Universal equation of state for wave turbulence in a quantum gas

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 glasses^2,3, active matter^4-7 and turbulence^8-11, 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 waves^13,14. 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.