Green—Kubo formula

The dlffuslvltles parallel to the pore walls at equilibrium were determined form the mean square particle displacements and the Green-Kubo formula as described In the previous subsection. The Green-Kubo Formula cannot be applied, at least In principle, for the calculation of the dlffuslvlty under flow. The dlffuslvlty can be still calculated from the mean square particle displacements provided that the part of the displacement that Is due to the macroscopic flow Is excluded. The presence of flow In the y direction destroys the symmetry on the yz plane. Hence the dlffuslvltles In the y direction (parallel to the flow) and the z direction (normal to the flow) can In principle be different. In order to calculate the dlffuslvltles the part of the displacement that Is due to the flow must of course be excluded. Therefore,... 

It is well known that each transport coefficients is given by a Green-Kubo formula or, equivalently, by an Einstein formula ... 

The starting point of the calculation is a Mori-type rephrasing of the Green-Kubo formula for the viscosity, which is given by Eq. (155). Thus... 

Equation [116] is the central result of this section and is one of the most important equations in nonequilibrium statistical mechanics. It is from this point that one makes contact with the more familiar relations relating transport coefficients to time correlation functions under the guise of the so-called Green-Kubo formulas.i2.39,40... 

Figure 11 The shear viscosity for liquid decane at T = 480 K and P = 170 atm calculated three different ways NEMD-NVT, NEMD-NPX and by means of the Green-Kubo formula (Eqs. [198]). The Green-Kubo value is actually at y = 0 but is placed on the y axis as a guide to the eye. The linear regime is reached when y 0.005 ps". Note the ensemble-dependent results in the shear thinning regime.

As discussed in the Introduction, considerable effort has been devoted to the study of transport properties. Specifically, viscosity estimates are of potential importance in the rational design of lubricants. In these simulations, two main approaches have been used. The earlier of the two focused on a linear response theory based Green-Kubo formula ... 

Integration of the double integral, Equation (7.91) leads to the Green-Kubo formula ... 

The derivation of the energy conductivity is based on the Green-Kubo formula. The energy exchange rate between these two sites is quantitatively measured by the... 

To relate Z t) with the friction coefficient we use the Green-Kubo formula and the Einstein relation [18, 19, 20] ... 

The search for an analytic expression for the velocity autocorrelation function, Z(t) = (l/3)(v(0).v(/)), where v(t) is the velocity of an arbitrary molecule at time t, has been of interest since the start of molecular simulation. The self-diffusion coefficient can be obtained from Z(t) using the Green-Kubo formula. 

Exercise. How does the use of periodic boundary conditions affect the evaluation of the Green-Kubo formula (99) The Einstein formula (lOl) ... 

In the following, the Green-Kubo formulas for self-diffusion, shear viscosity and thermal conductivity coefficients are compiled. A complete list of all the thermal transport coefficients of one- and two-component systems was given by Hoheisel Vogelsang (1988). 

Green-Kubo formulas and time correlation functions... 

The general form of Green-Kubo formulas has already been given in Section 9.1.1. However, for molecular liquids the microscopic formulation of the currents for self-diffusion, shear viscosity and thermal conductivity must be modified (Marechal Ryckaert 1983 Hoheisel 1993). 

Equation (15) is known as a Green-Kubo formula, as are many, but not all, relations of this form. Other important correspondences between transport properties and the integrals of time correlation functions include viscosity and pressure tensor thermal conductivity and energy current reaction rates... 

This equilibrium MD approach uses current fluctuations to calculate the thermal conductivity k via the fluctuation dissipation theorem. The MD approach is used to compute the autocorrelation function of the heat flux, which is related to the thermal conductivity by the Green-Kubo formula given by ... 

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