Green-Kubo formulation

An objective question, however, is whether the thermostat is able to produce correct time-correlation functions (at least in the hmit of a macroscopic system). Since transport coefficients (e.g., the diffusion constant) can be calculated either as ensemble averages (Einstein formulation) or as integrals of a time-correlation function (Green-Kubo formulation), at least such integrals should be correct if the thermostat leads to a canonical ensemble. When this is the case, it has been shown that the correlation functions themselves are also correct at least for some thermostats [91, 86]. 

Beside the Green-Kubo and the Einstein formulations, transport properties can be calculated by non-equilibrium Ml) (NEMD) methods. These involve an externally imposed field that drives the system out of the equilibrium. Similar to experimental approaches, the transport properties can be extracted from the longtime response to this imposed perturbation. E.g., shear flow and energy flux perturbations yield shear viscosity and thermal conductivity, respectively. Numerous NEMD algorithms can be found in the literature, e.g., the Dolls tensor [221], the Sllod algorithm [222], or the boundary-driven algorithm [223]. A detailed review of several NEMD approaches can be found, e.g., in [224]. 

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). 

The calculation of the thermal conductivity of gas hydrate using EMD and the Green-Kubo linear response theory was repeated recently. In that work, convergences of the relevant quantities were monitored carefully as a function of the model size. Subtleties in the numerical procedures were also carefully considered. The thermal conductivity of methane hydrate was found to converge within numerical accuracy for 3 x 3 x 3 and 4x4x4 supercells. In the calculation of the heat flux vector there is an interactive term that is a pairwise summation over the forces exerted by atomic sites on one another. The species (i.e., water and methane) enthalpy correction term requires that the total enthalpy of the system is decomposed into contributions from each species. Because of the partial transformation from pairwise, real-space treatment to a reciprocal space form in Ewald electrostatics, it is necessary to recast the diffusive and interactive terms in this expression in a form amenable for use with the Ewald method using the formulation of Petravic. ... 

The gas diffusion coefficients can be estimated either from MD simulations by using the Einstein (eqn (1.5)) or by means of the Green-Kubo (eqn (1.6)) formulations ... 

This condition was first recognized as a property of thermal Green functions by Kubo, Martin, and Schwinger (hence its name KMS). We begin by formulating it in its most naive form. Let si = and H... 

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