  We consider the free propagation geometry of a light beam (or fluid of light)
in a multimode waveguide. As a result of the effective photon-photon
interactions, the photon fluid thermalizes to an equilibrium state during its
conservative propagation. In this configuration, Rayleigh-Jeans (RJ)
thermalization and condensation of classical light waves have been recently
observed experimentally in graded index multimode optical fibers characterized
by a 2D parabolic trapping potential. As well-known, the properties of RJ
condensation differ substantially from those of Bose-Einstein (BE)
condensation: The condensate fraction decreases quadratically with the
temperature for BE condensation, while it decreases linearly for RJ
condensation. Furthermore, for quantum particles the heat capacity tends to
zero at small temperatures, and it takes a constant value in the classical
particle limit at high temperatures. This is in contrast with classical RJ
waves, where the specific heat takes a constant value at small temperatures,
and tends to vanish above the condensation transition in the normal
(uncondensed) state. Here, we reconcile the thermodynamic properties of BE and
RJ condensation: By introducing a frequency cut-off inherent to light
propagation in a waveguide, we derive generalized expressions of the
thermodynamic properties that include the RJ and BE limits as particular cases.
We extend the approach to encompass negative temperatures. In contrast to
positive temperatures, the specific heat does not display a singular behavior
at negative temperatures, reflecting the non-critical nature of the transition
to a macroscopic population of the highest energy level. Our work contributes
to understanding the quantum-to-classical crossover in the equilibrium
properties of light, within a versatile experimental platform based on
nonlinear optical propagation in multimode waveguides.
