We report preliminary measurements of the velocity autocorrelation function for a tagged particle in a lattice gas. These measurements agree with the Boltzmann-level theory. The Green-Kubo integration of these measurements agrees with theoretical predictions for the diffusion coefficient. To within the error bars of the simulations (3 /times/ 10/sup /minus/3/) we observe no long-time tails. 9 refs., 1 fig., 1 tab

Binder, P.M.

d'Humieres, D.

Poujol, L.

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calthebleA major purpose of the Techni-Information Center is to providebroadest dissemination possi-of information contained inDOE’s Research and DevelopmentReports to business, inckstry, theacademic community, and federal,state and local governments.Although a small portion of thisreport is not reproducible, it isbeing made available to expeditethe availability of information on theresearch discussed herein.1LA-IJR -89-3681..LOS Alamos Nal!onSl Laboratory IS Ooeraled by !hO Uruverslry o! Cwlof n!a !01 the UnlWd Stmes @PWtMOnl 0! Enafgy under conlracl W. 7405 -ErvG.36LA-uR--88-3681DE89 003475TITLE THE PROPERTIES OF TAGGED LATTICE FLUIDS:II. VELOCITY CORRELATION FUNCTIONSAUTHOR(S) P. M. Binder, D. d’Humieres and L. PoujolSU9MITTED TO Proceedings of the Workshop on Discrete Kinetic Theory, 1S1,Torino, Italy, September 1988 (World-Scientific) Ed. R. MonacoDISCLAIMERLos~$[qAkamosLos Alamos National LaboratoryLos Alamos,New Mexico 87545THE PROPERTIES OF TAGGED LATTICE FLUIDS:II. VELOCITY CORRELATION FUNCTIONSP.-M. Binderl’2, D.(~’Hurni&r=2 and L. Poujo12lCenter for Nmlinear Studies, hfS-B258,Los Alamcm, NM 87545, USA2L&mAoire de Physique de l’Ecole Normale Superieure,75231 Paru, FranceABSTRACTWe report prelim.inxy messurementa of the velocity autocorrelat ionfunction for a tagged particle in ● lattice gas. These m-urernentsagre with the Boltzman.n-bvel t hmry. The Green- Kubo integration ofthese measurermmts agraY with theoretkd predictions for the diflueioncoefficient. To within the error bam of the simulations (3x 10-3, weobserve no long-time ttila.1AS the theory of lattice gases develops, several measurements ot’ velocityautocorrelation functions, which w e an essential ingredient in the (Jreen- Kuboformulae for transport coefficients[’1, have been reported in the literature[2-41,The standard lattice gaa models in the square[21 and triangular [s: lattices,3enceforth HPP and FHP respectively, have collision rides which do not con-serve particle identity, For this reason, the velocity correlation functions thatColvin et al.[31 and Margolus et al.[’l report for lattice gases correspond tohydrodynamic or site correlations rather than to the oneparticle correlationfunctions needed in the Green-Kubo formulation. Both kinds of correlationusually have algebraic long-tirnc tails, the first through the dtxay of singluhydrodynamic modes, the second through the product of modes. Often thealgebraic exponent is the same for both, although the amplitude of the latteris considerably smaller; it hab been conjectured[ql that isotropy of the viscositytensor is a necessary condition for the equality in the exponents of both ~~i-relation functions 1, In this paper we report preliminary measurements of theone-particle velocity correlation funckwn for a lattice g~ model which we havemodified to conserve the individuality of one particie[”sl. This model is inter-esting in that the viscosity tensor is not isotropic 2, so that the long-time tails,if they exist, need not be the same aa those reported for the site correlationfunctionm in referencu 2 and 3,The model we study here ia model III of the previous paper in this volume,id est, the HPP model with one tqgged p~ticle which ?Mects left or rightat tvmbmly hcwi-cm collisions according to the parity of the time step, ‘rhisparticle is representative of any fluid particle in the system. For tl~ig model,1The logarithmic divergence measured in the vimmmty of hexagonal Iatticc gns+s and~ccompanying mode-coupling theory by Ku.ianoff, Mc Nanmra and Zanettl, (Inlvermty of( ‘hicsgo preprint ( 1987), are consistent with this ststement,7%w the article on invsrianto in Istti:e gases by L) d’tiurrdrm ●nd P Lallcmand in thisvolume‘).the Boit zmann level description of reference 8 appears to be adequate at alldensities for the calculation of the self-diffusion coefficient.We have performed simulations of this model for densities of 0,125, 0.25,0.50 and 0.75 of the maximum of 4 particles per node. 80000 independenttrajectories were run for each density value, in the 256 x 512-node periodicfield provided by the RAP lattice gas machine. To account for some of theinvariant of the system and guarantee no drift, we forced the momentum alongeach row and column to be exactly zero. No such provisions were taken towardscheckerboard or diagonal invariant such as those described by d’Humieres andLallemand in this volumeBased on the Zwanzig-Ailawadi estimateigl, this number of trajectories isenough for an ●ccurscy of me part in 200 in the velocity autocorrelationfunction. To this accuracy we could check that it obeys the equationqt) = [1 - C(l - C)Z]’ (1)presented in the previous paper in this volume, for about 24 time steps. Table1 contains the best fitt for [1 - C(1 – C)a] from simulations and the diffusioncoefficient value that results from the Green-Kubo integration of this value.The last column hu the Boltzmann-level result for the diffusion coefficient,which can dso be found in the previous paper. We observe excellent agreement,.,.bet ween theory ●nd simulations. The Gre&-Kubo results also agree very wellwith the Einstein equation measurements of the c!iffusinn coe!%cient report~in the preceding paper.Table 1: Density, bat exponential fit of the velocity correlation function,values of the diffusion coefficient from experimental Green-Kubo and themet-icrd methods.Density [i - C(l - C)2] D(Green-Kubo) D( Boltzmann )1/8 0.9077 5.17 +0.3 4.95 —1/4 0.8682 ‘3.313.54 +0.2 ,l/~ 0.8826 4.0 * 0.2 3.83/4 I0.9545 I 10.7 k 0.6 I 10,5Figure 1 is A plot of the logarithm of th~ measured velocity correlation ftmc-tion vs time for a particle density of 0.6. This density seems to provide thebest tradeoff betwem low enough density for the Boltzmann description to beaccurate, and high enough density to have meaningful results for at le~t afew meu-free paths. We have performed 22!5000 independent simulations forthis density, The exponent ial decay law inbeyed to within the error bars of0.003, with no evidence of long-time tails. It will be instructive for the readerto compare this plot with those for the site correlation function for the samemodel, given in references 2 and 4.We have presented preliminary measurements of the particle velocity au-tocorrelation iunction of a tagged version of the HPP m,odel. We obtain verygood agreement with theoretical results. We compare the Green- Kubo inte-gration of these measurement with the Bcdtzmann,- level predictions for thediffusion coefficient and obtain excellent agreement. We observe no long-timetails to the level of ●ccuracy of the simulations, which comes as no surpriucsince these tails are of the order 10””3- 10-4,We thank M. H. Ernst, Y. Pcmwau, D, Frenkel and G. Vichniac for useful dis-cussions, and CNRS, the U.S. Department of Energy, AFOSR, and the orga-nizers of the 1S1 Workshop on Discrete Kinetic Theory for financi !l support.4REFERENCES1. J. P. Rivet, Ph.D. thesis, University de Nice (1988)2. J. Hardy, O. de Pazzis and Y. Pomeau, J. Math. Phys. 14, 1746 (1973),J. Hardy, Y. Pomeau and 0. de Pazzis, Phys. Rev. Au, 1949 (1976)3. M. Calvin, A. J. C.Ladd and B. Alder, Phys.Rev.Lett. Q, 381 (1988)4. N. Margrdus, T. Toffoli and G. Vichniac, Phys.Rev.Lett. 5~,1694 ( 1986)5. U. Frisch, B. Haaslacher and Y. Porneau, Phys.Rev.Lett. 5&,1505 (1986)6. D. Bemardin, P. Binder, D.Dab, M. Ernst, I.1’Heureux, R. Kapral, X. Kong,M .Mareschal, A. Masters, J .P Rivet and G. van Velzen, “Loug-t ime tails inlattice g-”, report of the CECAM workshop on Cellular Automata andBrownian Motion (1988)7. P. M. Binder and D.d’Humi&eri, “Self-diffusion in a taggd-part icle latticegati”, Loa Alamoe preprint LA-UR-88-1341 (1988)8. P. M. Binder, ‘Properties of tagged .atiice fluids: I. Diffusion coefficients”,previous article in this voiume9. R. Zwanzig and N. K. Ailawadi, Phys. Rev, l& 280 (1969)5Figure 1: Natural logarithm of the particle velocity autocorrelation func-tion v-. time for a tagged-particle HPP gas, density =O.6, 225000 simulations..t

The properties of tagged lattice fluids: II. Velocity correlation functions

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The properties of tagged lattice fluids: II. Velocity correlation functions

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