Mesoscopic diffusion as a non-Markov process

We use both classical and quantum S-matrix theories to derive expressions for the ballistic diffusion current of a 1D mesoscopic crystalline solid. For both theories, we get the same local Fick’s law, thus proving that local Fick’s law and its associate local diffusion coefficient are incoherent results. The non-local Landauer diffusion coefficient is an incoherent result valid only for solids with a length larger than the coherence length. For solids with a length less than the coherence length, we show that quantum interference gives a new contribution for the non-local diffusion coefficient.