Crossover from ballistic to diffusive thermal transport in quantum   Langevin dynamics study of a harmonic chain connected to self-consistent   reservoirs
  Through an exact analysis using quantum Langevin dynamics, we demonstrate the crossover from ballistic to diffusive thermal transport in a harmonic chain with each site connected to Ohmic heat reservoirs. The temperatures of the two heat baths at the boundaries are specified from before whereas the temperatures of the interior heat reservoirs are determined self-consistently by demanding that in the steady state, on average, there is no heat current between any such (self-consistent) reservoir and the harmonic chain. Essence of our study is that the effective mean free path separating the ballistic regime of transport from the diffusive one emerges naturally. 
Dibyendu Roy