Green-Kubo formulas transport coefficients

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

Equation [116] is the central result of this section and is one of the most important equations in nonequilibrium statistical mechanics. It is from this point that one makes contact with the more familiar relations relating transport coefficients to time correlation functions under the guise of the so-called Green-Kubo formulas.i2.39,40... 

In the following, the Green-Kubo formulas for self-diffusion, shear viscosity and thermal conductivity coefficients are compiled. A complete list of all the thermal transport coefficients of one- and two-component systems was given by Hoheisel Vogelsang (1988). 

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 ... 

We observe that the level density is recovered if the observable is the identity, A = I, since i dt = Tp. The above formula may be generalized to situations where the observable itself depends on time, for example, A = R exp(+iHt/h)S exp(- iHt/h), where R and S are other observables, as is the case in the Green-Kubo-Yamamoto-Zwanzig formulas for the transport and reaction rate coefficients. 

The behavior of a system interacting with an external electromagnetic field has been discussed by Callen and Welton. Subsequently Kubo, 3 Landau and Lifshitz,i and others obtained a formula for electrical conductivity which is of the same tjq)e as the Green formulae for the other transport coefficients. Our approach is similar to that of these authors insofar as it is based on the consideration of external forces, but, as will be seen, there is a basic difference between electrical and non-electrical processes. 

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