Hi,
It is stated that the Green-Kubo method for calculating thermal conductivity is an equilibrium method.
What will be the Green-Kubo calculation under a temperature difference applied such that there will be one-dimensional heat conduction in the system?
As you have said, the Green-Kubo relation holds at or near equilinrium. Therefore in a non-equilibrium simulation, while you can still certainly (try to) measure the relevant correlation functions, they are no longer straightforwardly related to the transport coefficients of interest. On the other hand, a non-equilibrium simulation often lets you measure the flux-field relationship directly, as long as it is correctly set up (which is the hard part).