Recovering the Fourier law in harmonic chains: A Hamiltonian realization of the Debye/Visscher model

Quoting R. Peierls (1961) "there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity". And F. Bonetto, J.L. Lebowitz, L. Rey-Bellet: "Proving Fourier's law from first principles is an open problem in theoretical physics". To-day the scenario is basically the same: the Hamiltonian general foundation of Fourier heat conduction law in the thermodynamical limit has not yet been achieved. On the other hand, the Visscher phenomenological model for heat conduction was proved to yield the Fourier law in the thermodynamical limit. Here we suggest a pathway to remedy this situation, by providing a possible Hamiltonian foundation to the Visscher model. Hence, we identify the class of Hamiltonian model systems which, in the thermodynamical limit, would recover Fourier law. The dynamics of these Hamiltonian systems supports the idea that the physical mechanism to recover Fourier law is a linear response to a small temperature fluctuation around an equilibrium state, locally maintained by a strongly chaotic dynamics, in agreement with other models used to explain transport and bulk features of solids (e.g., Debye's law of specific heat). This result might have applications for nanoscale devices and in general systems with few degrees of freedoms, when efficient heat transmission is needed.