Green Kubo formalism valid for inhomogeneous systems? I'm interested in nano-composites and their effective properties and I use classical Molecular Dynamics as computation method.  My question is:


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*"Can I still use the Green Kubo formula to calculate to transport properties of the whole system?"

 A: In principle you could, but people don't do this very often for MD simulations of inhomogeneous systems.
The transport coefficients (diffusion coefficient, heat conductivity and etc.) are tensors actually. When these tensor acts on the corresponding gradients (temperature, concentration gradients and etc.), we obtained the corresponding flux. In homogeneous systems, all of the diagonal terms of the tensors have the same value, and this is the transport coefficients we usually talked about.
For inhomogeneous systems, those tensor elements become more complicated. Take the diffusion coefficient as an example, in slab geometry (the system is inhomogeneous in $z$ direction but homogeneous in $x$ and $y$ direction), as suggested by this paper(http://www.edocs.fu-berlin.de/docs/servlets/MCRFileNodeServlet/FUDOCS_derivate_000000003238/vonHansen-PRL-2013-111-118103.pdf), the diffusion tensor should have three diagonal elements $D_{xx}(z), D_{yy}(z)$ and $D_{zz}(z)$, where $D_{xx}(z) = D_{yy}(z)$ since there is no confinement in the $xy$ direction. It should be noticed that all these tensor elements have position dependence on $z$. 
As mentioned in this paper, because the diffusion in $x$ and $y$ directions can be still viewed as free diffusion, the corresponding diffusion tensors can be still calculated using the mean squared displacement (MSD). Therefore it is possible to compute them with Green-Kubo relation as well.
For the diffusion along the $z$ direction, this paper used some other methods to extract the corresponding diffusion tensor element $D_{zz}(z)$ because the diffusion is confined.
In my personal opinion, whether to use position-dependent tensor or a single constant to characterize transport depends on what kind of resolution you want. For some inhomogeneous systems (e.g. ion transport in amorphous polymer), we use a single diffusion coefficient to characterize the ion diffusion sometimes because we only care the overall diffusion and the net flux from the ion, and we don't care about the microscopic diffusive behavior. On the other hand, position-dependent tensors could provide more information about transport at microscopic level. 
The Green-Kubo relation is based on the linear response formalism in statistical mechanics.For inhomogeneous systems, in principle you could also write down the linear response formula and derive the corresponding Green-Kubo formula. I haven't done something like this, but I guess if you use position-dependent tensors to represent your transport coefficients, the corresponding Green-Kubo formula would be much more complicated than the homogeneous case.
