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argon above its melting temperature is a typical simple fluid .
consisting of spherical atoms that interact via short - range repulsion and long - range attraction , and are heavy enough for the quantum effects to be small , fluid argon and heavier noble gases are an excellent choice of a real system to be used for testing various approaches in classical theory of fluids .
an inter - particle interaction in argon is commonly represented by a well known 126 lennard - jones pair potential @xcite , @xmath0.\ ] ] the two parameters , @xmath1 and @xmath2 , are usually determined by fitting thermodynamic properties , derived from the potential ( [ eq : lennard - jones ] ) by theoretical or computational methods , to corresponding experimental data .
it is known that lennard - jones potential is only an approximation to real interaction in argon .
several experimental results obtained for argon at large pressures are better explained if a larger steepness , compared to lennard - jones , of argon - argon interaction potential at small inter - atomic separation distances is taken into account @xcite .
accurate argon 
argon interatomic potentials have been calculated by direct _ ab initio _ quantum chemical calculations @xcite or obtained by inversion of experimental data @xcite . moreover ,
many - body dispersion , exchange and induced polarization contributions to inter - atomic interactions are not small and noticeably influence thermodynamic properties of argon @xcite .
the most widely used of these contributions is triple - dipole dispersion interaction , derived by axilrod and teller @xcite and muto @xcite , and account of this contribution in addition to _ ab initio _ pair potential is sufficient to describe thermodynamic properties of argon with good accuracy @xcite .    by virtue of henderson theorem @xcite , which states that , for fluids with only pairwise interactions , and under given conditions of temperature and density , the pair potential which gives rise to a given radial distribution function @xmath3 is unique up to a constant , the thermodynamic properties of the system with many - body interactions can be described by a model system with an appropriate _ effective _ pair potential . generally , the effective potential depends on the thermodynamic state of the system and thermodynamic property to be described @xcite .
van der hoef and madden @xcite have demonstrated that the account of triple - dipole and dipole - dipole - quadrupole dispersion interactions moves the effective potential of argon towards lennard - jones form ( [ eq : lennard - jones ] ) .
moreover , the possibility of consistent description of many thermodynamic properties of argon , using lennard - jones potential in a wide domain of thermodynamic states @xcite , suggests that the state dependence of the effective potential is weak .
there is no analogous reason for kinetic properties of a system with many - body interactions to be equivalent to those of a system with a corresponding effective pair potential .
nevertheless , experimental data on self - diffusion , shear viscosity and thermal conductivity coefficients of argon have been shown to be accurately described by lennard - jones model with the parameters obtained by fitting thermodynamic data @xcite .
bulk viscosity is a noticeable exception .
bulk viscosity of argon has been measured experimentally @xcite , and its behavior can be qualitatively described by the results of a molecular dynamics simulation of a lennard - jones system @xcite . however
, when results of simulations with lennard - jones potential are rescaled in an attempt to describe experimental data liquid argon , bulk viscosity , contrary to other kinetic properties , appears strongly underestimated ( _ e.g. _ up to 50% in ref .  [ ) ] .    in view of the above
, i propose that the source of this discrepancy may lie in neglect of many - body interactions .
previous molecular dynamics simulations of systems consisting of 108 particles interacting via _ ab initio _ pair potential and axilrod - teller - muto ( atm ) interaction indicated that a triple - dipole interaction does not affect the bulk viscosity of liquid xenon near its triple point @xcite and dense gaseous krypton @xcite .
however , the error in the values of bulk viscosity obtained from molecular dynamics simulation of the systems with such a small number of particles can be quite large .
for example , the values of the reduced bulk viscosity of the lennard - jones systems consisting of 128 and 256 particles at the reduced temperature @xmath4 and the reduced density @xmath5 , reported in refs  , range from 0.89 to 1.47 , with the ratio of the latter to the former of 1.65 .
this paper presents the results of more accurate molecular dynamics simulations of a liquid consisting of 1372 argon atoms with _ ab initio_+atm interaction , which demonstrate that bulk viscosity , determined from green - kubo formulae , significantly changes with the account of three - body interaction , moving results towards experimental data .
nasrabad _ et al _ @xcite undertook a monte carlo simulation of argon using combination of _ ab initio _ pair interaction @xcite and atm triple - dipole dispersion interaction @xcite to test their ability to predict vapor - liquid equilibrium . although more accurate _
ab initio _
pair potentials for argon have become available recently @xcite , and other many - body contributions to inter - atom interaction can be calculated @xcite , we use the same interaction as nasrabad _
et al _ because , being able to predict accurately the phase diagram of argon @xcite , it is computationally more efficient .    specifically , the _ ab initio _
pair interaction potential used in the present work is described by a function @xcite @xmath6 where @xmath7 and numerical values of the parameters @xmath8 , @xmath9 , @xmath10 , @xmath11 , and @xmath12 are given in ref .
the atm triple - dipole interaction has form @xcite @xmath13 where the @xmath14 are the lengths of the sides , @xmath9 , @xmath10 , and @xmath15 are the angles of the triangle formed by three argon atoms , and @xmath16j@xmath17m@xmath18 for argon @xcite .    for simulations of argon using lennard - jones potential ( [ eq : lennard - jones ] )
the values @xmath19 and @xmath20k are used @xcite .
meier _ et al _ @xcite undertook a systematic study of the influence of the number of particles and the cutoff radius for pair interaction on the bulk viscosity of lennard - jones system . in view of their results , simulations were performed in a cubic box containing @xmath21 particles , and the cutoff radius for pair interactions was set to @xmath22 .
three - body interactions were cut off when the distance between any pair of the atoms in the triplet exceeded one quarter of the simulation box length ( around @xmath23 for the densities studied in this work ) .
usual periodic boundary conditions and minimum image convention were applied .
the simulations were started with the particles in a face - centered - cubic lattice , with randomly assigned velocities .
forces arising from three - body interactions were calculated using formulas given by allen and tildesley @xcite , and an expression for forces due to _ ab initio _
pair interaction was obtained by applying gradient operator to eq .
( [ eq : ab - initio ] ) .
newton s equations of motion were solved using velocity - verlet algorithm with the time step @xmath24 .
the runs were made at the experimental densities at various temperatures along the 40  atm isochore , taken from ref .
every simulation was initiated in the nvt ensemble and run for at least @xmath25 time steps to attain thermodynamic equilibrium .
after equilibration the thermostat was turned off and the nve ensemble was invoked to calculate bulk and shear viscosities .
the length of the production period was @xmath26 time steps for the system interacting via lennard - jones potential , and between @xmath27 and @xmath28 time steps for the system with _
ab initio_+atm interaction , depending on the state point .
bulk viscosity , @xmath29 , and shear viscosity , @xmath30 , were calculated using green - kubo formulas @xcite : @xmath31 @xmath32 where @xmath33 is volume , @xmath34 is boltzmann constant , @xmath35 is temperature , @xmath36 is time , @xmath37 is the deviation of the instantaneous pressure @xmath38 from its average value @xmath39 , @xmath40 is an off - diagonal element of the stress tensor , the angular brackets denote equilibrium ensemble averages over short trajectory sections of the phase - space trajectory of the system with multiple ( every time step ) time origins @xmath41 .
the stress tensor was calculated using formulae given by lee and cummings @xcite .
the integration in eqs  ( [ eq : zeta ] ) and ( [ eq : eta ] ) was carried out up to @xmath42 , where @xmath43 is simulation box length and @xmath44 is sound velocity taken from ref .
depending on the state point , the value of @xmath45 was between 4.80 and 11.25  ps .
the statistical error in time correlation functions was estimated using formula given by frenkel and smit @xcite , @xmath46 where @xmath47 is the length of the simulation , and the correlation time @xmath48 was approximated as the time during which time correlation function decays @xmath49 times .
[ fig : zeta ] and table  [ table : results ] present simulation results for the bulk viscosity obtained using _ ab initio_+atm ( eqs  ( [ eq : ab - initio ] ) and ( [ eq : triple - dipole ] ) ) and lennard - jones ( eq .  ( [ eq : lennard - jones ] ) )
interaction , respectively .
bulk viscosity , determined from green - kubo formulas , changes with the account of three - body interaction , moving towards experimental data .
however , this change is not sufficient to obtain numerical agreement with experiment , especially at lower densities . typical behavior of time correlation functions @xmath50 is shown in fig .
[ fig : tcf ] .    k. error bars connected with solid and dashed lines correspond to the simulation results with _ ab initio_+atm and lennard - jones interaction , respectively . experimental points are taken from refs  [ ] ( circles , pressure 40  atm ) and [ ] ( square with error bar , pressure 40  kg/@xmath51 ) . ]
[ cols="^,^,^,^,^,^,^,^,^ " , ]      used for calculation of bulk viscosity at density 1.258g/@xmath52 .
solid and dashed lines correspond to the simulation results with _ ab initio_+atm and lennard - jones interaction , respectively . ]
fernandez _ et al _ @xcite demonstrated that , contrary to bulk viscosity , the values of shear viscosity of argon obtained from molecular dynamics simulation of a lennard - jones system agree with experimental data .
lee and cummings @xcite and marcelli _ et al _ @xcite found that the influence of triple - dipole interaction on shear viscosity of argon is small .
the results of the present simulation , shown in fig .
[ fig : eta ] and table  [ table : results ] , agree with these findings .
k. error bars connected with solid and dashed lines correspond to the simulation results with _ ab initio_+atm and lennard - jones interaction , respectively .
dotted line corresponds to the interpolation data for pressure 40  atm taken from ref .
the message of this paper is that many - body interactions play a more substantial role in determining the value of the bulk viscosity than other transport coefficients .
the present results from the molecular dynamic simulation of liquid argon demonstrate that even account of a single many - body contribution , atm triple - dipole interaction , shifts the values of the bulk viscosity of argon towards experimental data .
larger sensitivity of the bulk viscosity to many - body interaction , compared to other transport coefficients , can be intuitively explained in the case of gaseous state .
bulk viscosity of a non - relativistic monoatomic gas calculated from the boltzmann equation , which takes into account only pair collisions of atoms , appears to be zero , in contrast to heat conductivity and shear viscosity which have non - zero values in the same approximation @xcite .
a non - zero value of bulk viscosity appears in the approximations corresponding to higher - order terms in the virial expansion @xcite , which correspond to the explicit account of at least three - atom collisions which , in turn , are sensitive to three - body inter - atomic interaction .
with the aim of locating the origin of discrepancy between experimental and computer simulation results on bulk viscosity of liquid argon , a molecular dynamic simulation of argon interacting via ab initio pair potential and triple - dipole three - body potential has been undertaken . 
 bulk viscosity , obtained using green - kubo formula , is different from the values obtained from modeling argon using lennard - jones potential , the former being closer to the experimental data . 
 the conclusion is made that many - body inter - atomic interaction plays a significant role in formation of bulk viscosity .