  We develop a theory of thermal transport of weakly interacting electrons in
quantum wires. Unlike higher-dimensional systems, a one-dimensional electron
gas requires three-particle collisions for energy relaxation. The fastest
relaxation is provided by the intrabranch scattering of comoving electrons
which establishes a partially equilibrated form of the distribution function.
The thermal conductance is governed by the slower interbranch processes which
enable energy exchange between counterpropagating particles. We derive an
analytic expression for the thermal conductance of interacting electrons valid
for arbitrary relation between the wire length and electron thermalization
length. We find that in sufficiently long wires the interaction-induced
correction to the thermal conductance saturates to an interaction-independent
value.
