Green-Kubo integration

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

Before leaving the topic of Green-Kubo integrals for transport properties, we mention briefly the characteristics of the electric current correlation functions that are used to compute the electrical conductivity. Figure 18 shows the electric current and velocity autocorrelation functions for [C2mim][Cl] at 486 K and 1 bar. The current fluctuations decay rapidly and appear to vanish... 

As A x was supposed stationary the integral is independent of time. The effect of the fluctuations is therefore to renormalize A0 by adding a constant term of order a2 to it. The added term is the integrated autocorrelation function of At. In particular, if one has a non-dissipative system described by A0, this additional term due to the fluctuations is usually dissipative. This relation between dissipation and the autocorrelation function of fluctuations is analogous to the Green-Kubo relation in many-body systems 510 but not identical to it, because there the fluctuations are internal, rather than added as a separate term as in (2.1). 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

The generalized Green-Kubo relations contain quantities integrated/averaged over the whole sample volume. Thus, the aspect of translational invariance/homogeneity does not become an issue in (5) yet. A system is translational invariant if the correlation between two points r and r depends on the distance r - r between the two points only. The correlation must not change if both points are shifted by the same amount. (Additionally, any quantity depending on one space point r only, must be constant.) A system would be isotropic if, additionally, the correlation only... 

The limit has been inserted to insure the convergence of this integral. This is an example of a Green-Kubo relation that relates the transport coefficient A of a property A to a time integral of the ordinary time-correlation function of its corresponding flux /<°> (e.g., see Zwanzig, 1965, Berne and Forster, 1971). 

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

Time correlation functions can be used in conjunction with the Green-Kubo relations to calculate the various transport coefficients in the system. For example, the self-diffusion coefficient D is related to the time integral of the velocity correlation function ... 

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 ... 

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