  One of the most straightforward ways to study thermal properties beyond
linear response is to monitor the relaxation of an arbitrarily large left-right
temperature gradient $T_L-T_R$. In one-dimensional systems which support
ballistic thermal transport, the local energy currents $\langle j(t)\rangle$
acquire a non-zero value at long times, and it was recently investigated
whether or not this steady state fulfills a simple additive relation $\langle
j(t\to\infty)\rangle=f(T_L)-f(T_R)$ in integrable models. In this paper, we
probe the non-equilibrium dynamics of the Hubbard chain using density matrix
renormalization group (DMRG) numerics. We show that the above form provides an
effective description of thermal transport in this model; violations are below
the finite-time accuracy of the DMRG. As a second setup, we study how an
initially equilibrated system radiates into different non-thermal states (such
as the vacuum).
