate the zeros. Ths fourth order polynomal s gven by: R (4 (p = 1 24 [120 + p( p[120 + (p 20p]]. 5 (19 To summarze, n order to calculate the dscrete p µ and ther respectve w, we frst fx R and solve the equatons n 2 x + n 2 y + n 2 z = R 2, (20a R (4 (p = 0, (20b to obtan the solutons for n x, n y, n z, and p. Wth these values, we buld the dscrete 4-vectors p µ lm = p0 l (1, n x,m /R, n y,m /R, n z,m /R, (21 where l = 1,..., 4 denotes the four zeros of the polynomal R (4 (p, and m = 0,..., M the trplets (n x, n y, n z m that satsfy the Dophantne equaton, assumng that M s the number of solutons. Here, for smplcty, we regroup the par of ndexes lm to, so that we can label the dscrete 4-momentums as p µ, where = 1,..., N wth N = 4 M. Next, we replace these values nto the equatons, w(p 0 J l (p µ J k (p µ d3 p p 0 w(p 0 p µ p ν p σ p λ d3 p p 0 w(p 0 p µ p ν p σ p λ p γ d3 p p 0 w(p 0 p µ p ν p σ p λ p γ p β d3 p p 0 N = w J l (p µ J k(p µ = δ lk, (22a N = w p µ pν p σ p λ, (22b N = w p µ pν p σ p λ p γ, (22c N = w p µ pν p σ p λ p γ pβ, (22d w = w j (f p k = p k j, (22e w 0, (22f and look for any soluton for w that fulflls the above relatons. Should none be found, we repeat the procedure wth a dfferent value of R. By performng ths teraton process, we found that R = 41 s suffcent to recover up to the thrd order moment of the Maxwell-Jüttner dstrbuton, and up to sxth order of ths dstrbuton n the Lorentz rest frame. The correspondng dscrete velocty vectors ϑ m are: (±6, ±2, ±1, (±6, ±1, ±2, (±2, ±6, ±1, (±1, ±6, ±2, (±1, ±2, ±6, (±2, ±1, ±6, (±5, 0, ±4, (±5, ±4, 0, (0, ±5, ±4, (±4, ±5, 0, (0, ±4, ±5, (±4, 0, ±5, (±4, ±3, ±4, (±3, ±4, ±4, and (±4, ±4, ±3; wth the values for p 0 l 0.743, 2.572, 5.731, and Consequently, ths gves a total of 4-momentum vectors N = 384. However, the last condton n q. (22 allows some weghts to become zero. Therefore, n our teraton procedure, we have taken the mnmal number of 4-momentum vectors p µ, by mposng the maxmum number of w to be zero. For ths reason, there are only 128 vectors p µ needed to fulfll the condtons n q. (22. In prncple, all the velocty vectors ϑ m are needed, but only some of the combnatons wth p 0 l are requred. The detaled lst of the ϑ m, p 0 l, and pµ, and ther respectve dscrete weght functons w are gven n the Supplementary Materal [21]. In Fg. 1 we report the confguraton of the velocty vectors ϑ to acheve the thrd order moment of the Maxwell-Jüttner dstrbuton functon. The ponts correspond to lattce nodes of a cubc lattce that, at the same tme, belong to the surface of the respectve sphere of radus R = 41. The relatvely large number of dscrete veloctes should not come as a surprse; n the case of non-relatvstc lattce Boltzmann, the number of dscrete veloctes also becomes hgh (at least 41 for achevng complete Gallean nvarance n the non-thermal case and 125 n the thermal case, see [18, 19]. Note that the specfed values of p 0 play the same role n defnng the quadrature as the reference temperature (energy n the non-relatvstc case [18, 19]. Fnally, we can wrte the dscrete verson of the equlbrum dstrbuton up to thrd order, 29 = w a n (T, U µ J n (p µ, (23 n=0 whch s shown n detals n Appendx B, q. (B2. Note that ths dstrbuton functon recovers the frst three moments of the Maxwell-Jüttner dstrbuton n the ultrarelatvstc regme, p µ d3 128 p p 0 = p µ = N µ, (24 =1 p µ p ν d3 128 p p 0 = p µ pν = T µν, (25 =1

6 6 where p µ p ν p λ d3 128 p p 0 = p µ pν p λ = P µνλ, (26 =1 N ν = nu ν, (27 T νµ = nt η νµ + 4nT U ν U µ, (28 beng the number of partcles 4-flow and the energymomentum tensor, respectvely, and P νµλ = 4nT 2 (η νµ U λ + η νλ U µ + η µλ U ν + 24nT 2 U ν U µ U λ, (29 wth n = 2T 3. However, the extenson to the case of massve partcles s straghtforward, by changng the coeffcents, a n, n qs. (8 and (23. D. Dscrete relatvstc Boltzmann equaton In the model of Anderson-Wttng for the collson operator, the relatvstc Boltzmann equaton takes the form gven by q. (1, p µ µ f = pµ U µ (f. (30 τ Ths collson operator s compatble wth the Landau- Lfshtz decomposton [14], whch mples fulfllment of the followng relatons U µ N µ = U µ U µ T µν = U µ fp µ d3 p p 0 = U µn µ = fp µ p ν d3 p p 0 = U µt µν = p µ d3 p p 0, (31a p µ p ν d3 p p 0, (31b Here, the subscrpt denotes the quanttes calculated wth the equlbrum dstrbuton. Therefore, upon ntegratng q. (30 n momentum space, we obtan µ N µ = 0, (32 whch s the conservaton of the number of partcles 4- flow. By multplyng by p ν and ntegratng, we obtan the conservaton of the momentum energy tensor µ T µν = 0. (33 In order to calculate the transport coeffcents, we need the thrd order moment, so that, upon multplyng q. (30 by p ν p β, we obtan µ P µνβ = 1 τ (U µp µνβ U µ P µνβ, (34 and by usng a Maxwellan teraton method [14], U µ P µνβ U µ P µνβ = τ µ P µνβ. (35 Note that we need at least the thrd order moment of the equlbrum dstrbuton, P µνβ, to compute the dsspaton coeffcents (namely, bulk and shear vscostes and heat conductvty. Ths requrement s fulflled n our dscrete and contnuum expansons of the equlbrum dstrbuton va qs. (24, (25, (26. However, to recover full dsspaton, we would also need to recover the thrd moment of the non-equlbrum dstrbuton, whch accordng to the 14 moments Grad s theory, can be wrtten as, P µνβ = P µνβ + b α P µνβα + d αλ P µνβαλ, (36 where b α and d αλ are coeffcents that carry the nformaton on the transport coeffcents [14]. Note that we need to recover terms up to the ffth order of the equlbrum dstrbuton. In prncple, ths could be done by the procedure descrbed on ths paper, but the resultng value for R could be unpractcally large. Nevertheless, at low veloctes, U µ (1, 0, 0, 0, the Maxwell-Jüttner dstrbuton can be approxmated by the weght functon w(p 0, and n analogy to the dscrete case, by w, and the fourth and ffth order are recovered va q. (22. As a result, at relatvely low veloctes, we expect the non-equlbrum thrd order tensor to be also fulflled. Therefore, the transport coeffcents for an ultrarelatvstc gas,.e. µ = 0 for the bulk vscosty, η = (2/3P τ for the shear vscosty, and λ = (4/5T P τ for the thermal conductvty, also apply to our model. To dscretze the relatvstc Boltzmann equaton, we frst mplement the tme transformaton descrbed n the prevous secton and ntegrate n tme q. (30 between t and t + δt. Ths yelds: f(x a + ϑ a δt, t + δt f(x a, t = pµ U µ τ p 0 (f δt. (37 By changng p µ p µ, f f and ϑ a ϑ a, we obtan f (x a +ϑ a δt, t +δt f (x a, t = pµ U µ τ p 0 (f δt. (38 Ths relatvstc lattce Boltzmann equaton presents an exact streamng at the left hand sde, and the collson operator at the rght hand sde looks exactly lke ts contnuum verson. Therefore, the conservaton laws for the number of partcles densty 4-flow, and the momentumenergy tensor, are also fulflled, as long as they are obtaned by usng the Landau-Lfshtz decomposton. Ths means that, frst, we need to calculate the momentumenergy tensor, 128 T αβ = f p α p β, (39 =1

7 7 and wth ths tensor, we solve the egenvalue problem, T αβ U β = T αβ U β = ɛu α, (40 obtanng the energy densty ɛ and the 4-vectors U α. Subsequently, the partcle densty can be calculated by n = U µ N µ = U 128 µn µ = f p µ U µ. (41 =1 The temperature T s obtaned by usng the equaton of state for the ultrarelatvstc gas, ɛ = 3nT. The transport coeffcents are the same as n the contnuum case, wth the lattce correcton resultng from second order Taylor expanson of the streamng term. All factored n, the coeffcents take the followng expresson µ = 0, η = (2/3P (τ δt /2, and λ = (4/5T P (τ δt /2. Note that revertng back the tme transformaton, we can wrte the transport coeffcents as η = (2/3P (τ δt/2/r, and λ = (4/5T P (τ δt/2/r. Summarzng, the present model does not present spurous dsspaton n the number of partcle conservaton equaton, n contrast to prevous RLB schemes [7, 8, 12], and also mproves the dsspatve terms gven by the mult-relaxaton tme scheme [17]. In addton, t realzes the expanson of the Maxwell-Jüttner dstrbuton on a cubc lattce, n contrast to Ref. [13]. We can also construct a relatvstc lattce Boltzmann model that recovers only up to second order (momentum-energy tensor, to compare wth the thrd order model and determne the nfluence of the thrd order moment n the expanson. Detals of the second order model can be found n Appendx A. v/c P/P BAMPS RLBD 3rd Order RLBD 2nd Order RLB z (fm BAMPS RLBD 3rd Order RLBD 2nd Order RLB z (fm FIG. 2: Velocty (top and pressure (bottom profles as functon of the z-coordnate for the case of a shockwave n quarkgluon plasma, wth η/s = 0.1. III. NUMRICAL VALIDATION In order to valdate our model, we solve the Remann problem for a quark-gluon plasma and compare the results wth BAMPS and two prevous relatvstc Boltzmann models. The frst one, proposed by Mendoza et al. [7, 8] and later mproved by Hupp et al. [12], whch we wll denote smply by RLB, and the second one, whch s a recent extenson of the RLB developed by L et al. [17] to nclude mult-relaxaton tme, whch we wll denote by MRT RLB. BAMPS was developed by Xu and Grener [10] and appled to the Remann problem n quark-gluon plasma by Bouras et al. [11]. Snce BAMPS solves the full relatvstc Boltzmann equaton, we take ts result as a reference to access the accuracy of our model. However, we keep n mnd that BAMPS also produces approxmate solutons. The present model s hereafter denoted by RLBD (RLB wth Dsspaton. For small ratos η/s, where s s the entropy densty, RLB and MRT RLB reproduced BAMPS results to a satsfactory degree of accuracy. However, for hgher η/s 0.1 and moderately fast fluds, γ 1.3, RLB faled to reproduce the velocty and pressure profles [12]. MRT RLB yelded good agreement wth the results at η/s = 0.1, but presented notable dscrepances for η/s = 0.5. The falure of both RLB and MRT RLB to solve the Remann problem for hgh vscous fluds can be ascrbed to ther nablty to recover the thrd order moment of the dstrbuton [12, 17]. In ths secton, we wll study the case of hgh η/s 0.1 n a regme of moderate veloctes. We perform the smulatons on a lattce wth cells, only half of whch are represented n our doman owng to symmetry condton (the other half s a mrror, n order to use perodc boundary condtons for smplcty. Therefore, our smulaton conssts of lattce stes, wth δx = 0.008fm and δt = fm/c for RLBD thrd order, and δt = 0.024fm/c for RLBD second order. The ntal condtons for the pressure are P 0 = 5.43 GeV fm and P 3 1 = GeV fm. In numercal unts, they 3 correspond to 1.0 and 0.062, respectvely. The ntal temperature z 0 s T 1 = 200MeV (n numercal unts 0.5, and T 0 = 400MeV for z < 0, whch corresponds to 1.0 n numercal unts. The entropy densty s s calculated accordng to the relaton, s = 4n n ln(n/n eq,

8 BAMPS RLBD 3rd Order RLBD 2nd Order RLB MRT RLB BAMPS T = 1.0 T = 2.5 T = 2.8 v/c v/c z (fm BAMPS RLBD 3rd Order RLBD 2nd Order RLB MRT RLB z (fm FIG. 4: Velocty profle as functon of the z-coordnate for the case of a shockwave n quark-gluon plasma, wth η/s = 0.5, by ncreasng the reference numercal temperature n the lattce, leadng to a smaller relaxaton tme τ. 0.1 P/P z (fm FIG. 3: Velocty (top and pressure (bottom profles as functon of the z-coordnate for the case of a shockwave n quarkgluon plasma, wth η/s = 0.5. where n eq s the densty calculated wth the equlbrum dstrbuton, n eq = d G T 3 /π 2, wth d G = 16 beng the degeneracy of the gluons. The velocty and pressure profles at t = 3.2 fm c wth vscosty-entropy densty ratos of η/s = 0.1, are shown n Fg. 2. In ths fgure, we compare the results wth BAMPS and RLB, where we can see that RLB presents a dscontnuty at z = 0, whle both second order and thrd order RLBD get closer to the BAMPS soluton. Snce the only dfference between second and thrd order RLBD s the thrd order moment of the dstrbuton, we conclude that at relatvely low η/s, the thrd order does not play a crucal role nether n the conservatve dynamcs nor dsspatve dynamcs of the system. However, note that at z 3fm, the thrd order model provdes an outstandng ft of the numercal results by BAMPS. On the other hand, by ncreasng the rato η/s, we see from Fg. 3 that, whle RLB gets worse and the second order RLBD fxes the dscrepancy only n part, the 3rd order RLBD mproves sgnfcantly the accuracy of the velocty and pressure profles. In Fg. 3, we also compare the results obtaned wth MRT RLB and BAMPS, for η/s = 0.5. Here, we see that there s agan an mprovement, ncludng the attanment of the rght value of the maxmum velocty (at z 1.5fm. In the pressure profle, RLBD gets closer to BAMPS than MRT RLB n the regon of the dscontnuty n the ntal condton (z 0. Note that there s a starcase shape n the results of RLBD for η/s = 0.5 n Fgs. 3. Ths s due to the large values taken by the sngle relaxaton tme n order to acheve such shear vscosty-entropy densty ratos, τ (n numercal unts, whch s beyond the hydrodynamc approxmaton and therefore hgher order moments (fourth and hgher orders of the dstrbuton functon would be requred, whch s not fulflled n our RLBD model. In order to prove ths statement, we have performed separate smulatons, see Fg. 4, where we observe that by ncreasng the value of the reference temperature of the lattce (typcally set at T = 1, so as to acheve the same shear vscosty, η = (2/3nT (τ 1/2/R, the value of τ decreases and the starcase dsappears. In partcular, for T 2.5, the results get closer to the ones wth BAMPS, and become ndependent of the reference temperature. Unfortunately, due the dscretzaton procedure used to develop ths model, whenever the reference temperature T > 4 the model becomes unstable, mostly lkely because the expanded equlbrum dstrbuton functon takes negatve values. IV. CONCLUSIONS AND DISCUSSIONS We have ntroduced a new relatvstc lattce Boltzmann model wth mproved dsspaton, as compared to RLB and MRT RLB. To ths purpose, we have performed an expanson of the Maxwell-Jüttner dstrbuton onto an orthonormal bass of polynomals n the 4-momentum space. In addton, n order to make the

9 9 model compatble wth a regular cubc lattce, we have performed the expanson n cartesan coordnates and appled a tme transformaton, such that partcles travel just the dstance necessary to reach lattce nodes, always at the speed of lght. The tme transformaton generates a sphere of radus R whch ntersects the cubc lattce, the ntersecton ponts beng lattce nodes by constructon. In addton, we have reproduced up to second order moment of the equlbrum dstrbuton, and up to thrd order moment, fndng R = 3 and R = 41 for second and thrd order moment compatblty, respectvely. The dscrete energy component of the 4-momentum, p 0, has been calculated by usng Gaussan quadrature, the nodes correspondng to the zeros of the next order polynomal. Wth ths confguraton, we need 90 vectors for recoverng second order and 384 for the thrd order moment case. However, only 66 and 128, respectvely, are actually needed to calculate the moments correctly. In order to valdate the model, we have compared our results wth BAMPS, as well as prevous RLB models. We have found that for η/s = 0.1, our model accurately descrbes the Remann problem n quark-gluon plasma, ncludng the expanson up to second order. However, for the case of η/s = 0.5, the second order model, although better than RLB, s less accurate than both MRT RLB and the thrd order model. The thrd order model yelds better results than the prevous RLB, but t develops a starcase shape as a consequence of the large value of the sngle relaxaton tme, whch les beyond the hydrodynamc regme. We have shown that the starcase pathology can be tamed by ncreasng the reference temperature n the model. Nevertheless, ncreasng the reference temperature beyond T = 4 hts aganst stablty lmts of the model. We may envsage that a mult-relaxaton tme extenson of the present model would further mprove the accuracy of the results. A smlar mprovement may be antcpated by mplementng hgher order expansons of the equlbrum dstrbuton. However, snce the transport coeffcents depend on the collson operator, ther calculaton wthn a mult-relaxaton tme model becomes ncreasngly nvolved. On the other hand, by performng expansons to nclude hgher order moments, the value of R mght become unpractcally large, wth several ensung dscretzaton ssues. Notwthstandng such potental dffcultes, these extensons are surely worth beng analyzed n depth for the future. Acknowledgments We acknowledge fnancal support from the uropean Research Councl (RC Advanced Grant FlowCCS. Work of I.V.K. was supported by the RC Advanced Grant LBM. FIG. 5: Drectons of the velocty vectors ϑ to recover up to the second order moment of the Maxwell-Jüttner dstrbuton, namely the momentum-energy tensor. The radus of the sphere s R = 3. The ponts represent lattce stes belongng to the surface of the sphere. Appendx A: Second order relatvstc lattce Boltzmann model To construct the second order lattce Boltzmann model, we use the procedure descrbed n ths paper. We have obtaned that R = 3 presents enough symmetres to fulfll the condtons n qs. (22, and the velocty vectors ϑ are gven by, (±3, 0, 0, (0, ±3, 0, (0, 0, ±3, (±2, ±1, ±2, (±1, ±2, ±2, and (±2, ±2, ±1. The values for the dscrete p 0 come from the soluton of the equaton, R (3 = 1 12 p0 (p = 0, (A1 nstead of R (4 for the case of the thrd order expanson. Ths gves the values p 0 l 0.936, 3.305, and The dscrete 4-momentum vectors p µ are constructed wth qs. (17, and (21, and they are n total, N = 3 30 = 90. However, as n the thrd order expanson, we have retaned the mnmal amount, out of 90, that are necessary to recover the second order moment, by mposng the maxmum number of w to be zero. Ths gves only 66 4-momentum vectors. The value of the weght functons for every momentum vector and the relaton wth the 30 drectons are gven n the Supplementary Materal [21]. In Fg. 5 we report the spatal confguraton of the vectors ϑ. The dscrete verson of the relatvstc Boltzmann equaton, q. (38, stll apples and the dscrete equlbrum dstrbuton functon s wrtten n detal n Appendx B,

10 10 q. (B1. However, due to the fact that the thrd order moment s not satsfed, an analytcal theory to calculate the transport coeffcents would be very complcated and goes beyond the scope of ths work. Therefore, we have calculated numercally only the shear vscosty, by matchng the results for low velocty wth the thrd order moment model. Ths, n order to compare the results of both expansons wth other models n the lterature. Ths gves a shear vscosty η 2nd (1/7P (τ δt/2/r. We could, n prncple, calculate the thrd order moment assocated wth the equlbrum dstrbuton gven by q. (B1, and, by applyng the Grad method, compute the other transport coeffcents. However, ths procedure would need to be performed entrely numercally, snce the weghts w and 4-momentum vectors p µ are only known numercally. Snce the man purpose of ths paper s to mprove the descrpton of dsspatve effects by performng the thrd order expanson and place t on a cubc lattce, we are not nterested n the bulk vscosty and the thermal conductvty for ths case, and leave ths task for future work. Appendx B: qulbrum Dstrbuton Functons The equlbrum dstrbuton functon capable to recover the frst and second order moments of the equlbrum dstrbuton s calculated by usng up to the second order polynomals n q. (7, namely the 14 polynomals J k wth k = 0,..., 13, obtanng = nw [ ( ( p 0 2 T 2 2U 02 U x2 U y2 1 2T U p 0 (T (T (U 0 (p x U x + p y 4T U y + p z U z 4U p x U x p y U y p z U z + 7U T (p ( 2 x 2 U U x2 + U y p x U x (p y U y + p z U z 4U 0 ( + p y 2 U 02 + U x2 + 2U y p y U y (p z U z 4U 0 + 8U 0 (U 0 p z U z 2 ] + 2T (5(p x U x + p y U y + p z U z 8U , (B1 For the case of the thrd order moment expanson, we repeat the same procedure, usng all the polynomals (k = 0,..., 29. Ths leads to the followng expressons:

11 11 = nw [ p 0 3 (T U T ( p 0 2 (T 3 ( ( ( T 2 4U 02 3 U x2 + U y T U ( 2U 02 (3p x U x + 3p y U y + 2p z U z + U x2 + U y2 + 1 (3p x U x + 3p y U y + p z U z ( + 36U 03 6U 0 3U x2 + 3U y2 + 4 ( ( + 3T 2 2U 0 (p x U x + p y U y + p z U z 22U U x2 + U y T (p x U x + p y U y + p z U z 14U p 0 ( (T ( ( 3 U 03 p x2 + p y2 24 U 0 p x ( 2 2U x2 + U y2 + 1 ( + 2p x U x (p y U y + p z U z + p y p y U x2 + 2p y U y2 + p y + 2pz U y U z U 02 (p x U x + p y U y + p z U z ( ( 2(p x U x + p y U y + p z U z + T 2 p x 2 U U x2 + U y p x U x (p y U y + p z U z 11U 0 ( + p y 2 U 02 + U x2 + 2U y p y U y (p z U z 11U 0 22p z U 0 U z + 56U ( ( + 2T (6(p x U x + p y U y + p z U z 25U T p x3 T 2 U x 3U U x2 + 3U y2 + 3 ( ( ( + p x2 T 3 U 02 2U x2 U y2 1 ( p y T U y + 6T U p z T U z U U x2 + U y2 + 1 ( ( + 3p x U (T ( x T p y 2 U 02 + U x2 + 2U y p y U y (p z U z 6U 0 12p z U 0 U z + 24U 02 4 ( + 14p y U y + 14p z U z 48U p y3 T 2 U y 3U U x2 + 4U y2 + 3 ( ( ( + p y2 T p z T U z U 02 + U x2 + 4U y (6T U 0 7 U 02 U x2 2U y2 1 ( ( ( + 6p y U y T 2T 3p z U 0 U z + 6U p z U z 24U 0 ( ( p z U z 2T 2 6U 02 1 ( ( 24U 0 T 2 2U T U ( T 2 2 ]. 24T U (B2 [1] K. S. Novoselov, A. K. Gem, S. V. Morozov, D. Jang, Y. Zhang, S. V. Dubonos, I. V. Grgoreva, and A. A. Frsov, Scence 306, 666 (2004. [2] K. Novoselov, A. Gem, S. Morozov, D. Jang, M. Katsnelson, I. Grgoreva, and S. Dubonos, Nature Letters 438, 197 (2005. [3] M. Müller, J. Schmalan, and L. Frtz, Phys. Rev. Lett. 103, (2009. [4]. Shuryak, Progress n Partcle and Nuclear Physcs 53, 273 (2004, ISSN , heavy Ion Reacton from Nuclear to Quark Matter. [5] P. K. Kovtun, D. T. Son, and A. O. Starnets, Phys. Rev. Lett. 94, (2005. [6] G. Polcastro, D. T. Son, and A. O. Starnets, Phys. Rev. Lett. 87, (2001. [7] M. Mendoza, B. M. Boghosan, H. J. Herrmann, and S. Succ, Phys. Rev. Lett. 105, (2010. [8] M. Mendoza, B. M. Boghosan, H. J. Herrmann, and S. Succ, Phys. Rev. D 82, (2010. [9] R. Benz, S. Succ, and Vergassola, Phys. Rep. 222, 145 (1992. [10] Z. Xu and C. Grener, Phys. Rev. C 71, (2005. [11] I. Bouras,. Molnar, H. Nem, Z. Xu, A. l, O. Fochler, C. Grener, and D. H. Rschke, Phys. Rev. Lett. 103, (2009. [12] D. Hupp, M. Mendoza, I. Bouras, S. Succ, and H. J. Herrmann, Phys. Rev. D 84, (2011, URL http: //lnk.aps.org/do/ /physrevd [13] P. Romatschke, M. Mendoza, and S. Succ, Phys. Rev. C 84, (2011, URL /PhysRevC [14] C. Cercgnan and G. M. Kremer, The Relatvstc Boltzmann quaton: Theory and Applcatons (Boston; Basel; Berln: Brkhauser, [15] X. He and L.-S. Luo, Phys. Rev. 56, 6811 (1997. [16] N. S. Martys, X. Shan, and H. Chen, Phys. Rev. 58, 6855 (1998, URL PhysRev

12 12 [17] Q. L, K. H. Luo, and X. J. L, Phys. Rev. D 86, (2012, URL /PhysRevD [18] S. S. Chkatamarla and I. V. Karln, Phys. Rev. Lett. 97, (2006. [19] S. S. Chkatamarla and I. V. Karln, Phys. Rev. 79, (2009. [20] J. Anderson and H. Wttng, Physca 74, 466 (1974. [21] See Supplementary Materal at.