Green-Kubo method

Let us imagine a set of particles whose spatial coordinates x obey the diffusion prescription of Eq. (125). Let us imagine that a time t = 0 an external perturbation is suddenly applied to these particles. Using the standard Green-Kubo method [61], we find [49] for the response of the system the following expression ... 

Sarman and Evans [24, 32] performed a comprehensive study of the flow properties of a variant of the Gay-Beme fluid. In order to make the calculations faster the Lennard-Jones core of the Gay-Beme potential was replaced by a 1/r core. This makes the potential more short ranged thereby decreasing the number of interactions and making the simulation faster. The viscosity coefficients were evaluated by EMD Green-Kubo methods both in the conventional canonical ensemble and in the fixed director ensemble. The results were cross checked by shear flow simulations. The studies covered nematic phases of both prolate ellipsoids with a length to width ratio of 3 1 and oblate ellipsoids with a length to width ratio of 1 3. The complete set of potential parameters for these model systems are given in Appendix II. 

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]. 

We note that in Ref. 87 the projection method was used to study the effect of friction on a process that, in the free case, is known to produce a Levy walk. As already discussed in Section VIII, the key point has been that the violation of the Green-Kubo relation must imply a different form of linear response, used later to justify the WS form of noncanonical equilibrium. It is worth mentioning that an interesting result of that article has been the following equation of motion ... 

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. 

The combined limitations of direct and Green-Kubo simulations mean that neither may be satisfactory if one is interested in the small, but nonzero, field limit. To accomplish this, two simulation techniques have been developed. The first is commonly known as the subtraction method because it is based on... 

Green-Kubo (equilibrium) approach, 284 nonequilibrium MD (NEMD) methods,... 

Another theme often explored with model alkanes is the relative merits of equilibrium MD (EMD) and non-equilibrium MD (NEMD) as methods for obtaining transport coefficients. Dysthe et al. explored some aspects of the methodology used to obtain transport coefficients by MD. They applied the Green-Kubo formalism to flexible multicentre models fi om linear and branched... 

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 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. ... 

For the flexible model simulations two different approaches were used when calculating those properties the first one was performing the simulations in the NVT ensemble, as has been done in several previous works, even they do not used the Green-Kubo relations the calculation but other statistical equivalent methods, and the second one was by simulating the system in the NVE ensemble, as previously done by some authors (Rey-Castro Vega, 2006 Rey-Castro et al., 2007). 

---