  We study the energy current and its fluctuations in quantum gapless 1d
systems far from equilibrium modeled by conformal field theory, where two
separated halves are prepared at distinct temperatures and glued together at a
point contact. We prove that these systems converge towards steady states, and
give a general description of such non-equilibrium steady states in terms of
quantum field theory data. We compute the large deviation function, also called
the full counting statistics, of energy transfer through the contact. These are
universal and satisfy fluctuation relations. We provide a simple representation
of these quantum fluctuations in terms of classical Poisson processes whose
intensities are proportional to Boltzmann weights.
