# Why Kubo formula can be applied to calculate conductivity?

It seems that Kubo formula is widely adopted to calculate conductivity, or at least Hall conductivity [for example, in the famous paper by TKNN: PRL 49 405-408 (1982)].

However, the derivation of Kubo formula (for example, see Sec. 4.3 in http://www.damtp.cam.ac.uk/user/tong/kintheory/four.pdf) relies on linearizing the evolution operator $$\overline{\exp}[-\mathrm{i}\int_{0}^{t}V(t')\mathrm{d}t']\approx 1-\mathrm{i}\int_{0}^{t}V(t')\mathrm{d}t',$$ where $$\overline{\exp}$$ denotes ordered exponential and $$V$$ is a Hermitian operator (typically perturbative Hamiltonian in the interaction picture in the derivation of Kubo formula).

The approximation above is indeed valid for small $$V$$ and small $$t$$. However, in my opinion, if $$t$$ is large enough, the approximation above loses validity no matter how small $$V$$ is. Therefore, Kubo formula should only be applicable for a small time scale.

In transport experiments, the measured system usually enters steady state and the time scale might be quite large. In this case, What justifies the application of Kubo formula in the theory of conductivity?

To me, the central idea behind Kubo's relation is that of matching''. We have a microscopic quantum many-body theory that, in principle, can be used to compute non-equilibrium properties of a system. In practice, however, these calculations are too difficult to be practical (and they contain "too much information" -- we don't actually want to know exact many-body wave functions). We therefore make use of the fact that the low energy, long distance limit is described by a simpler hydrodynamic theory. This theory can be established on general grounds, but it does contain effective parameters, transport coefficients like conductivities, viscosities, etc.