Green-Kubo

The final result that we wish to present in this connection is an example of the Green-Kubo time-correlation expressions for transport coefficients. These expressions relate the transport coefficients of a fluid, such as... 

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,... 

This can be obtained also from the velocity according to the Green-Kubo relation ... 

Sindzinger and Gillan have calculated the thermal conductivity for NaCl and KCl melts as well as for sohds on the basis of MD simulations in Ml thermal equilibrium using the Green-Kubo relations (Table 17). In a single molten salt system, the local fluxes jz and of charge and energy... 

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]. 

It is now shown that the steady-state probability density, Eq. (160), gives the Green-Kubo expression for the linear transport coefficient. Linearizing the exponents for small applied forces, Xr x, < 1, and taking the transport coefficient to be a constant, gives... 

In the intermediate regime, this may be recognized as the Green-Kubo expression for the thermal conductivity [84], which in turn is equivalent to the Onsager expression for the transport coefficients [2]. 

This result is a very stringent test of the present expression for the steady-state probability distribution, Eq. (160). There is one, and only one, exponent that is odd, linear in Xr, and that satisfies the Green-Kubo relation. 

The Green-Kubo result demands that this be equated to the negative of the natural rate of change of the first energy moment, Eq. (260), which means that... 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

In order to compute the discrete Green-Kubo expression for D we must evaluate correlation function expressions of the form (v L/v ). Consider... 

The colhsional contribution arises from grid shifting and accounts for effects on scales where the dimensionless mean free path is small, 1. The discrete Green-Kubo derivation leading to Eq. (52) involves a number of subtle issues that have been discussed by Ihle, Tiizel, and Kroll [26]. 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Computer simulations of transport properties using Green-Kubo relations... 

T. Ihle and D. M. Kroll, Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green—Kubo relations, Phys. Rev. E 67, 066705 (2003). 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

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

Onsager reciprocity relations as well as the Green-Kubo and Einstein formulas for these coefficients ... 

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 ... 

Because the fluid is in equilibrium, any ensemble average property should not change with time. Hence, the ensemble average of (u(tf)u(t")> depends only on the relative difference of time, t — t". That is, it is a stationary process. On transforming the time variables to f and r = tr — f" (rather like the centre of diffusion coefficient transformation of Chap. 9, Sect. 2), the Green—Kubo expression for the diffusion coefficient is obtained [453, 490],... 

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). 

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