  The non-equilibrium dynamics of stochastic light in a coherently-driven
nonlinear cavity resembles the equilibrium dynamics of a Brownian particle in a
scalar potential. This resemblance has been known for decades, but the
correspondence between the two systems has never been properly assessed. Here
we demonstrate that this correspondence can be exact, approximate, or break
down, depending on the cavity nonlinear response and driving frequency. For
weak on-resonance driving, the nonlinearity vanishes and the correspondence is
exact: The cavity dissipation and driving amplitude define a scalar potential,
the noise variance defines an effective temperature, and the intra-cavity field
satisfies Boltzmann statistics. For moderately strong non-resonant driving, the
correspondence is approximate: We introduce a potential that approximately
captures the nonlinear dynamics of the intra-cavity field, and we quantify the
accuracy of this approximation via deviations from Boltzmann statistics. For
very strong non-resonant driving, the correspondence breaks down: The
intra-cavity field dynamics is governed by non-conservative forces which
preclude a description based on a scalar potential only. We furthermore show
that this breakdown is accompanied by a phase transition for the intra-cavity
field fluctuations, reminiscent of a non-Hermitian phase transition. Our work
establishes clear connections between optical and stochastic thermodynamic
systems, and suggests that many fundamental results for overdamped Langevin
oscillators may be used to understand and improve resonant optical
technologies.
