Equation of State for Turbulence in the Gross-Pitaevskii Model

We report the numerical observation of a far-from-equilibrium equation of state (EOS) in the Gross-Pitaevskii (GP) model. We first show that the momentum distribution of the turbulent cascade is well described by wave-turbulent kinetic theory in the appropriate limits. Calculating the energy and particle fluxes Π_{ϵ}(k) and Π_{N}(k), we show that the turbulent state possesses the hallmarks of a direct energy cascade. Building on this, we show that the GP model encodes a universal EOS in the form of a relationship between the turbulent cascade's momentum distribution amplitude n_{0} and the energy flux ε in the steady state. We find that in our regime of "mixed" turbulence-where both vortices and waves play a significant role-n_{0}∝ε^{0.67(2)}, a result that is not captured by any existing theory of turbulence but that agrees with a recent experimental measurement for large energy fluxes. Finally, we find that the concept of quasi-static thermodynamic processes between equilibrium states extends to far-from-equilibrium steady states.