Green-Kubo expressions

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]. 

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 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... 

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],... 

It should be clear that at equilibrium, Pxy)o = 0- We can compare the result of Eq. [125] to the Green-Kubo expression for the shear viscosity given by Eq. [121]. We can equate the left-hand side of Eq. [125] to r), and we then obtain the following remarkable result ... 

Transport properties, such as diffusion coefficients, shear viscosity, fhermal or electrical conductivity, can be determined from the time evolution of the autocorrelation function of a particular microscopic flux in a system in equilibrium based on the Green-Kubo formalism [217, 218] or the Einstein equations [219], Autocorrelation functions give an insight into the dynamics of a fluid and their Fourier transforms can be related to experimental spectra. The general Green-Kubo expression for an arbitrary transport coefficient y is given by ... 

Green-Kubo expressions and time correlation functions... 

Table 1 Examples of Green-Kubo Expressions for Navier-Stokes Transport Coefficients...

Bhargava and Balasubramanian[142] carried out equilibrium MD simulations with a fixed charge model of 1,3-dimethylimidazolium chloride ([mmim][Cl]) at 425 K. Using a Green-Kubo expression (that is, the long-time integral of the stress-stress time correlation function) they obtained a viscosity for [mmim][Cl] that was about four times higher than the experimental value for [emim][Cl]. Apparently, there were no experimental data at this temperature for [mmim][Cl], but the authors assumed that the values would be similar to [emim][Cl]. 

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... 

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 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. 

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 ... 

W. H. Miller The expression for the reaction rate (in terms of a flux-flux autocorrelation function) obtained by myself, Schwartz, and Tromp in 1983 is very similar (though not identical) to the one given earlier by Yamamoto. It is also an example of Green-Kubo relations. 

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 ... 

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. 

Inserting this expression into the linear response relation (2.17) and comparing it to the Green-Kubo relation (3.3) gives the following thermal conductivity,... 

Using these expression (Green-Kubo relations), Zwanzig (1965b) investigated the frequency dependences of the viscosities of a fluid composed of molecules with internal degrees of freedom which are weakly coupled to the center of mass (translational motions). He found that the bulk viscosity is... 

The kinetic coefficients can be expressed in terms of ordinary time-correlation functions. Such relations are called Green-Kubo relations (see Section 1 l.B). 

This expression is a Green-Kubo-like expression for the quantum selfdiffusion constant in CMD. Working backwards from its usual derivation shows that the self-diffusion constant can also be obtained from the long-time behavior of the slope of the centroid mean-squared displacement ... 

The search for an analytic expression for the velocity autocorrelation function, Z(t) = (l/3)(v(0).v(/)), where v(t) is the velocity of an arbitrary molecule at time t, has been of interest since the start of molecular simulation. The self-diffusion coefficient can be obtained from Z(t) using the Green-Kubo formula. 

The foundations of DPD have been considered in a number of publica-tions. The rules of dissipative particle dynamics were derived from the underlying molecular interactions by a systematic coarse graining procedure. Evans derived expressions for the self-diffusion coefficient and shear viscosity of the DPD particles in the form of the Green-Kubo time correlation functions. DPD can be used to model arbitrarily shaped objects made up of fused spheres by... 

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 first systematic attempts to calculate the viscosity and thermal conductivity of dense liquids evaluated the Green-Kubo (G-K) equations, and the Einstein relations for diffusion. The G-K expressions relate the numerical values of linear nonequilibrium transport coefficients to the relaxation of appropriate fluctuations in the equilibrium state (see Section 9.2). These studies of about 20 years ago were most important in that they gave a quantitative insight into the behavior of nonequilibrium systems and opened the door to the modem view of statistical mechanics (Hansen MacDonald 1986 McQuarrie 1976 Egelstaff 1992). From the practical standpoint, however,... 

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