We present a comprehensive study of the statistical features of a three-dimensional time-reversible Navier-Stokes (RNS) system, wherein the standard viscosity $ u$ is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter $mathcal{R}_r$, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale $ell_{rm f}$ and the total energy $E_0$. We find that the system exhibits a transition from a high-enstrophy phase at small $mathcal{R}_r$, where truncation effects tend to produce partially thermalized states, to a hydrodynamical phase with low enstrophy at large $mathcal{R}_r$. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous phase transition, where $mathcal{R}_r$ indeed behaves as a thermodynamic control parameter, e.g., a temperature, the enstrophy plays the role of an order parameter, while the symmetry breaking parameter $h$ is (one over) the truncation scale $k_{rm max}$. We find that the signatures of the phase transition close to the critical point $mathcal{R}_r^star$ can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti, Phys.Lett.A, 223, 1996 could hold true. Our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes (NS) equations with viscosity chosen so as to match the average value of the reversible viscosity.
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