Equation of State for the Direct Turbulent-Wave Cascade in a 2D Bose Gas

The nonequilibrium equation of state provides an effective thermodynamic-like description of far-from-equilibrium systems that exhibit steady states. Here, we experimentally construct the equation of state for the direct energy cascade in a turbulent two-dimensional Bose gas. Under continuous energy injection at a large length scale and matching dissipation at a small length scale, a nonthermal stationary state emerges, featuring a power-law cascade spectrum. We establish that the steady-state cascade is sustained by a scale-invariant energy flux that is independent of the dissipation scale, in accordance with the zeroth law of turbulence. Our equation of state takes the form of a power law which links the cascade amplitude with the underlying energy flux and holds universally across different drive strengths, gas densities, and interaction strengths.