Abstract
Boyle’s 1662 observation that the volume of a gas is, at constant temperature, inversely proportional to pressure, offered a prototypical example of how an equation of state (EoS) can succinctly capture key properties of a many-particle system. Such relationships are now cornerstones of equilibrium thermodynamics1. Extending thermodynamic concepts to far-from-equilibrium systems is of great interest in various contexts, including glasses2,3, active matter4,5,6,7 and turbulence8,9,10,11, but is in general an open problem. Here, using a homogeneous ultracold atomic Bose gas12, we experimentally construct an EoS for a turbulent cascade of matter waves13,14. Under continuous forcing at a large length scale and dissipation at a small one, the gas exhibits a non-thermal, but stationary, state, which is characterized by a power-law momentum distribution15 sustained by a scale-invariant momentum-space energy flux16. We establish the amplitude of the momentum distribution and the underlying energy flux as equilibrium-like state variables, related by an EoS that does not depend on the details of the energy injection or dissipation, or on the history of the system. Moreover, we show that the equations of state for a wide range of interaction strengths and gas densities can be empirically scaled onto each other. This results in a universal dimensionless EoS that sets benchmarks for the theory and should also be relevant for other turbulent systems.
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Data availability
The data that support the findings of this study are available in the Apollo repository (https://doi.org/10.17863/CAM.96408). Source data are provided with this paper. Any additional information is available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank C. Castelnovo, J. Dalibard, N. Dogra, K. Fujimoto, M. Gałka, G. Krstulovic, N. Navon, D. Proment and M. Zwierlein for helpful discussions. This work was supported by EPSRC (grant no. EP/N011759/1 and no. EP/P009565/1), ERC (QBox and UniFlat) and STFC (grant no. ST/T006056/1). T.A.H. acknowledges support from the EU Marie Skłodowska-Curie programme (grant no. MSCA-IF- 2018 840081). A.C. acknowledges support from the NSF Graduate Research Fellowship Program (grant no. DGE2040434). C.E. acknowledges support from Jesus College, Cambridge. R.P.S. acknowledges support from the Royal Society. Z.H. acknowledges support from the Royal Society Wolfson Fellowship.
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L.H.D. led the data collection and analysis, with most significant contributions from G.M. and T.A.H. All authors (L.H.D., G.M., T.A.H., J.A.P.G., J.E., A.C., C.E., R.P.S. and Z.H.) contributed significantly to the experimental setup, the interpretation of the results and the production of the manuscript. Z.H. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Steady-state momentum distributions and comparison with the perturbative WWT theory.
a, Steady-state nk for three different combinations of a and n, with fluxes ϵ chosen such that the cascade amplitudes are similar. The blue shading shows the k range where we fit all our data. For the data shown here, 1/ξ = 0.48 (red), 0.87 (orange) and 1.22 μm−1 (blue). Fitted with γ as a free parameter, these spectra give γ = 3.3 (red), 3.0 (orange) and 3.1 (blue). The solid line shows n0k−3(kξ)−0.2 and the dashed line shows \({n}_{0}{k}^{-3}{\rm{ln}}{(k/{k}_{0})}^{-1/3}\) with the same n0 (chosen such as to offset the curves from the data for clarity), ξ corresponding to the orange data, and k0 = 0.4 μm−1. The error bars show standard errors of measurement. b, Histogram of extracted γ for all spectra corresponding to the 153 points in Fig. 4, when fitting nk ∝ k−γ with γ as a free parameter. The purple bar indicates the Rb data point. c, Comparison of all the data shown in Fig. 4 with the perturbative WWT theory27 (solid line), without any free parameters. The error bars show standard fitting errors.
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Dogra, L.H., Martirosyan, G., Hilker, T.A. et al. Universal equation of state for wave turbulence in a quantum gas. Nature 620, 521–524 (2023). https://doi.org/10.1038/s41586-023-06240-z
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DOI: https://doi.org/10.1038/s41586-023-06240-z
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