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arXiv:2602.06131

A Nonequilibrium Equation of State for a Turbulent 2D Bose Gas

Yi Jiang, Nikolai Maslov, Andrey Karailiev, Christoph Eigen, Martin Gazo, Zoran Hadzibabic

cond-mat.quant-gascond-mat.stat-mechphysics.atom-phquant-phSubmitted Feb 5, 2026• Announced Feb 9, 2026• 0 citations
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Abstract

Nonequilibrium equations of state can provide an effective thermodynamic-like description of far-from-equilibrium systems. We experimentally construct such an equation for a direct energy cascade in a turbulent two-dimensional Bose gas. Our homogeneous gas is continuously driven on a large length scale and, with matching dissipation on a small length scale, exhibits a nonthermal but stationary power-law momentum distribution. Our equation of state links the cascade amplitude with the underlying scale-invariant energy flux, and can, for different drive strengths, gas densities, and interaction strengths, be recast into a universal power-law form using scalings consistent with the Gross-Pitaevskii model.

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4 pages, 3 figures

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Nonequilibrium equations of state can provide an effective thermodynamic-like description of far-from-equilibrium systems. We experimentally construct such an equation for a direct energy cascade in a turbulent two-dimensional Bose gas. Our homogeneous gas is continuously driven on a large length scale and, with matching dissipation on a small length scale, exhibits a nonthermal but stationary power-law momentum distribution. Our equation of state links the cascade amplitude with the underlying scale-invariant energy flux, and can, for different drive strengths, gas densities, and interaction strengths, be recast into a universal power-law form using scalings consistent with the Gross-Pitaevskii model.
We observe an inverse turbulent-wave cascade, from small to large length scales, in a driven homogeneous 2D Bose gas. Starting with an equilibrium condensate, we drive the gas isotropically on a length scale much smaller than its size, and observe a nonthermal population of modes with wavelengths larger than the drive one. At long drive times, the gas exhibits a steady nonthermal momentum distribution. At length scales increasing from the drive one to the system size, this distribution features in turn: (i) a power-law spectrum with an exponent close to the analytical result for a particle cascade in weak-wave turbulence, and (ii) a spectrum reminiscent of a nonthermal fixed point associated with universal coarsening in an isolated 2D gas. In further experiments, based on anisotropic driving, we reveal the complete qualitative picture of how the steady-state cascade forms.
We realize a turbulent cascade of wave excitations in a homogeneous 2D Bose gas and probe on all relevant time and length scales how it builds up from small to large momenta, until the system reaches a steady state with matching energy injection and dissipation. This all-scales view directly reveals the two theoretically expected cornerstones of turbulence formation-the emergence of statistical momentum-space isotropy under anisotropic forcing and the spatiotemporal scaling of the momentum distribution at times before any energy is dissipated.
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.
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We numerically study two-dimensional quantum turbulence with a Gross-Pitaevskii model. With the energy initially accumulated at large scale, quantum turbulence with many quantized vortex points is generated. Due to the lack of enstrophy conservation in this model, direct energy cascade with a Kolmogorov-Obukhov energy spectrum $E(k)\ensuremath{\propto}{k}^{\ensuremath{-}5/3}$ is observed, which is quite different from two-dimensional incompressible classical turbulence in the decaying case. A positive value for the energy flux guarantees a direct energy cascade in the inertial range (from large to small scales). After almost all the energy at the large scale cascades to the small scale, the compressible kinetic energy realizes the thermodynamic equilibrium state without quantized vortices.
When a Bose-Einstein condensate (BEC) is driven out of equilibrium, density waves interact nonlinearly and trigger turbulent cascades. In a turbulent BEC, energy is transferred toward small scales by a direct cascade, whereas the number of particles displays an inverse cascade toward large scales. In this work, we study analytically and numerically the direct and inverse cascades in wave-turbulent BECs. We analytically derive the Kolmogorov-Zakharov spectra, including the log correction to the direct cascade scaling and the universal prefactor constants for both cascades. We test and corroborate our predictions using high-resolution numerical simulations of the forced-dissipated Gross-Pitaevskii model in a periodic box and the corresponding wave-kinetic equation. Theoretical predictions and data are in excellent agreement, without adjustable parameters. Moreover, in order to connect with experiments, we test and validate our theoretical predictions using the Gross-Pitaevskii model with a confining cubic trap. Our results explain previous experimental observations and suggest new settings for future studies.