How to get conductivity from Green function $\mathcal{G}(x_1,x_2,\tau)$ of inhomogeneous system?

Question

I'd like to study an inhomogeneous system, i.e., momentum is not a good quantum number therein. Therefore, I tried to calculate temperature Green functions like $\mathcal{G}(x_1,x_2;\tau)$, or its twofold Fourier transformation $\mathcal{G}(p_1,p_2;\tau)$.

But how can I get any transport property, e.g., conductivity, from these Green functions? I checked Mahan's overwhelming book, however, it only deals with the formalism of $\mathcal{G}(p;\tau)$ for homogeneous systems. Thanks in advance for any useful information.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user huotuichang

Comments

If your system is inhomogeneous or has messy boundaries, then the conductivity is probably not given by just one number but rather would be anisotropic.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user Nanite

Answer 1

Losing translation invariance is one of the big problem when studying disordered system (for example). In that case, one usually averages over the disorder to render the system translation invariant (on average), with its own technical difficulties. The applications for transport properties are explained in the Mahan or most good text book.

If the inhomogeneity comes in your case from another source, there might be other possibilities. You might also have to generalize the definition of the conductivity in your special case, but it is hard to tell without more details.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user Adam