  We report on the theoretical derivation of macroscopic thermal properties
(specific heat, thermal conductivity) of an electrically insulating rod
connected to two reservoirs, from the linear superposition of its mechanical
mode Brownian motions. The calculation is performed for a weak thermal
gradient, in the classical limit (high temperature). The development is kept
basic as far as geometry and experimental conditions are concerned, enabling an
almost fully analytic treatment. In the modeling, each of the modes is subject
to a specific Langevin force, which enables to produce the required temperature
profile along the rod. The theory is predictive: the temperature gradient (and
therefore energy transport) is linked to motion amplitude cross-correlations
between nearby mechanical modes. This arises because energy transport is
actually mediated by mixing between the modal waves, and not by the modes
themselves. This result can be tested on experiments, and shall extend the
concepts underlying equipartition and fluctuation-dissipation theorems. The
theory links intimately the macroscopic size of the clamping region where the
mixing occurs to the microscopic lengthscale of the problem at hand: the phonon
mean-free-path. This clamping region, which is key, has received recently a
renewed attention in the field of nanomechanics with topical works on "phonon
shields" and "soft clamping". We believe that our work should impact the domain
of thermal transport in nanostructures, with future developments of the theory
toward the quantum regime.
