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

0.19 MB·English

by  S. Viscardy

Transport and Helfand moments in the Lennard-Jones fluid. I. Shear viscosity
S. Viscardy, J. Servantie, and P. Gaspard
Center for Nonlinear Phenomena and Complex Systems,
Universit´e Libre de Bruxelles,
7 Campus Plaine, Code Postal 231, B-1050 Brussels, Belgium
0
0 Weproposeanewmethod,theHelfand-moment method, tocomputetheshearviscosity byequi-
2 librium molecular dynamics in periodic systems. In this method, the shear viscosity is written as
an Einstein-like relation in terms of the variance of the so-called Helfand moment. This quantity,
n
a is modified in order to satisfy systems with periodic boundary conditions usually considered in
J molecular dynamics. We calculate the shear viscosity in the Lennard-Jones fluid near the triple
point thanks to this new technique. We show that the results of the Helfand-moment method are
1
1 in excellent agreement with the results of the standard Green-Kubomethod.

] PACSnumbers: 02.70.Ns;05.60.-k;05.20.Dd
h
c
e I. INTRODUCTION formulas. The application of the pure Green-Kubo tech-
m
niquebyequilibriummoleculardynamicstotheLennard-
- Jones fluid has been performed a short time after by
t Since Maxwell’s firstpaper [1, 2] onthe kinetic theory
a Levesqueet al [9] andlater by Schoenand Hoheisel[10].
of gases, shear viscosity as well as the other transport
t However,untilthemiddleofthe eightiesandtheworkof
s properties are known to find their origin in the micro-
. SchoenandHoheisel[10],aswellastheoneofErpenpeck
t scopicmotionofatomsandmoleculescomposingmatter.
a using the Monte-CarloMetropolismethod [11], nonequi-
However,itisonlysincethefiftiesthatexactformulasare
m
librium molecular dynamics was predominantly used for
known to calculate the transport coefficients in terms of
- the computation of shear viscosity [12, 13, 14, 15, 16].
d the microscopic dynamics. These so-called Green-Kubo
n formulas give each transport coefficient as the time inte- At the end of the eighties and the beginning of the
o gral of the autocorrelation function of some specific mi-
nineties, the generalized Einstein relations started to be
c croscopic flux associated with the transport property of
used for calculating the transport coefficients. An alter-
[
interest [3, 4, 5, 6] . Today, the Green-Kubo technique native equilibrium moleculardynamics method has been

1 allows us to calculate numerically the transport coeffi- proposed in which the variance of the time integral of
v cients by simulating the molecular dynamics of systems
the microscopic flux is calculated [17, 18]. Actually it
3 with a finite number of particles and periodic boundary
is the analog of the method of Alder et al.’s [8] for soft
5
conditions.
sphere potential systems. Recently, this technique has
2
1 On the other hand, Einstein classicwork onBrownian beenappliedbyMeieret al. [19],andbyHessandEvans
0 motion showed that transport properties such as diffu- [20], the latter having rather considered an equilibrium
7 sion can also be understood in terms of random walks. ensemble of time averages of the flux. In this context,
0 It was Helfand [7] who identified in 1960 the fluctuating two important points were discussed. The first concerns
t/ quantities which, by their random walk, are associated the so-called McQuarrie expression. In his book [21],
a witheachtransportcoefficients. Thesefluctuatingquan- McQuarriepresentedHelfand’sformulafortheshearvis-
m
titiesaretheso-calledHelfandmomentsandarethecen- cosity, but with a slightly different form. This difference
- troidsofthe conservedquantitywhichistransported. In impliedasimplificationoftheexpression,apparentlygiv-
d
n principle,eachtransportcoefficientcanthusbe obtained ing an important advantage compared to the original
o from the linear increase of the statistical variance of the formula [22, 23, 24, 25]. The second point concerned
c corresponding Helfand moment by the so-called general- the validity of the generalized Einstein relations in peri-
v: ized Einstein relations. Nevertheless,it is not yet known odic systems [17, 18, 24, 26]. Arguing that the periodic
i today how these Helfand moments should be defined in boundaryconditionsimplythatthevarianceoftheorigi-
X molecular dynamics with periodic boundary conditions, nalexpressionoftheHelfandmoment isboundedintime,
r which limits their use in numerical simulations. the generalized Einstein relations were considered to be
a
The purpose of the present paper is to derive an ana- impractical in periodic systems. In this paper, we will
lyticalexpressionoftheHelfandmomentassociatedwith show that a generalized Einstein relation is available for
viscosityformoleculardynamicswithperiodicboundary viscosity after the addition of two terms to the original
conditions,andto apply the Helfand-momentmethod to Helfand moment to take into account the periodicity of
the calculation of shear viscosity in the Lennard-Jones the system. The Helfand-moment method presents the
fluid near the triple point. important advantage to define shear viscosity as a non-
Alreadyinthefirstnumericalcalculationofviscosityin negativequantity,satisfyingthepositivityoftheentropy
1970[8], the algorithmof Alder et al. was basedon gen- production. We have previously applied such a method
eralized Einstein relations derived from the Green-Kubo to a system of two hard disks with periodic boundary
2
conditions [27]. In the present paper, we calculate the be useful. In the sixties, Helfand proposedquantities as-
shear viscosity in a Lennard-Jones fluid near the triple sociated with the different transport processes in order
point. We compare the results obtained by the Helfand- to establish Einstein-like relations for the transport co-
moment method with our own Green-Kubo values and efficients [7]. In particular, we have for shear viscosity
those found in the literature. that
Inaddition, the Helfand-momentmethod playsanim-
1 2
portantroleintheescape-rateformalism. Thisformalism η = lim G(η)(t)−G(η)(0) . (3)
establishes direct relationships between the characteris- N,V,t→∞2kBTVt(cid:28)h i (cid:29)
tic quantities of the microscopic chaos (Lyapunov expo-
It can be shown [7] that the Einstein-Helfand relation is
nents and fractal dimensions) and the transport coeffi-
equivalenttotheGreen-Kuboformula(2)bydefiningthe
cients [28, 29, 30, 31]. A few years ago, such a relation
Helfand moment as:
has been studied for the viscosity in the two-hard-disk
N
model [32]. Furthermore with the use of the Helfand
G(η)(t)= p (t)y (t), (4)
moment, it should be possible to construct at the mi- xa a
croscopic level the hydrodynamic modes, which are the Xa=1
solutions of the Navier-Stokes equations. This approach if the corresponding microscopic flux is the time deriva-
calledthehydrodynamic-modemethodhasbeensuccess- tive of the Helfand moment:
fully applied for diffusion [33].
d
The paper is organized as follows. In Section II, we J(η)(t)= G(η)(t). (5)
dt
outlinethetheoreticalbackgroundofthegeneralizedEin-
stein formula. Section III is devoted to the presentation We notice that the limit t → ∞ should be related to
of our Helfand-moment method used for the calculation the thermodynamic limit N,V →∞. Indeed, for a fluid
oftheshearviscosityinthispaper. InSectionIV,wedis- of particles confined in a finite box, the quantity (4) is
cuss the so-called McQuarrie expressionand the validity bounded so that the coefficient (3) would vanish if the
of the generalized Einstein formula for periodic systems. limit t → ∞ was taken before the thermodynamic limit
The results of the molecular dynamics simulations are N,V → ∞. Therefore, the number N of particles and
given in Section V. The comparison with our Green- the volume V should be large enough in order that the
Kubo results and previous researches are done. Finally, varianceoftheHelfandmomentdisplaysalinearincrease
conclusions are drawn in Section VI. over a sufficiently long time interval t, allowing the coef-
ficient η to be well defined. The larger the system, the
longer the time interval. It is in this sense that the limit
II. EINSTEIN-HELFAND FORMULA N,V,t→∞ should be considered in Eq. (3).
Another remark is that, compared with the Green-
One century ago, Einstein theoretically established a Kubo formula (2), Eq. (3) presents the advantage to
relationshipbetweenthediffusioncoefficientofBrownian define the shear viscosity as a positive quantity, satisfy-
motion and the random walk of the Brownian particle ing the conditions of non-negative entropy production.
duetoitscollisionswiththemoleculesofthesurrounding
fluid [34]:
III. HELFAND MOMENT IN PERIODIC
SYSTEMS
[x(t)−x(0)]2
D = lim , (1)
t→∞D 2t E Often, the molecular dynamics is simulated with pe-
riodic boundary conditions. In this case, the particles
where x is the position of the colloidalparticle and t the
exiting at one boundary are reinjected at the opposite
time. Thereafter, different studies led to the well-known
boundary. Duetotheperiodicityofthesystem,particles
Green-Kubo formula for the shear viscosity η obtained
in the simulation box may interact with image particles
by Green [3, 4], Kubo [5] and Mori [6]:
as well as the particles inside the original unit cell. As a
1 ∞ consequence, the images of particle b may contribute to
η = lim dt J(η)(t)J(η)(0) , (2) the force F applied by the particle b on the particle a:
N,V→∞kBTV Z0 D E ab ∂u(r )
where J(η) is the microscopic flux associated with the Fab =− ∂rab (6)
ab
shear viscosity η. The viscosity coefficient is obtained βββX(a,b)
in the thermodynamic limit where N,V → ∞ while the
with
particle density n = N/V remains constant. In the fol-
lowing, this condition is always assumed together with r =r −r −Lβββ(a,b) (7)
ab a b
the limit N,V →∞. The simplicity of Eq. (1) obtained
by Einstein presents a particularinterest. The extension where L the length of the simulation box and βββ(a,b) de-
ofsucharelationtotheothertransportcoefficientscould terminesthecelltranslationvector[18]. Therangeofthe
3
interaction potential must be smaller than L/2 to guar- boundary conditions:
anteethattheparticleainteractsonlywithoneoftheim-
N
agesofb in Eq. (6). The interactingpairis foundby the G(η)(t) = p (t)y (t)
ax a
minimum-image convention, kr −r −Lβββ(a,b)k < L/2.
a b a=1
X
Here, we define the quantity
− p(s) ∆y(s) θ(t−t )
ax a s
Lb|a(t)=Lβββ(a,b)(t) (8) Xa Xs
1 t
− dτ F (r )L (13)
which is the vector to be added to rb in order to satisfy 2 x ab b|ay
the minimum-image convention. Xa6=bZ0
For a dynamics which is periodic in the box, the posi- where p(s) =p (t ) is the momentum at the time of the
ai ai s
tionsshouldjumptosatisfytheminimum-imageconven-
jump t andθ(t−t ) the Heavisidestepfunction defined
s s
tion. As aconsequence,the positionsandmomentaused
as
to calculatethe viscositybythe Green-Kubomethodac-
1 for t>t ,
tually obey the modified Newtonian equations θ(t−t )= s (14)
s 0 for t<t .
s
(cid:26)
dr p
dta = ma + ∆r(as) δ(t−ts), Wjume pnsotiniceortdhearttothesatqiusfaynttihtye mLbin|aiymuhmas-imdiasgceonctoinnuvoeuns-
s
X tion. Let us point out that L changes when the force
dp b|ay
dta = F(rab), (9) Fx(rab) vanishes, so that the last term varies continu-
bX(6=a) ously in time and does not present any jump. We notice
thatthe last two terms of Eq. (13) involves the particles
where ∆r(s) is the jump of the particle a at time t with near the boundaries of the box. The second term is due
a s
k∆r(s)k = L. We notice that the modified Newtonian to the jumps of the particles to or from the neighbor-
a
ing boxes. The third termconcernsthe pairs ofparticles
equations(9) conserveenergy,totalmomentumandpre-
interacting between neighboring cells. The Helfand mo-
serve phase-space volumes (Liouville’s theorem).
mentofEq. (13)canbeusedtoobtaintheshearviscosity
Moreover, we see that the periodic boundary condi-
coefficientforsystemswithperiodicboundariesthanksto
tions imply that the Helfand moment of Eq. (4) is
the Einstein-like relation (3).
boundedandcannotbedifferentiatednearthetimest of
s
the jumps. In order to have a well-defined quantity, one
should remove the discontinuities at the jumps, so that
IV. DISCUSSION
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 Since the beginning of the nineties, some confusions
have been propagated in the literature concerning the
useofthe mean-squareddisplacementequationforshear
G(η)(t)= p (t)y (t)+I(t). (10)
ax a viscosity. First, it concerns the so-called McQuarrie ex-
Xa pression. On the other hand, several works have been
done which have prematurely concluded that the mean-
According to Eq. (5), the time derivative of the Helfand
square displacement equation for shear viscosity is inap-
moment must be the microscopic flux
plicable for systems with periodic boundary conditions.
p p 1 These confusions and criticisms are reported in partic-
J(η)(t)= ax ay + F (r )y , (11)
m 2 x ab ab ular by Erpenbeck in Ref. [26]. Since these questions
Xa Xa6=b are central in this paper, this section is devoted to such
problems in order to avoid any misconception.
definedwiththe positiony oftheminimum-imagecon-
ab
vention. In order to satisfy the equality (5) in periodic
systems,weshowinAppendixAthatthetermI(t)must
A. McQuarrie expression for shear viscosity
be given by
Inhiswell-knownandremarkablebookStatistical Me-
I(t) = − p(asx)∆ya(s)θ(t−ts) chanics, McQuarrie [21] reported the work achieved by
Xa Xs Helfand [7]. The derivation he proposed is quite differ-
1 t ent but he obtained the same intermediate relation as
− dτ F (r )L . (12)
2 x ab b|ay Helfand, that is [35]
a6=bZ0
X
N
1
where both r and L depend on the time τ in the η = lim [x (t)−x (0)]2p (t)p (0) .
integralof theablasttermb|.ayWe then obtainour generalex- N,V,t→∞2kBTVt*a,b=1 a b ay by +
X
pressionfortheHelfandmomentinsystemswithperiodic (15)
4
Thereafter in his book, McQuarrie let as an exercise the known Alder et al. method initially developed for hard-
derivation from Eq. (15) of the final expression which is ball systems [8] is not based on the Helfand expressions,
printed in Ref. [21] as follows butinsteadonthemean-squaredisplacementofthetime
integral of the microscopic flux [26].
N
1 The main doubt on the use of Helfand moments in
η = lim [x (t)p (t)−x (0)p (0)]2 ,
MQ N,V,t→∞2kBTVt* a ay a ay + periodic systems comes from the fact that the original
Xa=1 expression (4) is bounded and would lead to a vanish-
(16)
ing shear viscosity in the long-time limit. By this argu-
while Helfand obtained
ment, Allen concluded that the only correctway to han-
1 N 2 dleGxy(t)istowriteitas 0tG˙xy(τ)dτ,andexpressG˙xy
ηH = lim xa(t)pay(t)−xa(0)pay(0) .in pairwise, minimum-image form [17]. In other words,
N,V,t→∞2kBTVt*"a=1 # + the Alder et al. method wRould be the only valid method
X
(17)
for studying viscosity, that is, through a method inter-
The difference between both expressions is in the posi-
mediate between the Helfand and Green-Kubo methods.
tion of the sum over particles, and it seems that such a
Let us mention that this opinion was recently followed
difference is due to atyping error. Nevertheless,the Mc-
by Hess, Kr¨oger and Evans [20, 36] as well as by Meier,
Quarrie expression (16) at first sight presents a certain
LaeseckeandKabelac[19,37]havingconsideredsystems
advantage compared to Helfand’s one (17). Indeed, the
with soft-potential interactions. However, this does not
sumovertheparticlescancomeoutoftheaverage. Con-
precludethepossibilitytomodify theoriginalexpression
sequently, one would obtain a sum of averagesno longer
(4) of the Helfand moment in order to recover the mi-
depending on the different particles. If Eq. (16) would
croscopic flux (11). This is precisely what we have done
hold, Eq. (16) could be rewritten as
here above with our Helfand-moment method by adding
N the following two terms
η = lim [x (t)p (t)−x (0)p (0)]2 .
MQ 1 1y 1 1y
N,V,t→∞2kBTVt N
D (1E8) − p(s) ∆y(s) θ(t−t )
In other words, the McQuarrie relation seems to present ax a s
a=1 s
the advantage that shear viscosity would be evaluated XX
1 t
through a single-particle expressionwhereas Helfand ex- − dτ F (r )L (19)
pressed the viscosity by a collective approach. 2 a6=bZ0 x ab b|ay
X
The first time that Eq. (16) has been considered was
in the work of Chialvo and Debenedetti [22]. Without to the original one. Albeit the first original term is
givinga theoreticalproofofthe validity ofthe lastequa- bounded in time, the two new terms increase without
tion or the equivalence with Eq. (17), they provided a bound intime because ofthe jumps andthe interactions
numerical comparison between both methods and con- between the particle and the image particles (due to the
cluded that the difference between η and η is small. minimum-image convention). Therefore, they can con-
H MQ
Later,Chialvo,CummingsandEvans[23]alsocompared tribute to the linear growth in time of the variance of
the McQuarrie and Helfand expressions and speculated the Helfand moment. The Helfand-moment method we
an equivalencefor theoretical reasons. Thereafter, Allen, proposehereiscompletelyequivalenttotheGreen-Kubo
BrownandMasters[24]showedbycomparisonwiththeir formulaandpresentsthe advantagetoexpressthetrans-
Green-Kubo results that the numerical calculations for portcoefficientsbyEinstein-likerelations,directlyshow-
shearviscosityobtainedbyChialvoandDebenedetti[22] ing their positivity.
were incorrect. Moreover, Allen devoted a comment in
Ref. [23], and concluded that the McQuarrie expression
is not valid and is not able to give shear viscosity [25]. V. NUMERICAL RESULTS
This conclusion was confirmed later by Erpenbeck [26],
which settled the question. As aforementioned, this is We carriedoutmoleculardynamicssimulationstocal-
not really surprising since Eq. (16) seems quite clearly culate the shear viscosity by the Helfand-moment and
to be the result of a typing error. This discussion em- the Green-Kubo methods. We use the standard 6-12
phasizes the fact that viscosity is a collective transport Lennard-Jones potential
property, implying the intervention of all the particles.
σ 12 σ 6
u(r)=4ǫ − (20)
r r
B. Periodic systems and Helfand-moment method (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
We use the reduced units defined in Table I.
The other point which was questioned is whether a Allthecalculationsweperformaredonewiththecutoff
mean-squared displacement equation for shear viscosity r = 2.5σ. The equations of motion are integrated with
c
isusefulforsystemssubmittedtoperiodicboundarycon- thevelocityVerletalgorithm[38]oftimestep∆t=0.003.
ditions[17,24,26]. First,itwaspointedoutthatthewell The initial positions of the atoms form a fcc lattice and
5
Quantity Units for N =1372 atoms near the triple point at the reduced
temperature T∗ = kBǫT temperature T∗ =0.722anddensity n∗ =0.8442. As we
numberdensity n∗ =nσ3 see, the two method are in perfect agreement.
time t∗ =tpmǫσ2
distance r∗ = σr
shear viscosity η∗ =η√σm2ǫ 3.35
TABLE I: Reducedunits of the Lennard-Jonesfluid. 3.3
3.25
*
the initial velocities are given by a Maxwell-Boltzmann hy 3.2
distribution. Thereafter, the system is equilibrated over cosit 3.15
3×105 time steps to reach thermodynamic equilibrium. vis
Aftertheequilibrationstage,theproductionstagestarts. 3.1
At each time step, the microscopic flux (11) and the
3.05
Helfand moment (13) are calculated. Every 300 time
units (105 time steps), we compute the time autocorre-
3
lationfunctionofthe flux andthe mean-squaredisplace-
ment of the Helfand moment for this piece of trajectory 2.95
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
and average them with the previous results. Thanks to
1/N
this method we can calculate with a very large statistics
since we do not need to keep in memory the whole tra-
FIG. 2: Viscosity at the phase point T∗ = 0.722 and n∗ =
jectory. Depending on the size of the systems (N =108-
0.8442asafunctionoftheinverseofthenumberN ofatoms.
1372), the number of pieces of trajectory varies between The circles are the results of the numerical simulations and
2000 and 6000, hence the total number of time steps is thedashed line thelinear extrapolation.
between2×108 and6×108. Statisticalerrorisobtained
from the mean-square deviation of the correlation func- We estimated the shear viscosity by a linear fit on the
tionorofthemean-squaredisplacementonthetrajectory mean-square displacement of the Helfand moment. The
pieces. fit is done in the region between 5 and 10 time units to
guaranteethatthelinearregimeisreached. Wedepictin
4 Fig. 2 the shear viscosity versus the inverse N−1 of the
system size. The linear extrapolation gives the following
3.5
estimate of shear viscosity for an infinite system,
3 η∗ =3.291±0.057 (21)
* 2.5 This result is in agreement with the previous works as
hy reported in Table II. Indeed, the previous extrapolation
sit 2 found by Erpenbeck [11] η∗ =3.345±0.068,Palmer [39]
o
sc η∗ = 3.25±0.08, and Meier et al. [19] η∗ = 3.258±
vi 1.5
0.033areallinagreementwithourresult(21)withinthe
statistical error.
1
0.5
Green-Kubo VI. CONCLUSIONS
Helfand
0
0 1 2 3 4 5 6 7
In this paper, we propose a new method for the com-
time
putation of shear viscosity by molecular dynamics. The
Helfand-moment method is anadaptationofthe Helfand
FIG. 1: Viscosity at the phase point T∗ = 0.722 and n∗ =
formula (3) for systems with periodic boundary condi-
0.8442 for N = 1372. The plain line is the derivative of the
mean-squaredisplacementoftheHelfandmomentandthecir- tions by adding two terms (12) to the original expres-
cles theintegral ofthemicroscopic fluxautocorrelation func- sion of the Helfand moment (4). The method consists
tion. in the calculation of the mean-square displacement of
the Helfand moment (13). The variance of this quan-
We depict in Fig. 1 the time derivative of the mean- tity gives the shear viscosity by the generalized Einstein
squaredisplacementoftheHelfandmoment(13)andthe relation (3). We have discussed its validity in the light
timeintegraloftheautocorrelationfunctionofthemicro- of the discussions found in the literature of the begin-
scopic flux (11). In Fig. 1, the calculation is performed ning of the nineties. Thanks to this new method, we
6
have computed the shear viscosity in the Lennard-Jones where we have used the modified Newton equations (9).
fluid near the triple point. We showed that the Helfand- The term implying the interparticle force F(r ) may be
ab
moment method gives the same results as the standard modified into
Green-Kubo method. Moreover, our extrapolated value
of shear viscosity is in statistical agreement with those
found in the literature. More than stating as an alterna-
1 1
tive method to the standard Green-Kubo in equilibrium F (r )y (t)= F (r )y + F (r )y .
x ab a x ab a x ba b
2 2
moleculardynamics,theHelfand-momentmethodisuse- a6=b a6=b a6=b
X X X
ful and plays a central role in the escape-rate formalism (A2)
and the hydrodynamic-mode method. Indeed, in these Since the force F is central, we obtain F (r ) =
x ab
theories, the Helfand moment allows us to put in evi- −F (r ), which implies that
x ba
dence fractal structures at the microscopic level, which
arerelatedtothetransportprocesses[28,29,30,31,32].
We remarkthatthe methodcanbe similarlyextended
1
to the bulk viscosity ζ. As for the shear viscosity, two F (r )y (t)= F (r )(y −y ). (A3)
x ab a x ab a b
terms must be added to the original expression of the 2
a6=b a6=b
X X
Helfand moment associatedwith the bulk viscosity. For-
mally, this last coefficient is expressed as follows:
ζ+ 4η = lim 1 G(ζ)(t)−hG(ζ)(t)i 2 , whichstilldiffersfromthecorrespondingtermappearing
3 N,V,t→∞2kBTVt(cid:28)h i (cid:29)(22) iinmathgeecmonicvreonstcioopnicbeflcuaxus(e11y) −deyfin=edywit+hLthe macinciomrduimng-
a b ab b|ay
where its Helfand moment is defined as: to Eqs. (7) and (8). Consequently, Eq. (A1) becomes
G(ζ)(t) = p (t)x (t)
ax a
a
X
− p(asx) ∆x(as) θ(t−ts) dG(dηt)(t) = J(η)(t)+ 12 Fx(rab)Lb|ay
a s
XX a,b6=a
1 t X
− 2 a6=bZ0 dτ Fx(rab)Lb|ax, (23) + pax(t)∆ya(s)δ(t−ts)+ dId(tt).
X a s
XX
(A4)
and with G(ζ)(0) = 0. In the companion paper, we will
present a similar method for the calculation of thermal
conductivity [45].
Comparing with Eq. (5), we should have
Acknowledgments
We thank K. Meier for useful discussions. This re- dI(t)
= − p (t)∆y(s)δ(t−t )
search is financially supported by the “Communaut´e dt ax a s
a s
franc¸aise de Belgique” (contract “Actions de Recherche XX
1
Concert´ees” No. 04/09-312) and the National Fund − F (r )L (A5)
2 x ab b|ay
for Scientific Research (F. N. R. S. Belgium, contract
a,b6=a
X
F. R. F. C. No. 2.4577.04).
APPENDIX A: DERIVATION OF THE HELFAND
MOMENT FOR THE SHEAR VISCOSITY IN
PERIODIC SYSTEMS whereupon I(t) can be expressed as
By taking the time derivative of the Helfand moment
(10), we have:
I(t) = − p(s)∆y(s)θ(t−t )
ax a s
dG(η)(t) p (t)p (t)
ax ay a s
= XX
dt m 1 t
Xa − 2 dτ Fx(rab)Lb|ay . (A6)
+ pax(t)∆ya(s)δ(t−ts) aX,b6=aZ0
a s
XX
dI(t)
+ F (r )y (t)+ (A1)
x ab a
dt
a6=b
X
7
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E-print: cond-mat/0601210. (2003).
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8
Authors year method rc∗ut T∗ N η∗ ∆η∗
Levesqueet al. [9] 1973 GK (MD) N.C. 0.722 864 4.03 0.3
Ashurstand Hoover[13] 1973 SSW N.C. 0.722 864 3.88 0.0026
Ashurstand Hoover[40] 1975 SSW N.C. 0.722 ∞ 2.9 0.1
Hooveret al. [14] 1980 SSW 2.5 0.715 864 3.0 0.15
OSW 0.722 108 3.18 0.1
Levesquea) 1980 GK (MD) N.C. 0.728 108 2.97 N.C.
0.715 256 2.92 N.C.
0.722 864 3.85 N.C.
Pollock a) 1980 GK (MD) N.C. 0.722 256 2.6 0.1
0.722 500 3.2 0.2
Evans[15] 1981 LE 2.5 0.722 108-256 b) 3.17 0.03
Heyes[41] 1983 DT 2.5 0.73 500 3.08 0.24
Schoen and Hoheisel [10] 1985 GK (MD) 2.5 0.73 500 3.18 0.15
Erpenbeck[11] 1988 GK (MC) 2.5 0.722 108 2.912 0.071
864 3.200 0.160
∞ 3.345 0.068
Heyes[42] 1988 GK (MD) N.C. 0.72 108 3.2 0.16
256 3.5 0.18
500 3.4 0.17
Ferrario et al. [43] 1991 GK (MD) 2.5 0.725 500 3.02 0.07
0.7247 864 3.24 0.10
0.725 864 3.11 - 3.27 c) 0.10
2048 3.24 0.04
4000 3.28 0.13
Palmer [39] 1994 TCAF 2.5 0.722 ∞ 3.25 0.08
Stassen and Steele [44] 1995 GK (MD) 3.4 0.722 256 3.297 N.C.
Meier et al. [19] 2004 GEF 2.5 0.722 108 2.984 0.089
3.25 256 3.105 0.093
4.0 500 3.188 0.096
5.0 864 3.314 0.099
2.5 1372 3.277 0.098
5.5 2048 3.224 0.097
5.5 4000 3.275 0.098
∞ 3.258 0.033
This work 2007 HM (MD) 2.5 0.722 108 3.057 0.045
256 3.232 0.060
500 3.270 0.050
864 3.247 0.033
1372 3.268 0.055
∞ 3.291 0.029
TABLE II: Results found in the literature for the shear vis-
cosity in the Lennard-Jones fluid near the triple point. The
reduced density equals n∗ =0.8442, except for the result re-
ported by Heyes (1988) [42] (n∗ = 0.848), and by Stassen
and Steele [44] (n∗ = 0.8445) . Abbreviations: DT, differ-
ence in trajectories method. GEF, generalized Einstein re-
lation with integration of the flux. GK (MC), Green-Kubo
results with Monte-Carlo method. GK (MD), Green-Kubo
results with molecular dynamics. OSW, oscillatory shearing
walls. SSW,steady shearing walls. TCAF, method based on
the transverse-current autocorrelation functions. HM (MD)
the present Helfand-moment method with molecular dynam-
ics (MD).The infinitesign meansthatthevalueoftheshear
viscosity is obtained by extrapolation for N → ∞. N. C.
means that the valuehas not been communicated.
a) Values unpublished but communicated by Hoover et al.
[14].
b) No size dependence.
c) Valuesobtained for different thermostatting rates.