An Equation of State for Turbulence in the Gross-Pitaevskii model

TL;DR

This paper demonstrates a universal equation of state for turbulence in the Gross-Pitaevskii model, linking momentum distribution amplitude and energy flux, and extends thermodynamic concepts to far-from-equilibrium states.

Contribution

It introduces a novel EOS for turbulence in the GP model, revealing a new scaling law and extending thermodynamic ideas to non-equilibrium steady states.

Findings

Momentum distribution matches wave-turbulent kinetic theory

Identifies a universal relationship between $n_0$ and $\epsilon$

Confirms thermodynamic processes apply far from equilibrium

Abstract

We report the numerical observation of a far-from-equilibrium equation of state (EOS) in the Gross-Pitaevskii 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 $\Pi_\varepsilon(k)$ and $\Pi_N(k)$, 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 $\epsilon$ in the steady state. We find that in our regime of `mixed' turbulence - where both vortices and waves play a significant role - $n_0\propto \epsilon^{0.67(2)}$, 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.