Calculating the electrical conductivity of a metal nanostructure

The general system is shown in the following picture. There are two electrodes and a potential applied to them, so there is current flowing in the system. The electrodes are connected with an arbitrary shape nanostructure (nanowire). The dimensions of $h$ and $d$ are from $10$ to $100\ nm$.

I want to find the current distribution for this system. I am using FEM to solve the partial differential equation for currents: $$\nabla\cdot (\sigma(\vec{r}) \nabla\phi(\vec{r})) = 0$$

The problem is the conductivity $\sigma(\vec{r})$, which clearly is lower in the nanostructure than the bulk value. In Zhang - Nano/Microscale Heat Transfer, there is an overview how to calculate the conductivity for straight infinite nanowire, I can try to apply that for individual segments of the nanoconnection, but I feel that it would be very inaccurate.

I have heard that I could calculate the exact conductivity distribution in the system from Boltzmann transport equation, is this true? Or is there another way to find the conductivity distribution in the system?

Edit: The conductivity can be calculated from mean free path (MFP). I guess one possibility would be to pick a point in the system and shoot electrons randomly in all directions. If the electrons reach a surface, then they will scatter and if they reach the bulk MFP, they will scatter from atoms. And then average all the scattering lengths to find the MFP in that point. Would this give an accurate estimation of the MFP distribution (and also the conductivity distribution)?