Green-Kubo, relations

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... 

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. 

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). 

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). 

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. 

Thus, we find that a dilemma is created. We have to make a choice between two possible ways out. The first is that the Green-Kubo relation of Eq. (180) does not apply, and the second is that the Green-Kubo relation of Eq. (180) does apply, but the common sense expectation that a weak perturbation creates a finite current is violated. 

Let us make a final comment, concerning the violation of the Green-Kubo relation. There is a close connection between the breakdown of this fundamental prescription of nonequilibrium statistical physics and the breakdown of the agreement between the density and trajectory approach. We have seen that the CTRW theory, which rests on trajectories undergoing abrupt and unpredictable jumps, establishes the pdf time evolution on the basis of v /(f), whereas the density approach to GME, resting on the Liouville equation, either classical or quantum, and on the convenient contraction over the irrelevant degrees of freedom, eventually establishes the pdf time evolution on the basis of a correlation function, the correlation function in the dynamical case... 

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 ... 

In the molecular dynamics calculations the trajectories of methane molecules in the pore are followed using the equation of motion with appropriate temperature control. A diffuse reflection condition is applied at the pore wall. For the EMD simulations a collective transport coefficient obtained from autocorrelation of the fluctuating axial streaming velocity via a Green-Kubo relation [S]... 

In the earliest paper on FRs, a footnote remarked that it is possible to obtain the Green-Kubo relations from them if the fluctuations have a Gaussian distribution. ... 

When we evaluate the Green-Kubo relations for the transport coefficients we solve the equations of motion for the molecules. They are often modelled as rigid bodies. Therefore we review some of basic definition of rigid body dynamics [10]. The centres of mass of the molecules evolve according to the ordinary Newtonian equations of motion. The motion in angular space is more complicated. Three independent coordinates a, =(a,a,-2, ,3), i = 1, 2,. . N where N is the number of molecules, are needed to describe the orientation of a rigid body. (Note that a, is not a vector because it does not transform like a vector when the coordinate system is rotated.) The rate of change of a, is... 

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,... 

Comparison of the thermal conductivities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (3.3). The entries for finite field have been obtained by applying the heat flow algorithm (3.5). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

S. Sarman, Green-Kubo Relations for the Viscosity of Biaxial Nematic Liquid Crystals, J. Chem. Phys. 105 (1996) 4211 ... 

Liquid simulation studies have been essential in assessing the applicability of various fluctuation relations to real physical systems. These are important relations in nonequilibrium statistical mechanics that are valid far from equilibrium and can be used to derive Green-Kubo relations for transport coefficients.223,224 They show how thermodynamic irreversibility emerges from... 

Equilibrium MD simulations of self-diffusion coefficients, shear viscosity, and electrical conductivity for C mim][Cl] at different temperatures were carried out [82] The Green-Kubo relations were employed to evaluate the transport coefficients. Compared to experiment, the model underestimated the conductivity and self-diffusion, whereas the viscosity was over-predicted. These discrepancies were explained on the basis of the rigidity and lack of polarizability of the model [82], Despite this, the experimental trends with temperature were remarkably well reproduced. The simulations reproduced remarkably well the slope of the Walden plots obtained from experimental data and confirmed that temperature does not alter appreciably the extent of ion pairing [82],... 

The structure of a water/silicon interface was studied (Ursenbach et al, 1997), in addition to a water/copper interface (Halley et al, 1998) and a water/palladium interface (Klesingeza/., 1998). Finally, two studies have used Car-Parrinello simulations in conjunction with the Green-Kubo relations to calculate viscosities in liquid metals (Alfe and Gillan, 1998 Stadler et al, 1999). 

The formally exact general result for T in (4b) can be applied to compute the thermodynamic transverse stress, (7(y) = Oxy /V. Equation (4b) leads to an exact non-linear Green-Kubo relation ... 

The generalized Green-Kubo relations contain quantities integrated/averaged over the whole sample volume. Thus, the aspect of translational invariance/homogeneity does not become an issue in (5) yet. A system is translational invariant if the correlation between two points r and r depends on the distance r - r between the two points only. The correlation must not change if both points are shifted by the same amount. (Additionally, any quantity depending on one space point r only, must be constant.) A system would be isotropic if, additionally, the correlation only... 

The mode coupling approximations Introduced above can now be applied to the exact generalized Green-Kubo relations (5d). Steady state expectation values are approximated by projection onto pair density modes, giving... 

The familiar shear modulus of linear response theory describes thermodynamic stress fluctuations in equilibrium, and is obtained from (5b, lid) by setting y = 0 [1, 3, 57], While (5b) then gives the exact Green-Kubo relation, the approximation (lid) turns into the well-studied MCT formula (see (17)). For finite shear rates, (lid) describes how afflne particle motion causes stress fluctuations to explore shorter and shorter length scales. There the effective forces, as measured by the gradient of the direct correlation function, = nc = ndck/dk, become smaller, and vanish asympotically, 0 the direct correlation function is connected... 

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... 

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