Nineteenth-century thermodynamics described gases with equations of state: pressure, volume, temperature linked by constitutive relations. Van der Waals added particle interactions — attractive at long range, repulsive at short — and got phase transitions, critical points, Joule-Thomson cooling.
Lian, Xiong, Chen, and Lu show the same mathematics works for light in a multimode optical waveguide.
In a multimode system, each mode carries some fraction of the total optical power. At equilibrium, power distributes across modes according to a statistical mechanical principle: the mode occupation follows from maximizing entropy subject to conservation constraints. The linear theory (no interactions between modes) gives an ideal-gas-like equation of state. The distribution is thermal.
Nonlinear inter-mode coupling changes this. Within a mean-field approximation, the nonlinearity renormalizes the linear spectrum — each mode's effective frequency shifts proportionally to the total power in other modes. This is exactly the van der Waals correction: the “volume” available to each mode decreases, and the “pressure” (optical power per mode) increases at high densities.
The resulting equation of state predicts power localization — the optical analog of condensation, where a macroscopic fraction of the power concentrates in a few modes. It predicts heating and cooling during optical Joule-Thomson expansion — the analog of gas cooling when expanding through a throttle, where the modes redistribute power as the system relaxes through a constriction.
The mapping is not a metaphor. The mathematical structure is identical: a free energy with a mean-field correction, a partition function with an effective potential, critical exponents determined by the same universality. The tools that van der Waals used in 1873 to describe steam apply without modification to photons in a nonlinear fiber.