The transport property of thermal conductivity has been introduced as a phenomenological law by Joseph Fourier in 1822. Its molecular origins were later discovered with Boltzmann's and Maxwell's works on dilute gases. In metallic solids, the thermal conduction is mainly determined by the transport of electrons and thus related to the electric conductivity according to the Wiedemann-Franz law, as explained by Drude in 1900. In contrast, the thermal conductivity of non-metallic solids results from the mutual scattering of sound waves due to the anharmonicities of the interatomic forces, as shown by Peierls in 1929. In all these systems, the microscopic mechanism of thermal conduction is the scattering of the energy carriers, which induces local thermalization in a single stage process, and subsequent uniformization of temperature. Yet, this one-stage scenario does not exhaust the possible microscopic mechanisms. We will show that local thermalization can proceed in two stages in a further class of systems where local randomization takes place before the start of energy transfer leading to thermalization. On the basis of this two-stage mechanism of local thermalization, we provide a derivation of Fourier's law from first principles.