  In this work we address the recent experiment of Altimiras and collaborators,
where an electron distribution function at the quantum Hall (QH) edge at
filling factor 2 has been measured with high precision. It has been reported
that the energy of electrons injected into one of the two chiral edge channels
with the help of a quantum point contact (QPC) is equally distributed between
them, in agreement with earlier predictions, one being based on the Fermi gas
approach, and the other utilizing the Luttinger liquid theory. We argue that
the physics of the energy relaxation process at the QH edge may in fact be more
rich, providing the possibility for discriminating between two physical
pictures in experiment. Namely, using the recently proposed non-equilibrium
bosonization technique we evaluate the electron distribution function and find
that the initial "double-step" distribution created at a QPC evolves through
several intermediate asymptotics, before reaching eventual equilibrium state.
At short distances the distribution function is found to be asymmetric due to
non-Gaussian current noise effects. At larger distances, where noise becomes
Gaussian, the distribution function acquires symmetric Lorentzian shape.
Importantly, in the regime of low QPC transparencies T the width of the
Lorentzian scales linearly with T, in contrast to the case of equilibrium Fermi
distribution, whose width scales as square root of T. Therefore, we propose to
do measurements at low QPC transparencies. We suggest that the missing energy
paradox may be explained by the nonlinear dispersion of the spectrum of edge
states.
