Emergent Bose-Einstein statistics in classical non-equilibrium systems with scale selection

Non-equilibrium systems are ubiquitous in nature, from driven quantum matter and biological life forms to atmospheric and interstellar gases. Identifying and characterizing universal aspects of their dynamics continues to pose major conceptual challenges due to the absence of conserved quantities like energy. Here, we investigate the statistics of a broad class of non-equilibrium systems in which an intrinsic length-scale selection mechanism effectively constrains the dynamics of the microscopic degrees of freedom. As specific examples, we compare experimental observations of chaotic Faraday surface waves on water with simulations of a generalized Navier-Stokes equation for active fluids and quantum particle simulations in a random potential. Strikingly, we find that in all three cases the Fourier amplitudes of the energy density follow Bose-Einstein statistics. Furthermore, we show that the time evolution of these systems can be approximated by a sequence of randomly sampled monochromatic Gaussian fields, suggesting a unified view of non-equilibrium and equilibrium systems with length scale selection.