Viscosity Green-Kubo expression

It should be clear that at equilibrium, Pxy)o = 0- We can compare the result of Eq. [125] to the Green-Kubo expression for the shear viscosity given by Eq. [121]. We can equate the left-hand side of Eq. [125] to r), and we then obtain the following remarkable result ... 

Transport properties, such as diffusion coefficients, shear viscosity, fhermal or electrical conductivity, can be determined from the time evolution of the autocorrelation function of a particular microscopic flux in a system in equilibrium based on the Green-Kubo formalism [217, 218] or the Einstein equations [219], Autocorrelation functions give an insight into the dynamics of a fluid and their Fourier transforms can be related to experimental spectra. The general Green-Kubo expression for an arbitrary transport coefficient y is given by ... 

Bhargava and Balasubramanian[142] carried out equilibrium MD simulations with a fixed charge model of 1,3-dimethylimidazolium chloride ([mmim][Cl]) at 425 K. Using a Green-Kubo expression (that is, the long-time integral of the stress-stress time correlation function) they obtained a viscosity for [mmim][Cl] that was about four times higher than the experimental value for [emim][Cl]. Apparently, there were no experimental data at this temperature for [mmim][Cl], but the authors assumed that the values would be similar to [emim][Cl]. 

Using these expression (Green-Kubo relations), Zwanzig (1965b) investigated the frequency dependences of the viscosities of a fluid composed of molecules with internal degrees of freedom which are weakly coupled to the center of mass (translational motions). He found that the bulk viscosity is... 

The foundations of DPD have been considered in a number of publica-tions. The rules of dissipative particle dynamics were derived from the underlying molecular interactions by a systematic coarse graining procedure. Evans derived expressions for the self-diffusion coefficient and shear viscosity of the DPD particles in the form of the Green-Kubo time correlation functions. DPD can be used to model arbitrarily shaped objects made up of fused spheres by... 

Let us first consider nonequilibrium properties of dense fluids. Linear response theory relates transport coefficients to the decay of position and velocity correlations among the particles in an equilibrium fluid. For example, the shear viscosity ti can be expressed in Green-Kubo formalism as a time integral of a particular correlation function ... 

The first systematic attempts to calculate the viscosity and thermal conductivity of dense liquids evaluated the Green-Kubo (G-K) equations, and the Einstein relations for diffusion. The G-K expressions relate the numerical values of linear nonequilibrium transport coefficients to the relaxation of appropriate fluctuations in the equilibrium state (see Section 9.2). These studies of about 20 years ago were most important in that they gave a quantitative insight into the behavior of nonequilibrium systems and opened the door to the modem view of statistical mechanics (Hansen MacDonald 1986 McQuarrie 1976 Egelstaff 1992). From the practical standpoint, however,... 

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