Green-Kubo expressions theory

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

This leads us to express the response on the basis of the perturbed v /(f) and, if the perturbation is very weak, on the basis of the unperturbed v /(f), thereby making us move in a direction different from the path adopted by the conventional approach to the response to external perturbation. If the function /(f) has an inverse power-law form, the external perturbation may have the effect of truncating this inverse power-law form. We notice that a weak perturbation affects the low modes of the system of interest, which are responsible for the long-time property of the function v /(f), if it has an inverse power-law form. Thus, a power-law truncation may well be realized, with a consequent significant departure from the prediction of the Green-Kubo theory. 

A published derivation of the Green-Kubo or fluctuation-dissipation expressions from the combination of the FR and the central limit theorem (CLT) was finally presented in 2005. This issue had been addressed previously and the main arguments presented, but subtleties in taking limits in time and field that lead to breakdown of linear response theory at large fields, despite the fact that both the FR and CLT apply, " were not fully resolved. ... 

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

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 thermal conductivity of a material can be calculated directly from equilibrium molecular dynamics (EMD) simulation based on the linear response theory Green-Kubo relationship. " The fluctuation-dissipation theorem provides a connection between the energy dissipation in irreversible processes and the thermal fluctuations in equilibrium. The thermal conductivity tensor. A, can be expressed in terms of heat current autocorrelation correlation functions (HCACFs),/, ... 

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. ... 

As noted above, the SCF theory of the conductivity deserves further attention. Nevertheless in the case of the simple mean field theory of Section 3.8.5, Edwards (1970b) notes that the conductivity is identically zero for the localized states of E < 0. The details proceed rather simply as before. In the Kubo-Greenwood-Peierls (Kubo (1956) (1957)) expression for theton-ductivity, it is necessary to obtain the average of the product of two Green s functions... 

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