  Nonlinear multimode optical systems have attracted substantial attention due
to their rich physical properties. Complex interplay between the nonlinear
effects and mode couplings makes it difficult to understand the collective
dynamics of photons. Recent studies show that such collective phenomena can be
effectively described by a Rayleigh-Jeans thermodynamics theory which is a
powerful tool for the study of nonlinear multimode photonic systems. These
systems, in turn, offer a compelling platform for investigating fundamental
issues in statistical physics, attributed to their tunability and the ability
to access negative temperature regimes. However, to date, a theory for the
nonequilibrium transport and fluctuations is yet to be established. Here, we
employ the full counting statistics theory to study the nonequilibrium
transport of particle and energy in nonlinear multimode photonic systems in
both positive and negative temperature regimes. Furthermore, we discover that
in situations involving two reservoirs of opposite temperatures and chemical
potentials, an intriguing phenomenon known as the loop current effect can
arise, wherein the current in the positive energy sector runs counter to that
in the negative energy sector. In addition, we numerically confirm that the
fluctuation theorem remains applicable in optical thermodynamics systems across
all regimes, from positive temperature to negative ones. Our findings closely
align with numerical simulations based on first-principles nonlinear wave
equations. Our work seeks to deepen the understanding of irreversible
non-equilibrium processes and statistical fluctuations in nonlinear many-body
photonic systems which will enhance our grasp of collective phenomena of
photons and foster a fruitful intersection between optics and statistical
physics.
