  We show that the orbital angular momentum (OAM) of a light field can be
thermalized in a nonlinear cylindrical multimode optical waveguide. We find,
that upon thermal equilibrium, the maximization of the optical entropy leads to
a generalized Rayleigh-Jeans distribution that governs the power modal
occupancies with respect to the discrete OAM charge numbers. This distribution
is characterized by a temperature that is by nature different from that
associated with the longitudinal electromagnetic momentum flow of the optical
field. Counterintuitively and in contrast to previous results, we demonstrate
that even under positive temperatures, the ground state of the fiber is not
always the most populated in terms of power. Instead, because of OAM, the
thermalization processes may favor higher order modes. A new equation of state
is derived along with an extended Euler equation -- resulting from the
extensivity of the entropy itself. By monitoring the nonlinear interaction
between two multimoded optical wavefronts with opposite spins, we show that the
exchange of angular momentum is dictated by the difference in OAM temperatures,
in full accord with the second law of thermodynamics. The theoretical analysis
presented here is corroborated by numerical simulations that take into account
the complex nonlinear dynamics of hundreds of modes. Our results may pave the
way towards high power optical sources with controllable orbital angular
momenta, and at a more fundamental level, could shed light on the physics of
other complex multimoded nonlinear bosonic systems that display additional
conservation laws.
