Fourier’s law: Insight from a simple derivation

The onset of Fourier’s law in a one-dimensional quantum system is addressed via a simple model of weakly coupled quantum systems in contact with thermal baths at their edges. Using analytical arguments we show that the crossover from the ballistic (invalid Fourier’s law) to diffusive (valid Fourier’s law) regimes is characterized by a thermal length scale, which is directly related to the profile of the local temperature. In the same vein, dephasing is shown to give rise to classical Fourier’s law, similarly to the onset of Ohm’s law in mesoscopic conductors.

Figure 1

(a) Schematic representation of the model. The system consists of identical, weakly connected subsystems. The left- and right-most subsystems are held at temperatures $T_{\text{L}}$ and $T_{\text{R}}$, respectively. (b) Representation of the transition process in $S_1$ (see text).

Figure 2

(Color online) The (normalized) local temperature $\left[T\left(x\right)-T_{\text{L}}\right]/\delta T$ as a function of the position variable $x$. This figure should be compared with Fig. 1 of Ref. [5].