# Derivation of the Landauer formula for phonons using Nonequilibrium Green's functions

I am currently trying to understand this paper: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.96.255503

I really like their derivation of the Landauer formula for phonons using Nonequlibrium Green's functions (NEGF) because of its compactness. I am having trouble understanding it though since it has been abbreviated a lot. Can someone with a deeper knowledge of quantum physics and/or Green's functions help me understand?

Especially I am wondering about how they get to equation number (3):

$J_{th} = -\lim_{t'-t} \sum_{i\in L, j \in S, \alpha\beta=x,y,z} \frac{k_{i\alpha,j\beta}}{2} [\mathrm{i} \hbar \frac{d}{dt'} G_{i\alpha,j\beta}^<(t, t') +h.c.]$.

How do they get the limit and the time derivation from $J_{th} = -\langle \dot H_L \rangle = -\langle [H_L, H_{sys}]\rangle$? I don't see that coming from the commutator. Any ideas?