Green-Kubo Method Research Articles

Martingale product estimators for sensitivity analysis in computational statistical physics

Abstract We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name ‘martingale product estimators’. We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so that its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green–Kubo (GK) estimator, motivated, in part, by its variance growing linearly in time. We compare on illustrative numerical tests the new estimators with results obtained by the GK method.

Dynamics and lifetimes of resonant phonons in the quasiparticle and nonquasiparticle regimes

The dynamics and lifetimes of a phonon with model resonant interactions from the quasiparticle to nonquasiparticle regime were investigated employing Green's function and the Green-Kubo method. In the weak-coupling case, the dynamics of the resonant phonon are analogous to a damped harmonic oscillator and its lifetime $\ensuremath{\tau}$ is in accordance with the standard phonon transport theory $\ensuremath{\tau}=\frac{1}{2\mathrm{\ensuremath{\Gamma}}}$ ($\mathrm{\ensuremath{\Gamma}}$ being the imaginary part of phonon self-energy). In the strong-coupling nonquasiparticle regime, however, the resonant phonon ``propagates'' in a complex form of wave packets and the actual phonon lifetime ${\ensuremath{\tau}}_{\mathrm{GK}}$ as determined from the Green-Kubo formula significantly deviates from the standard $\ensuremath{\tau}=\frac{1}{2\mathrm{\ensuremath{\Gamma}}}$ relation. Taking the four-phonon resonant ${\mathrm{AgCrSe}}_{2}$ and three-phonon resonant PbSe model systems as examples, the phonon nonquasiparticle dynamics and their lifetimes in real materials are further investigated by first-principle calculations. Substantial discrepancies between ${\ensuremath{\tau}}_{\mathrm{GK}}$ and $\ensuremath{\tau}$ are found for the strongly resonant phonons at high-symmetrical points in both materials. Meanwhile, the lifetime ${\ensuremath{\tau}}_{\mathrm{GK}}$ of the phonons that are not subjected to resonant phonon interactions almost recovers to the conventional theoretical result $\ensuremath{\tau}$ despite the non-Lorentzian spectral feature of these phonons. It is suggested that $\ensuremath{\omega}{\ensuremath{\tau}}_{\mathrm{GK}}$, instead of the conventional $\ensuremath{\omega}\ensuremath{\tau}$, can be applied as a criterion to distinguish phonon quasiparticles and nonquasiparticles in addition to the phonon spectral functions.

Bulk viscosity of dilute monatomic gases revisited

Extensive research has been carried out in the past to estimate the bulk viscosity of dense monatomic fluids; however, little attention has been paid to estimate the same in the dilute gas regime. Here, the term ‘dilute’ is used in reference to kinetic theory of gases and refers to low density conditions where average inter-molecular spacing is much greater than the size of the molecules. In this work, we have performed molecular dynamics simulations and used the Green–Kubo method to make precise estimations for the bulk viscosity of argon gas at atmospheric conditions. The investigated temperature and pressure range is 300 to 750 K and 0.5 to 1.5 bar respectively. It is observed that the estimated bulk viscosity is O(10−10) Pa s, which is several orders of magnitude smaller than that of other diatomic and polyatomic gases, but nonetheless, not an absolute zero as typically assumed. It implies that Stokes’ hypothesis is true for dilute monatomic gases only in an approximate rather than absolute sense. The variation of bulk viscosity with pressures at constant temperatures has also been studied and is found to be of quadratic nature. The obtained bulk viscosity values are also reported in the reduced Lennard-Jones units to enable extension of the present results to other noble gases as well. It has also been observed that the bulk viscosity of Lennard-Jones monatomic gases becomes temperature independent at very low densities.

Linear viscoelastic shear and bulk relaxation moduli in poly(tetramethylene oxide) (PTMO) using united-atom molecular dynamics

A quantitative understanding and prediction of the viscoelastic relaxation behavior of polymer melts is required to build up predictive multiscale models, which can be utilized for the practical design of viscoelastic materials. We report the application of the Green–Kubo method to compute the common shear relaxation modulus and the rarely reported bulk relaxation modulus for a realistic united atom model of poly(tetramethylene oxide) (PTMO), which is one key components of the segmented polyurethane (PU) elastomers. The temperature and molecular weight dependent viscoelastic behavior is investigated in detail by computing the bulk relaxation modulus K(t) along with shear relaxation modulus G(t). Our results provide new rheological data for PTMO melt based on the united atom model, which incorporates the chemical details of the polymer backbone. The predicted stress relaxation are mapped onto master curves using the time–temperature superposition principle with horizontal and vertical shift factors. The computed shift factors agree with the Williams–Landel–Ferry equation and appear to be similar for the shear and bulk relaxation modulus. The underlying dynamics for the shear and bulk relaxation modulus appear to stem from the same mechanistic origin as both processes seem to sample similar kinetic signature of the reference time scale of the underlying thermally activated processes in the liquid. Our results demonstrate furthermore, that MD simulations with the chemical details beyond coarse-grained models can sample the Rouse dynamics and the transitions towards entanglement, which is evident by the emergence of a plateau in shear relaxation modulus and scaling relations in the diffusion behavior of the monomer at larger molecular weights.

Effects of branching and polydispersity on thermal conductivity of paraffin waxes

• Increased branching of paraffin molecules leads to decreased thermal conductivity. • Polydispersity has a minor effect on thermal conductivity at 250 K, but none at 450 K. • For branched paraffin, there is a strong concomitance of crystallinity and thermal conductivity. • Increased thermal conductivity of polydisperse paraffin is explained purely by increasing average chain length. Paraffin waxes are promising phase change materials, abundantly available at very low cost. Having large latent heat, these materials can be used for thermal energy storage. However, when used in heat batteries, paraffin’s low thermal conductivity prevents fast charging and discharging. This calls for the design of tailored hybrid materials with improved properties, the present study concentrates on properties of pure paraffin wax. Using fully atomistic molecular-dynamics (MD) simulations, we study the effects of polydispersity and branching on the thermal conductivity of paraffin waxes, in molten (450 K) and solid (250 K) state. Both branching and polydispersity affect the density and especially the crystallinity of the solid. Branching has a pronounced effect on crystallisation caused by inhibited alignment of the polymer backbones while the effect of polydispersity is less pronounced. The thermal conductivity (TC) has been simulated using the reverse non-equilibrium molecular-dynamics method, as well as the equilibrium Green-Kubo approach. Increased branching, added to backbones comprised of twenty monomers, results in decreasing TC of up to 30%, polydispersity only has an effect in the semi-crystalline state. Comparison to available experiments shows good agreement which validates the model details, applied force field and the calculation methods. We show that at comparable computational costs, the reverse non-equilibrium MD approach produces more reliable results for TC, as compared to the equilibrium Green-Kubo method. The major contribution to TC by acoustic phonon transport along the backbone was shown by analysing extreme cases. The phonon density of states (PDOS) of samples with high branching or with small chain length displayed diminished peaks in the acoustic range as compared to the PDOS of samples with low branching or larger chain length, respectively. The suggested MD approach can definitely be used to investigate specific material modifications aimed at increasing the overall TC.

Thermophysical property prediction of biodiesel mixtures at extreme conditions using molecular dynamics simulation

Liquid biofuels such as biodiesel are playing an increasing role in renewable energy utilisation, but accurate predictions of fuel properties at extreme conditions remain challenging. In this study, molecular dynamics (MD) simulation with classical force field is performed to obtain the thermophysical properties of biodiesel over extended ranges of temperatures and pressures. The predicted properties include the critical temperature, critical density, critical pressure, surface tension, viscosity, thermal conductivity and diffusion coefficient. The specific MD setup for fuel thermophysical property prediction is examined. It is observed that the long-range dispersion interaction is essential to accurately predicting the surface tension, and averaging over a sufficient number of independent replication simulations is required to eliminate the statistical error when using the Green-Kubo method for viscosity calculation. The reliability of MD approach is validated against experimental data at normal temperature and pressure. The equilibrium MD simulation has an advantage over nonequilibrium MD simulation when building databases of fuel properties. Moreover, the properties of biodiesel are compared with conventional diesel to elucidate the effects of fuel composition and molecular structure of the fuel surrogates on fuel thermophysical properties. For biodiesel utilisation, higher values of critical temperature, surface tension and viscosity, lower diffusivity together with the increased aggregation tendency in bath gas indicate the needs of further optimisation of injection system to achieve the desired mixing in combustion applications.

On the estimation of bulk viscosity of dilute nitrogen gas using equilibrium molecular dynamics approach

In this work, we present a study for the estimation of bulk viscosity using the equilibrium molecular dynamics-based Green–Kubo method. We have performed a parametric study to find optimal hyper-parameters to estimate bulk viscosity using the Green–Kubo method. Although similar studies exist for shear viscosity, none has been reported so far specifically for bulk viscosity. The expected uncertainty in bulk viscosity for a given length and number of molecular dynamics trajectories used in statistical averaging is determined. The effect of system size, temperature, and pressure on bulk viscosity has also been studied. The study reveals that the decay of autocorrelation function for bulk viscosity is slower than that for shear viscosity and hence requires a longer correlation length. A novel observation has been made that the autocorrelation length required for convergence in the Green–Kubo method for both shear and bulk viscosity of dilute nitrogen gas is of the same mean collision time length units irrespective of simulation pressure. However, when the temperature is varied, the required autocorrelation length remains unaffected for shear viscosity but increases slightly with temperature for bulk viscosity. The results obtained from the Green–Kubo method are compared with experimental and numerical results from the literature with special emphasis on their comparison with the results from the nonequilibrium molecular dynamics-based continuous expansion/compression method. Although the primary focus and novelty of this work are the discussion on bulk viscosity, a similar discussion on shear viscosity has also been added.

Extraction of Thermal Properties of Organic Ablative Materials Using Molecular Dynamics Simulations

In this study, molecular dynamics method is used for extracting thermal properties of materials relevant to the ablative type of thermal protection system. The methodology and results are discussed for extracting thermal conductivity and specific heat capacity of ablative materials for their virgin and char states in a manner that is defined in the experimental data sets. To select a suitable nonreactive force field, Green–Kubo simulations are performed over phenol, toluene, and benzene materials using Amber, Dreiding, and all-atom optimized potentials for liquid simulations (OPLSAA) force fields. Next, crosslinked and non-crosslinked polymers are created, and pyrolysis simulations are performed over them using ReaxFF to obtain virgin and char states of the polymers. Thermal conductivity is extracted using the Green–Kubo method, whereas specific heat capacity is extracted by using the enthalpy gradient obtained by performing isobaric simulations over virgin and char materials. Later, by introducing the concept of an artificial composite material, the extracted thermal properties are used in macroscale one-dimensional heat conduction simulations with pyrolysis using an in-house finite element method (FEM)-based thermal response solver.

Molecular dynamics study of effects of point defects on thermal conductivity in cubic silicon carbide

Silicon carbide (SiC) has been widely used in nuclear technology due to its excellent properties. In the irradiation environment, the energetic incident particles can cause the atoms in the material to deviate from the position of the crystal lattice, thereby producing the vacancies, interstitial atoms, anti-site atoms and other point defects. These defects will change the thermal properties of the material and degrade the service performance of the material. Therefore, in this work the equilibrium molecular dynamics method (Green-Kubo method) is used to study the effect of point defects on the heat transfer properties of cubic SiC (<i>β</i>-SiC or 3<i>C</i>-SiC) with the help of the Tersoff-type potential. The point defects considered include Si interstitial atoms (Si<sub>I</sub>), Si vacancies (Si<sub>V</sub>), Si anti-site atoms (Si<sub>C</sub>), C interstitial atoms (C<sub>I</sub>), C vacancies (C<sub>V</sub>) and C anti-site atoms (C<sub>Si</sub>). It is found that the thermal conductivity (<i>λ</i>) decreases with the increase of the point defect concentration (<i>c</i>). The excessive thermal resistance (Δ<i>R</i> = <i>R</i><sub>defect </sub>– <i>R</i><sub>perfect</sub>, <i>R</i> = 1/<i>λ</i>, <i>R</i><sub>defect</sub> is the thermal resistance of the defective material, and <i>R</i><sub>perfect</sub> is the thermal resistivity of the material without defects) has a linear relation with the concentration of point defects in the considered range (0.2%–1.6%), and its slope is the thermal resistivity coefficient. It can be found that the thermal resistivity coefficient of vacancy and interstitial atoms are higher than that of anti-site atoms; the thermal resistivity coefficient of point defects at high temperature is higher than at low temperature; the thermal resistivity coefficient of Si vacancies and Si interstitial atoms are higher than that of C vacancies and C interstitial atoms. These results are helpful in predicting the thermal conductivity of silicon carbide under irradiation and controlling the thermal conductivity of silicon carbide.

Molecular dynamics approach to study the effects of MgCl2 salt atomic ratio on the phase transition phenomenon in the phase change materials

In general, phase change materials reduce energy consumption in different structures by increasing the thermal conductivity or decreasing the phase change time. On the other hand, hydrated salts are the most important group of phase change materials with high phase change temperatures and are extensively studied in latent thermal energy storage systems. Today, these atomic structures are used in different usages such as thermal transfer compounds, commercial and industrial applications, for cooling and heating on a small scale, etc. In the current computational study, we utilized Molecular Dynamics procedure to explain the thermal and phase transition (from liquid to solid-state) behavior of water liquid in the presence of MgCl2 salt as salts hydrate compound. Computationally, physical parameters like temperature, potential/total energy, phase transition time, thermal conductivity, and velocity/density profile of H2O-MgCl2 nanofluid were reported for time=15 ns. Molecular Dynamics outputs indicated that the phenomenon of phase transition in simulated nanofluid happens in a shorter time using a 4% atomic ratio salt structure. This atomic behavior of simulated structure causes thermal behavior improvement is defined nanofluid. Numerically, by MgCl2 salt ratio optimizing, H2O-MgCl2 nanofluid thermal conductivity converged to 0.67 W/m.k. Technically, thermal conductivity calculation in current research is done by using the Green-Kubo method.

Thermal Conductivity of Polybutadiene Rubber from Molecular Dynamics Simulations and Measurements by the Heat Flow Meter Method.

Thermal conductivities of polybutadiene rubbers crosslinked by 2.4 and 2.8 phr of sulfur have been found to be functions of temperature via molecular dynamics (MD) simulations using the Green–Kubo method. From an analysis of the heat flux autocorrelation functions, it has been revealed that the dominant means of heat transport in rubbers is governed by deformations of polymeric chains. Thermal conductivities of rubber samples vulcanized by 2.4 and 2.8 phr of sulfur have been measured by the heat flow meter method between 0 C and 60 C at atmospheric pressure. The temperature dependencies of the thermal conductivities of rubbers and their glass transition temperatures derived from MD simulations are in good agreement with the literature and experimental data. Details are discussed in the paper.

The computational investigation of thermal conductivity of 11S globulin protein for biological applications: Molecular dynamics simulation

From the globulin protein group, there is the globulin 11S family investigated in a protein fraction within an isolated structure. The thermal behavior of this protein is important for biological applications. Computer simulations based on the atomic arrangement of this bio-structure predict the thermal behavior. In the current computational study, the thermal behavior of 11S globulin using equilibrium molecular dynamics approaches (Green-Kubo method) is calculated. In this method, 11S globulin protein by C, H, N, O, S, and P atoms by using the DREIDING force-field. Numerically, our MD results for thermal conductivity (k) of 11S globulin protein in standard condition (T0 = 300 K and P0 = 1 bar) is 0.63 W/mK. Also, simulations indicate that the initial temperature and pressure are important effects on the thermal behavior of the modeled protein. By initial temperature/pressure enhancing in simulated structures, the thermal behavior is improved/decreased after t = 10 ns.

Investigate the effect of temperature, length, width, and thickness of substrate on thermal conductivity of graphene copper nanocomposites

The novel thermal conductivity of graphene-copper (GN/Cu) nanocomposites, which include thermal transport of phonons and electrons, has attracted tremendous attention for the development of next-generation nanoelectronic and optoelectronic device applications. To study the effect of temperature, length, width, and substrate thickness on thermal conductivity, Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software has been taken to perform molecular dynamics (MD) simulation. The Green Kubo methods with Embedded Atom Method (EAM), Reactive Empirical Bond Order (REBO), andLenard-Jones (L-J) potentials were used to analyze interactions of atoms. First of all, the thermal conductivity of copper and graphene at 300k temperature has been observed. The thermal conductivity of (GN, GN/Cu (10 0), GN/Cu (110), and GN/Cu (111)) was observed by a wide range of temperature (300K to 1000K). Thermal conductivities of GN, GN/Cu (1 0 0), GN/Cu (1 1 0) and GN/Cu (1 1 1) at 300k were obtained as 1500 Wm-1K -1 ,1204Wm-1K -1 ,1005 Wm -1K -1 ,897 Wm -1K -1 respectively. It was witnessed that the thermal conductivity of GN/Cu (1 0 0), GN/Cu (1 1 0), and GN/Cu (1 1 1) decreased by 80.26%, 67%, 59.8% as compared to graphene by changing temperature. It was also observed that thermal conductivity of GN/Cu (1 0 0), GN/Cu (1 1 0), and GN/Cu (1 1 1) decreased by 231.30%, 221.01%, 231.26% with the change in thickness of the substrate. It was calculated that thermal conductivity of GN/Cu (1 0 0), GN/Cu (1 1 0), GN/Cu (1 1 1) increased by 290.7%, 323.98%, 523.1% within length variation from 7nm to 11.0nm.It was also calculated that thermal conductivity of GN/Cu (1 0 0), GN/Cu (1 1 0), GN/Cu (1 1 1) increased by 290.7%, 323.98%, 523.1% within width variation from 0.8 nm to 2.0 nm. Finally, it was concluded that thermal conductivity increased with the increase of length as well as width. It was investigated that the above-mentioned parameters significantly improve the thermal conductivity of GN/Cu (111), which is beneficial for high-performance nanoelectronic and optoelectronic devices include solar cells.

Effect of Modification on the Fluid Diffusion Coefficient in Silica Nanochannels.

The diffusion behavior of fluid water in nanochannels with hydroxylation of silica gel and silanization of different modified chain lengths was simulated by the equilibrium molecular dynamics method. The diffusion coefficient of fluid water was calculated by the Einstein method and the Green–Kubo method, so as to analyze the change rule between the modification degree of nanochannels and the diffusion coefficient of fluid water. The results showed that the diffusion coefficient of fluid water increased with the length of the modified chain. The average diffusion coefficient of fluid water in the hydroxylated nanochannels was 8.01% of the bulk water diffusion coefficient, and the diffusion coefficients of fluid water in the –(CH2)3CH3, –(CH2)7CH3, and –(CH2)11CH3 nanochannels were 44.10%, 49.72%, and 53.80% of the diffusion coefficients of bulk water, respectively. In the above four wall characteristic models, the diffusion coefficients in the z direction were smaller than those in the other directions. However, with an increase in the silylation degree, the increased self-diffusion coefficient due to the surface effect could basically offset the decreased self-diffusion coefficient owing to the scale effect. In the four nanochannels, when the local diffusion coefficient of fluid water was in the range of 8 Å close to the wall, Dz was greater than Dxy, and beyond the range of 8 Å of the wall, the Dz was smaller than Dxy.

Green–Kubo formula for Boltzmann and Fermi–Dirac statistics

Shear viscosity of nuclear matter is extracted via the Green-Kubo formula and the Gaussian thermostated SLLOD algorithm (the shear rate method) in a periodic box by using an improved quantum molecular dynamic (ImQMD) model without mean field, also it is calculated by a Boltzmann-type equation. Here a new form of the Green-Kubo formula is put forward in the present work. For classical limit at nuclear matter densities of $0.4\rho_{0}$ and $1.0\rho_{0}$, shear viscosity by the traditional and new form of the Green-Kubo formula as well as the SLLOD algorithm are coincident with each other. However, for non-classical limit, shear viscosity by the traditional form of the Green-Kubo formula is higher than those obtained by the new form of the Green-Kubo formula as well as the SLLOD algorithm especially in low temperature region. In addition, shear viscosity from the Boltzmann-type equation is found to be less than that by the Green-Kubo method or the SLLOD algorithm for both classical and non-classical limits.

Predicting shear viscosity of 1,1-diphenylethane at high pressures by molecular dynamics methods

In this paper, we use the molecular dynamics methods to predict the pressure-viscosity dependencies of model lubricant 1,1-diphenylethane up to 800 MPa along three isotherms: 37.8 °C, 60 °C and 98.9 °C. The precise COMPASS class II force field is used to determine atomic interactions in the model. The Green-Kubo method is used to calculate the shear viscosities. The time decomposition method is applied for accurate calculation of the Green-Kubo integral limit. For the pressures below 340 MPa, the results match the available experimental data. The faster-than-exponential behavior of the pressure-viscosity dependence is observed for all the temperatures which agrees with the experimental data from Scott Bair.

Molecular dynamics simulations of choline chloride and phenyl propionic acid deep eutectic solvents: Investigation of structural and dynamics properties

The prediction of deep eutectic composition is hard and so far, has been distinguished by trial and error. Therefore, in this work, molecular dynamics simulations were performed for specifying the composition of the eutectic point of phenyl propionic acid (Phpr) and choline chloride (ChCl) mixtures. The distinctive properties of the Phpr and ChCl eutectic mixture at the composition of the eutectic point were investigated and were compared to other eutectic mixtures with the different mole fractions of Phpr and ChCl. Structural properties such as radial distribution function (RDF), coordination number, hydrogen-bond number, interaction energies, and dipole moment of species, as well as dynamical properties such as mean square displacement (MSD), viscosity, and self-diffusion coefficient were analyzed. The obtained results of structural properties indicated that each chloride anion is surrounded by two Phpr molecules for deep eutectic point states that is in good agreement with available experimental reports. Moreover, the viscosity of studied mixtures evaluated by the Green-Kubo method was found to be consistent with the reported experimental data. Besides, the stress-autocorrelation function (SACF) and convergency of viscosity with time were calculated. Finally, the eutectic point could be detected by the changes in the trends of total van der Waals interaction energies and the viscosity.

Thermodiffusion and ion transport in doped ceria by molecular dynamics simulations

We simulated thermodiffusion and ionic conductivity in doped ceria by molecular dynamics. Four approaches to estimate the ionic conductivity were applied using the auto-correlation function and collective displacement of oxygen ions, the tracer diffusion coefficient from the mean squared displacement, and the displacement in an electric field. The Green-Kubo method was applied to determine the mass-heat coupling cross-coefficients LOq and LqO of the Onsager transport matrix. This is the first calculation of these coefficients in an ionic compound. The simulated parameters are negative throughout the applied temperature range suggesting net transport of anions from a cold to a hot region in a temperature gradient. The results emphasize that the coupling between mass and heat transport in an oxygen ion electrolyte could have an impact for practical applications.

Thermal transport by electrons and ions in warm dense aluminum: A combined density functional theory and deep potential study

We propose an efficient scheme that combines density functional theory (DFT) with deep potentials (DPs), to systematically study convergence issues in the computation of the electronic thermal conductivity of warm dense aluminum (2.7 g/cm3 and temperatures ranging from 0.5 eV to 5.0 eV) with respect to the number of k-points, the number of atoms, the broadening parameter, the exchange-correlation functionals, and the pseudopotentials. Furthermore, we obtain the ionic thermal conductivity using the Green–Kubo method in conjunction with DP molecular dynamics simulations, and we study size effects on the ionic thermal conductivity. This work demonstrates that the proposed method is efficient in evaluating both electronic and ionic thermal conductivities of materials.

Viscuit and the fluctuation theorem investigation of shear viscosity by molecular dynamics simulations: The information and the noise.

The shear viscosity, η, of model liquids and solids is investigated within the framework of the viscuit and Fluctuation Theorem (FT) probability distribution function (PDF) theories, following Heyes et al. [J. Chem. Phys. 152, 194504 (2020)] using equilibrium molecular dynamics (MD) simulations on Lennard-Jones and Weeks-Chandler-Andersen model systems. The viscosity can be obtained in equilibrium MD simulation from the first moment of the viscuit PDF, which is shown for finite simulation lengths to give a less noisy plateau region than the Green-Kubo method. Two other formulas for the shear viscosity in terms of the viscuit and PDF analysis are also derived. A separation of the time-dependent average negative and positive viscuits extrapolated from the noise dominated region to zero time provides another route to η. The third method involves the relative number of positive and negative viscuits and their PDF standard deviations on the two sides for an equilibrium system. For the FT and finite shear rates, accurate analytic expressions for the relative number of positive to negative block average shear stresses is derived assuming a shifted Gaussian PDF, which is shown to agree well with non-equilibrium molecular dynamics simulations. A similar treatment of the positive and negative block average contributions to the viscosity is also shown to match the simulation data very well.