  We study heat transport in a gas of one-dimensional fermions in the presence
of a small temperature gradient. At temperatures well below the Fermi energy
there are two types of relaxation processes in this system, with dramatically
different relaxation rates. As a result, in addition to the usual thermal
conductivity, one can introduce the thermal conductivity of the gas of
elementary excitations, which quantifies the dissipation in the system in the
broad range of frequencies between the two relaxation rates. We develop a
microscopic theory of these transport coefficients in the limit of weak
interactions between the fermions.
