Transport and Helfand moments in the Lennard-Jones fluid. I. Shear viscosity

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1 Journal of Chemical Phyic 16 (7) (7 page) Tranport and Helfand moment in the Lennard-Jone fluid. I. Shear vicoity S. Vicardy, J. Servantie, and P. Gapard Center for Nonlinear Phenomena and Complex Sytem, Univerité Libre de Bruxelle, Campu Plaine, Code Potal 31, B-15 Bruel, Belgium We propoe a new method, the Helfand-moment method, to compute the hear vicoity by equilibrium molecular dynamic in periodic ytem. In thi method, the hear vicoity i written a an Eintein-like relation in term of the variance of the o-called Helfand moment. Thi quantity, i modified in order to atify ytem with periodic boundary condition uually conidered in molecular dynamic. We calculate the hear vicoity in the Lennard-Jone fluid near the triple point thank to thi new technique. We how that the reult of the Helfand-moment method are in excellent agreement with the reult of the tandard Green-Kubo method. PACS number:.7.n; 5.6.-k; 5..Dd I. INTRODUCTION Since Maxwell firt paper [1, ] on the kinetic theory of gae, hear vicoity a well a the other tranport propertie are known to find their origin in the microcopic motion of atom and molecule compoing matter. However, it i only ince the fiftie that exact formula are known to calculate the tranport coefficient in term of the microcopic dynamic. Thee o-called Green-Kubo formula give each tranport coefficient a the time integral of the autocorrelation function of ome pecific microcopic flux aociated with the tranport property of interet [3 6]. Today, the Green-Kubo technique allow u to calculate numerically the tranport coefficient by imulating the molecular dynamic of ytem with a finite number of particle and periodic boundary condition. On the other hand, Eintein claic work on Brownian motion howed that tranport propertie uch a diffuion can alo be undertood in term of random walk. It wa Helfand [7] who identified in 196 the fluctuating quantitie which, by their random walk, are aociated with each tranport coefficient. Thee fluctuating quantitie are the o-called Helfand moment and are the centroid of the conerved quantity which i tranported. In principle, each tranport coefficient can thu be obtained from the linear increae of the tatitical variance of the correponding Helfand moment by the o-called generalized Eintein relation. Neverthele, it i not yet known today how thee Helfand moment hould be defined in molecular dynamic with periodic boundary condition, which it their ue in numerical imulation. The purpoe of the preent paper i to derive an analytical expreion of the Helfand moment aociated with vicoity for molecular dynamic with periodic boundary condition, and to apply the Helfand-moment method to the calculation of hear vicoity in the Lennard-Jone fluid near the triple point. Already in the firt numerical calculation of vicoity in 197 [8], the algorithm of Alder et al. wa baed on generalized Eintein relation derived from the Green-Kubo formula. The application of the pure Green-Kubo technique by equilibrium molecular dynamic to the Lennard- Jone fluid ha been performed a hort time after by Leveque et al [9] and later by Schoen and Hoheiel [1]. However, until the middle of the eightie and the work of Schoen and Hoheiel [1], a well a the one of Erpenpeck uing the Monte-Carlo Metropoli method [11], nonequilibrium molecular dynamic wa predominantly ued for the computation of hear vicoity [1 16]. At the end of the eightie and the beginning of the ninetie, the generalized Eintein relation tarted to be ued for calculating the tranport coefficient. An alternative equilibrium molecular dynamic method ha been propoed in which the variance of the time integral of the microcopic flux i calculated [17 19]. Actually it i the analog of the method of Alder et al. [8] for oft phere potential ytem. Recently, thi technique ha been applied by Meier et al. [], and by He and Evan [1], the latter having rather conidered an equilibrium enemble of time average of the flux. In thi context, two important point were dicued. The firt concern the o-called McQuarrie expreion. In hi book [], McQuarrie preented Helfand formula for the hear vicoity, but with a lightly different form. Thi difference implied a implification of the expreion, apparently giving an important advantage compared to the original formula [3 6]. The econd point concerned the validity of the generalized Eintein relation in periodic ytem [17, 18, 5, 7]. Arguing that the periodic boundary condition imply that the variance of the original expreion of the Helfand moment i bounded in time, the generalized Eintein relation were conidered to be impractical in periodic ytem. In thi paper, we will how that a generalized Eintein relation i available for vicoity after the addition of two term to the original

2 Helfand moment to take into account the periodicity of the ytem. The Helfand-moment method preent the important advantage to define hear vicoity a a nonnegative quantity, atifying the poitivity of the entropy production. We have previouly applied uch a method to a ytem of two hard dik with periodic boundary condition [8]. In the preent paper, we calculate the hear vicoity in a Lennard-Jone fluid near the triple point. We compare the reult obtained by the Helfandmoment method with our own Green-Kubo value and thoe found in the literature. In addition, the Helfand-moment method play an important role in the ecape-rate formalim. Thi formalim etablihe direct relationhip between the characteritic quantitie of the microcopic chao (Lyapunov exponent and fractal dimenion) and the tranport coefficient [9 3]. A few year ago, uch a relation ha been tudied for the vicoity in the two-hard-dik model [33]. Furthermore with the ue of the Helfand moment, it hould be poible to contruct at the microcopic level the hydrodynamic mode, which are the olution of the Navier-Stoke equation. Thi approach called the hydrodynamic-mode method ha been uccefully applied for diffuion [34]. The paper i organized a follow. In Section II, we outline the theoretical background of the generalized Eintein formula. Section III i devoted to the preentation of our Helfand-moment method ued for the calculation of the hear vicoity in thi paper. In Section IV, we dicu the o-called McQuarrie expreion and the validity of the generalized Eintein formula for periodic ytem. The reult of the molecular dynamic imulation are given in Section V. The comparion with our Green- Kubo reult and previou reearche are done. Finally, concluion are drawn in Section VI. II. EINSTEIN-HELFAND FORMULA One century ago, Eintein theoretically etablihed a relationhip between the diffuion coefficient of Brownian motion and the random walk of the Brownian particle due to it colliion with the molecule of the urrounding fluid [35]: 1 D = t t [x(t) x()], (1) where x i the poition of the colloidal particle and t the time. Thereafter, different tudie led to the well-known Green-Kubo formula for the hear vicoity η obtained by Green [3, 4], Kubo [5] and Mori [6]: η = N,V 1 k B T V J (η) (t)j (η) (), () where J (η) i the microcopic flux aociated with the hear vicoity η. The vicoity coefficient i obtained in the thermodynamic it where N, V while the particle denity n = N/V remain contant. In the following, thi condition i alway aumed together with the it N, V. The implicity of Eq. (1) obtained by Eintein preent a particular interet. The extenion of uch a relation to the other tranport coefficient could be ueful. In the ixtie, Helfand propoed quantitie aociated with the different tranport procee in order to etablih Eintein-like relation for the tranport coefficient [7]. In particular, we have for hear vicoity that η = 1 [ G (η) (t) G ()] (η). (3) k B T V t It can be hown [7] that the Eintein-Helfand relation i equivalent to the Green-Kubo formula () by defining the Helfand moment a: G (η) (t) = N p xa (t)y a (t), (4) if the correponding microcopic flux i the time derivative of the Helfand moment: J (η) (t) = d G(η) (t). (5) We notice that the it t hould be related to the thermodynamic it N, V. Indeed, for a fluid of particle confined in a finite box, the quantity (4) i bounded o that the coefficient (3) would vanih if the it t wa taken before the thermodynamic it N, V. Therefore, the number N of particle and the volume V hould be large enough in order that the variance of the Helfand moment diplay a linear increae over a ufficiently long time interval t, allowing the coefficient η to be well defined. The larger the ytem, the longer the time interval. It i in thi ene that the it N, V, t hould be conidered in Eq. (3). Another remark i that, compared with the Green- Kubo formula (), Eq. (3) preent the advantage to define the hear vicoity a a poitive quantity, atifying the condition of non-negative entropy production. III. HELFAND MOMENT IN PERIODIC SYSTEMS Often, the molecular dynamic i imulated with periodic boundary condition. In thi cae, the particle exiting at one boundary are reinjected at the oppoite boundary. Due to the periodicity of the ytem, particle in the imulation box may interact with image particle a well a the particle inide the original unit cell. A a conequence, the image of particle b may contribute to the force F ab applied by the particle b on the particle a: F ab = β (a,b) u(r ab ) r ab (6)

3 3 with r ab = r a r b L β (a,b) (7) where L the length of the imulation box and β (a,b) determine the cell tranlation vector [18]. The range of the interaction potential mut be maller than L/ to guarantee that the particle a interact only with one of the image of b in Eq. (6). The interacting pair i found by the minimum-image convention, r a r b L β (a,b) < L/. Here, we define the quantity L b a (t) = L β (a,b) (t) (8) which i the vector to be added to r b in order to atify the minimum-image convention. For a dynamic which i periodic in the box, the poition hould jump to atify the minimum-image convention. A a conequence, the poition and momenta ued to calculate the vicoity by the Green-Kubo method actually obey the modified Newtonian equation dr a dp a = p a m + = b( a) r () a δ(t t ), F(r ab ), (9) where r () a i the jump of the particle a at time t with r () a = L. We notice that the modified Newtonian equation (9) conerve energy, total momentum and preerve phae-pace volume (Liouville theorem). Moreover, we ee that the periodic boundary condition imply that the Helfand moment of Eq. (4) i bounded and cannot be differentiated near the time t of the jump. In order to have a well-defined quantity, one hould remove the dicontinuitie at the jump, o that the Helfand moment can grow without bound. In order to do that, we add a term I(t) to the original Helfand moment (4) to get G (η) (t) = a p ax (t)y a (t) + I(t). (1) According to Eq. (5), the time derivative of the Helfand moment mut be the microcopic flux J (η) (t) = a p ax p ay m + 1 F x (r ab ) y ab, (11) defined with the poition y ab of the minimum-image convention. In order to atify the equality (5) in periodic ytem, we how in Appendix A that the term I(t) mut be given by I(t) = a 1 p () ax y () a θ(t t ) dτ F x (r ab ) L b ay. (1) where both r ab and L b ay depend on the time τ in the integral of the lat term. We then obtain our general expreion for the Helfand moment in ytem with periodic boundary condition: G (η) (t) = N p ax (t) y a (t) a 1 p () ax y () a θ(t t ) dτ F x (r ab ) L b ay (13) where p () ai = p ai (t ) i the momentum at the time of the jump t and θ(t t ) the Heaviide tep function defined a { 1 for t > t, θ(t t ) = (14) for t < t. We notice that the quantity L b ay ha dicontinuou jump in order to atify the minimum-image convention. Let u point out that L b ay change when the force F x (r ab ) vanihe, o that the lat term varie continuouly in time and doe not preent any jump. We notice that the lat two term of Eq. (13) involve the particle near the boundarie of the box. The econd term i due to the jump of the particle to or from the neighboring boxe. The third term concern the pair of particle interacting between neighboring cell. The Helfand moment of Eq. (13) can be ued to obtain the hear vicoity coefficient for ytem with periodic boundarie thank to the Eintein-like relation (3). IV. DISCUSSION Since the beginning of the ninetie, ome confuion have been propagated in the literature concerning the ue of the mean-quared diplacement equation for hear vicoity. Firt, it concern the o-called McQuarrie expreion. On the other hand, everal work have been done which have prematurely concluded that the meanquare diplacement equation for hear vicoity i inapplicable for ytem with periodic boundary condition. Thee confuion and criticim are reported in particular by Erpenbeck in Ref. [7]. Since thee quetion are central in thi paper, thi ection i devoted to uch problem in order to avoid any miconception. A. McQuarrie expreion for hear vicoity In hi well-known and remarkable book Statitical Mechanic, McQuarrie [] reported the work achieved by Helfand [7]. The derivation he propoed i different but he obtained the ame intermediate relation a Helfand,

4 that i [36] η = N 1 [x a (t) x b ()] p ay (t)p by (). k B T V t a,b=1 (15) Thereafter in hi book, McQuarrie let a an exercie the derivation from Eq. (15) of the final expreion which i printed in Ref. [] a follow η McQ = while Helfand obtained η H = 1 N k B T V t (16) [ N ] 1 x a (t)p ay (t) x a ()p ay (). k B T V t (17) The difference between both expreion i in the poition of the um over particle, and it eem that uch a difference i due to a typing error. Neverthele, the Mc- Quarrie expreion (16) at firt ight preent a certain advantage compared to Helfand one (17). Indeed, the um over the particle can come out of the average. Conequently, one would obtain a um of average no longer depending on the different particle. If Eq. (16) would hold, it could be rewritten a a ingle-particle expreion wherea the vicoity i a collective effect according to Helfand expreion (17). In the early ninetie, the validity of Eq. (16) wa invetigated [3 7], until Allen paper [6] in which it ha been concluded that Eq. (16) i not valid. Thi concluion wa later confirmed by Erpenbeck [7], which ettled the quetion. A aforementioned, thi hould not be urpriing ince Eq. (16) i likely the reult of a miprint. Thi dicuion emphaize the fact that vicoity i a collective tranport property, implying the intervention of all the particle. B. Periodic ytem and Helfand-moment method The other point which wa quetioned i whether a mean-quared diplacement equation for hear vicoity i ueful for ytem ubmitted to periodic boundary condition [17, 5, 7]. Firt, it wa pointed out that the well known Alder et al. method initially developed for hardball ytem [8] i not baed on the Helfand expreion, but intead on the mean-quare diplacement of the time integral of the microcopic flux [7]. The main doubt on the ue of Helfand moment in periodic ytem come from the fact that the original expreion (4) i bounded and would lead to a vanihing hear vicoity in the long-time it. By thi argument, Allen concluded that the only correct way to handle G xy (t) i to write it a Ġxy(τ) dτ, and expre Ġxy in pairwie, minimum-image form [17]. In other word, the Alder et al. method would be the only valid method for tudying vicoity, that i, through a method intermediate between the Helfand and Green-Kubo method. Let u mention that thi opinion wa recently followed by He, Kröger and Evan [1, 37] a well a by Meier, Laeecke and Kabelac [, 38] having conidered ytem with oft-potential interaction. However, thi doe not preclude the poibility to modify the original expreion (4) of the Helfand moment in order to recover the microcopic flux (11). Thi i preciely what we have done [x a (t)p ay (t) x a ()p ay ()] here above with our Helfand-moment method by adding the, following two term 1 N p () ax y a () θ(t t ) 4 dτ F x (r ab ) L b ay (18) to the original one. Albeit the firt original term i bounded in time, the two new term increae without bound in time becaue of the jump and the interaction between the particle and the image particle (due to the minimum-image convention). Therefore, they can contribute to the linear growth in time of the variance of the Helfand moment. The Helfand-moment method we propoe here i completely equivalent to the Green-Kubo formula and preent the advantage to expre the tranport coefficient by Eintein-like relation, directly howing their poitivity. V. NUMERICAL RESULTS We carried out molecular dynamic imulation to calculate the hear vicoity by the Helfand-moment and the Green-Kubo method. We ue the tandard 6-1 Lennard-Jone potential [ (σ ) 1 ( σ ) ] 6 u(r) = 4ɛ (19) r r We ue the reduced unit defined in Table I. Quantity Unit temperature T = k BT ɛ number denity n = nσ 3 time t = t p ɛ mσ ditance r = r σ hear vicoity η = η σ mɛ TABLE I: Reduced unit of the Lennard-Jone fluid. All the calculation we perform are done with the cutoff r c =.5σ. The equation of motion are integrated with the velocity Verlet algorithm [39] of time tep t =.3. The initial poition of the atom form a fcc lattice and the initial velocitie are given by a Maxwell-Boltzmann

5 5 ditribution. Thereafter, the ytem i equilibrated over time tep to reach thermodynamic equilibrium. After the equilibration tage, the production tage tart. At each time tep, the microcopic flux (11) and the Helfand moment (13) are calculated. Every 3 time unit (1 5 time tep), we compute the time autocorrelation function of the flux and the mean-quare diplacement of the Helfand moment for thi piece of trajectory and average them with the previou reult. Thank to thi method we can calculate with a very large tatitic ince we do not need to keep in memory the whole trajectory. Depending on the ize of the ytem (N = ), the number of piece of trajectory varie between and 6, hence the total number of time tep i between 1 8 and Statitical error i obtained from the mean-quare deviation of the correlation function or of the mean-quare diplacement on the trajectory piece. 4 vicoity η* /N FIG. : Vicoity at the phae point T =.7 and n =.844 a a function of the invere of the number N of atom. The circle are the reult of the numerical imulation and the dahed line the linear extrapolation. vicoity η* Green-Kubo Helfand time FIG. 1: Vicoity at the phae point T =.7 and n =.844 for N = 137. The plain line i the derivative of the mean-quare diplacement of the Helfand moment and the circle the integral of the microcopic flux autocorrelation function. We depict in Fig. 1 the time derivative of the meanquare diplacement of the Helfand moment (13) and the time integral of the autocorrelation function of the microcopic flux (11). In Fig. 1, the calculation i performed for N = 137 atom near the triple point at the reduced temperature T =.7 and denity n =.844. A we ee, the two method are in perfect agreement. We etimated the hear vicoity by a linear fit on the mean-quare diplacement of the Helfand moment. The fit i done in the region between 5 and 1 time unit to guarantee that the linear regime i reached. We depict in Fig. the hear vicoity veru the invere N 1 of the ytem ize. The linear extrapolation give the following etimate of hear vicoity for an infinite ytem, η = 3.91 ±.57 () Thi reult i in agreement with the previou work a reported in Table II. Indeed, the previou extrapolation found by Erpenbeck [11] η = ±.68, Palmer [4] η = 3.5 ±.8, and Meier et al. [] η = 3.58 ±.33 are all in agreement with our reult () within the tatitical error. VI. CONCLUSIONS In thi paper, we propoe a new method for the computation of hear vicoity by molecular dynamic. The Helfand-moment method i an adaptation of the Helfand formula (3) for ytem with periodic boundary condition by adding two term (1) to the original expreion of the Helfand moment (4). The method conit in the calculation of the mean-quare diplacement of the Helfand moment (13). The variance of thi quantity give the hear vicoity by the generalized Eintein relation (3). We have dicued it validity in the light of the dicuion found in the literature of the beginning of the ninetie. Thank to thi new method, we have computed the hear vicoity in the Lennard-Jone fluid near the triple point. We howed that the Helfandmoment method give the ame reult a the tandard Green-Kubo method. Moreover, our extrapolated value of hear vicoity i in tatitical agreement with thoe found in the literature. More than tating a an alternative method to the tandard Green-Kubo in equilibrium molecular dynamic, the Helfand-moment method i ueful and play a central role in the ecape-rate formalim and the hydrodynamic-mode method. Indeed, in thee theorie, the Helfand moment allow u to put in evidence fractal tructure at the microcopic level, which are related to the tranport procee [9 33]. We remark that the method can be imilarly extended to the bulk vicoity ζ. A for the hear vicoity, two term mut be added to the original expreion of the Helfand moment aociated with the bulk vicoity. For-

6 6 mally, thi lat coefficient i expreed a follow: ζ η = 1 k B T V t where it Helfand moment i defined a: G (ζ) (t) = a a 1 [ G (ζ) (t) G (ζ) (t) ], p ax (t) x a (t) p () ax x () a θ(t t ) (1) dτ F x (r ab ) L b ax, () and with G (ζ) () =. In the companion paper, we will preent a imilar method for the calculation of thermal conductivity [46]. Acknowledgment We thank K. Meier for ueful dicuion. Thi reearch i financially upported by the Communauté françaie de Belgique (contract Action de Recherche Concertée No. 4/9-31) and the National Fund for Scientific Reearch (F. N. R. S. Belgium, contract F. R. F. C. No ). APPENDIX A: DERIVATION OF THE HELFAND MOMENT FOR THE SHEAR VISCOSITY IN PERIODIC SYSTEMS By taking the time derivative of the Helfand moment (1), we have: where we have ued the modified Newton equation (9). The term implying the interparticle force F(r ab ) may be modified into F x (r ab )y a (t) = 1 F x (r ab )y a + 1 F x (r ba )y b. (A) Since the force F i central, we obtain F x (r ab ) = F x (r ba ), which implie that F x (r ab )y a (t) = 1 F x (r ab ) (y a y b ). (A3) which till differ from the correponding term appearing in the microcopic flux (11) defined with the minimumimage convention becaue y a y b = y ab +L b ay according to Eq. (7) and (8). Conequently, Eq. (A1) become dg (η) (t) = J (η) (t) a a,b a F x (r ab )L b ay p ax (t) y a () δ(t t ) + di(t). Comparing with Eq. (5), we hould have di(t) = p ax (t) y a () δ(t t ) a 1 F x (r ab )L b ay a,b a whereupon I(t) can be expreed a (A4) (A5) dg (η) (t) = p ax (t)p ay (t) m a + p ax (t) y a () δ(t t ) a + F x (r ab )y a (t) + di(t) (A1) I(t) = a 1 a,b a p () ax y () a θ(t t ) dτ F x (r ab ) L b ay. (A6) [1] J. Maxwell, Phil. Mag. 19, 19 (186). [] S. Vicardy, E-print: cond-mat/611. [3] M. S. Green, J. Chem. Phy. 19, 136 (1951). [4] M. S. Green, Phy. Rev. 119, 89 (196). [5] R. Kubo, J. Phy. Soc. Jpn. 1, 57 (1957). [6] H. Mori, Phy. Rev. 11, 189 (1958). [7] E. Helfand, Phy. Rev. 119, 1 (196). [8] B. J. Alder, D. M. Ga, and T. E. Wainwright, J. Chem. Phy. 53, 3813 (197). [9] D. Leveque, L. Verlet, and J. Kürkijarvi, Phy. Rev. A 7, 169 (1973). [1] M. Schoen and C. Hoheiel, Mol. Phy. 56, 653 (1985). [11] J. J. Erpenbeck, Phy. Rev. A 38, 655 (1988). [1] A. W. Lee and S. F. Edward, J. Phy. C 5, 191 (197). [13] W. T. Ahurt and W. G. Hoover, Phy. Rev. Lett. 31, 6 (1973). [14] W. G. Hoover, D. J. Evan, R. B. Hickman, A. J. C. Ladd, W. T. Ahurt, and B. Moran, Phy. Rev. A, 169 (198). [15] D. J. Evan, Phy. Rev. A 3, 1988 (1981). [16] C. Trozzi and G. Ciccotti, Phy. Rev. A 9, 916 (1984). [17] M. P. Allen, in Computer Simulation in Chemical Phyic, edited by M. P. Allen and D. J. Tildeley (Kluwer, Amterdam, 1993), pp [18] J. M. Haile, Molecular Dynamic Simulation (John Wiley & Son, New York, 1997).

7 7 [19] P. E. Smith and W. F. van Gunteren, Chem. Phy. Lett. 15, 315 (1993). [] K. Meier, A. Laeecke, and S. Kabelac, J. Chem. Phy. 11, 3671 (4). [1] S. He and D. J. Evan, Phy. Rev. E 64, 117 (1). [] D. McQuarrie, Statitical Mechanic (Univerity Science Book, Saualito, ). [3] A. A. Chialvo and P. G. Debenedetti, Phy. Rev. A 43, 489 (1991). [4] A. A. Chialvo, P. T. Cumming, and D. J. Evan, Phy. Rev. E 47, 17 (1993). [5] M. P. Allen, D. Brown, and A. J. Mater, Phy. Rev. E 49, 488 (1994). [6] M. Allen, Phy. Rev. E 5, 377 (1994). [7] J. J. Erpenbeck, Phy. Rev. E 51, 496 (1995). [8] S. Vicardy and P. Gapard, Phy. Rev. E 68, 414 (3). [9] J. R. Dorfman and P. Gapard, Phy. Rev. E 51, 8 (1995). [3] P. Gapard and J. R. Dorfman, Phy. Rev. E 5, 355 (1995). [31] P. Gapard, Chao, Scattering and Statitical Mechanic (Cambridge Univerity Pre, Cambridge, 1998). [3] J. R. Dorfman, An Introduction to Chao in Nonequilibrium Statitical Mechanic (Cambridge Univerity Pre, Cambridge, 1999). [33] S. Vicardy and P. Gapard, Phy. Rev. E 68, 415 (3). [34] P. Gapard, Phy. Rev. E 53, 4379 (1996). [35] A. Eintein, Ann. d. Phy. 17, 549 (195), tranlated and reprinted in Invetigation on the theory of the brownian movement (Dover, New York, 1956). [36] Eq. (3.13) in Helfand paper [7] and Eq. (1-34) in Mc- Quarrie book []. [37] S. He, M. Kröger, and D. Evan, Phy. Rev. E 67, 41 (3). [38] K. Meier, A. Laeecke, and S. Kabelac, J. Chem. Phy. 1, (5). [39] W. C. Swope, H. C. Anderen, P. H. Beren, and K. R. Wilon, J. Chem. Phy. 76, 637 (198). [4] B. J. Palmer, Phy. Rev. E 49, 359 (1994). [41] W. T. Ahurt and W. G. Hoover, Phy. Rev. A 11, 658 (1975). [4] D. M. Heye, J. Chem. Soc. Faraday Tran. 79, 1741 (1983). [43] D. M. Heye, Phy. Rev. B 37, 5677 (1988). [44] M. Ferrario, G. Ciccotti, B. L. Holian, and J. P. Ryckaert, Phy. Rev. A 44, 6936 (1991). [45] H. Staen and W. A. Steele, J. Chem. Phy. 1, 93 (1995). [46] S. Vicardy, J. Servantie, and P. Gapard, J. Chem. Phy. 16, (7) companion paper.

8 8 Author year method r cut T N η η Leveque et al. [9] 1973 GK (MD) N. C Ahurt and Hoover [13] 1973 SSW N. C Ahurt and Hoover [41] 1975 SSW N. C Hoover et al. [14] 198 SSW OSW Leveque a) 198 GK (MD) N. C N. C N. C N. C. Pollock a) 198 GK (MD) N. C Evan [15] 1981 LE b) Heye [4] 1983 DT Schoen and Hoheiel [1] 1985 GK (MD) Erpenbeck [11] 1988 GK (MC) Heye [43] 1988 GK (MD) N. C Ferrario et al. [44] 1991 GK (MD) c) Palmer [4] 1994 TCAF Staen and Steele [45] 1995 GK (MD) N. C. Meier et al. [] 4 GEF Thi work 7 HM (MD) TABLE II: Reult found in the literature for the hear vicoity in the Lennard-Jone fluid near the triple point. The reduced denity equal n =.844, except for the reult reported by Heye (1988) [43] (n =.848), and by Staen and Steele [45] (n =.8445). Abbreviation: DT, difference in trajectorie method. GEF, generalized Eintein relation with integration of the flux. GK (MC), Green-Kubo reult with Monte-Carlo method. GK (MD), Green-Kubo reult with molecular dynamic. OSW, ocillatory hearing wall. SSW, teady hearing wall. TCAF, method baed on the tranvere-current autocorrelation function. HM (MD) the preent Helfand-moment method with molecular dynamic (MD). The infinite ign mean that the value of the hear vicoity i obtained by extrapolation for N. N. C. mean that the value ha not been communicated. a) Value unpublihed but communicated by Hoover et al. [14]. b)