Thermodynamic-limit phase transition and Gallavotti ensemble equivalence in the three-dimensional time-reversible Navier-Stokes system

The three-dimensional time-reversible Navier-Stokes (RNS) equations, in which the constant viscosity is replaced by a state-dependent 'reversible viscosity' thermostat that keeps the total kinetic energy fixed, display a transition between a partially thermalized 'warm' phase and a hydrodynamic phase as the control parameter (the ratio of energy injection at the forcing scale to the conserved total energy) is increased. Shukla, Dubrulle, Nazarenko, Krstulovic and Thalabard (Phys. Rev. E 100, 043104, 2019) argued that this transition is a second-order phase transition in the thermodynamic limit of infinite spectral resolution, and proposed the joint critical-point asymptotics as the regime in which Gallavotti's reversible-irreversible ensemble equivalence conjecture should hold — but their evidence was limited to 128^3 resolution and low-order, one-point statistics. The subsequent literature (Jaccod-Chibbaro 2021; Gallavotti 2021-2022; Margazoglou et al. 2022; Costa-Barral-Dubrulle 2023 on logarithmic lattices; Maji et al. 2023 in 2D; the 2025 'efficiency' program) leaves the core open: for the full three-dimensional system there is still no strict-limit verification of the second-order character of the transition, no assessment of equivalence in the prescribed joint limit including intermittent statistics, no theory of the bursty transition-window fluctuations, and no first-principles derivation of the transition properties and critical exponents.

Background

In the standard incompressible Navier-Stokes (NS) equations, viscous dissipation breaks time-reversal symmetry, even though the microscopic molecular dynamics is reversible. Gallavotti (Phys. Lett. A 223, 91, 1996) proposed restoring symmetry at the macroscopic level by replacing the constant kinematic viscosity with a fluctuating, state-dependent 'reversible viscosity' (a deterministic thermostat analogous to Gauss' principle of least constraint): the coefficient is adjusted in time so that a macroscopic quadratic quantity — the total kinetic energy, or alternatively the enstrophy, i.e. the volume integral of the squared vorticity — is exactly conserved. The resulting time-reversible Navier-Stokes (RNS) equations are expected, by Gallavotti's equivalence conjecture, to share the same invariant (steady-state) statistics as the ordinary irreversible NS equations for suitable observables in an appropriate turbulent limit, in analogy with the equivalence of microcanonical and canonical ensembles in equilibrium statistical mechanics.

Numerical simulation necessarily truncates the Fourier spectrum at a maximum wavenumber $k_{\max}$ (a Galerkin truncation). Truncated inviscid dynamics generically 'thermalizes' at small scales: energy piles up near $k_{\max}$ in near-equipartition (Gibbs) states instead of cascading. For reversible thermostatted systems in shell models, Biferale, Cencini, De Pietro, Gallavotti and Lucarini (Phys. Rev. E 98, 012202, 2018) showed that the order in which the Reynolds number and the truncation are taken to their limits is critical for whether reversible and irreversible ensembles coincide.

Shukla, Dubrulle, Nazarenko, Krstulovic and Thalabard (Phys. Rev. E 100, 043104, 2019; arXiv:1811.11503) gave this problem a thermodynamic formulation for the full 3D RNS system with energy-conserving thermostat under Taylor-Green forcing (a standard single-scale sinusoidal forcing and initial-condition protocol, applied here at forcing wavenumber $k_{\mathrm{f}}=\sqrt{3}$ ). They introduced the dimensionless control parameter $\mathcal{R}_{\mathrm{r}}=f_0\ell_{\mathrm{f}}/E_0$ (forcing amplitude $f_0$ times forcing length scale $\ell_{\mathrm{f}}$ , divided by the conserved total energy $E_0$ ), and found two statistical regimes: a 'warm' phase at small $\mathcal{R}_{\mathrm{r}}$ , with partial ultraviolet thermalization and high enstrophy, and a 'hydrodynamic' phase at large $\mathcal{R}_{\mathrm{r}}$ , with compact spectral support and statistics akin to ordinary turbulence. The normalized enstrophy $\widetilde{\Omega}$ (enstrophy divided by its equipartition value) acts as an order parameter, and $h=k_0/k_{\max}$ — with $k_0$ the infrared (box) wavenumber, equal to $1$ in the source's normalization, so that $h\to 0$ is equivalent to $k_{\max}\to\infty$ — acts as the inverse resolution, playing the role of a symmetry-breaking field. A 'reversible Leith' (RL) spectral diffusion model — a one-dimensional nonlinear-diffusion model of the energy spectrum equipped with the reversible thermostat — exhibits a Landau-type mean-field free energy whose signatures match the numerics, leading the authors to conjecture that the transition is continuous (second-order) with critical point $\mathcal{R}_{\mathrm{r}}^\star\approx 2.75$ . They further argued that Gallavotti's equivalence should be tested in the joint critical-point asymptotics $\mathcal{R}_{\mathrm{r}}\to\mathcal{R}_{\mathrm{r}}^\star$ from above and $h\to 0$ , where anomalous energy dissipation (non-vanishing dissipation as viscosity vanishes) should coexist with vanishing thermostat effects; a single matched RNS-NS comparison at $\mathcal{R}_{\mathrm{r}}=3.48$ supported this heuristically. The paper's own conclusions, however, flag the evidence as 'only suggestive': resolutions up to $128^3$ , finite-time integration, and only low-order one-point statistics. Subsequent work by the same and adjacent groups — logarithmic-lattice projections (a reduced Fourier geometry retaining symmetries and conservation laws at a fraction of the cost), enstrophy-constrained variants, and 2D analogues — has strengthened the evidence but explicitly leaves the full three-dimensional questions open.

Problem Statement

For the three-dimensional time-reversible Navier-Stokes (RNS) system with energy-conserving reversible-viscosity thermostat, determine the thermodynamic nature of the warm-to-hydrodynamic transition and the validity of Gallavotti's dynamical ensemble equivalence in the turbulent limit. Concretely, with control parameter $\mathcal{R}_{\mathrm{r}}=f_0\ell_{\mathrm{f}}/E_0$ , order parameter $\widetilde{\Omega}$ (normalized enstrophy), and resolution field $h=k_0/k_{\max}$ :

Establish whether the transition near the critical point $\mathcal{R}_{\mathrm{r}}^\star\approx 2.75$ (the value estimated at finite resolution in the source) is a genuine second-order (continuous) phase transition in the strict thermodynamic limit $k_{\max}\to\infty$ for the full 3D dynamics — beyond the suggestive $128^3$ evidence and beyond logarithmic-lattice reductions — including finite-size scaling of the order parameter and its fluctuations with $k_{\max}$ .
Determine whether the invariant measures of the RNS system and of the standard NS equations (with viscosity matched to the time-averaged reversible viscosity) become equivalent in the joint critical-point asymptotics $\mathcal{R}_{\mathrm{r}}\to\mathcal{R}_{\mathrm{r}}^\star$ from above and $h\to 0$ , for inertial-range observables including anomalous energy dissipation and higher-order (intermittent) statistics — i.e. whether these statistics converge to identical limits, and over what range of observables/wavenumbers the equivalence holds.
Characterize the enhanced enstrophy fluctuations and the bursty multistability of the finite- $k_{\max}$ transition window (intermittent switching between the coexisting warm and hydrodynamic-like states): their origin and scaling, whether they persist or disappear as $k_{\max}\to\infty$ , and how forcing-induced injected-power fluctuations affect the selection between warm and hydrodynamic steady branches.
Derive the transition properties and critical exponents from first principles for the full RNS dynamics — beyond the heuristic reversible-Leith/Landau mean-field picture — including whether non-mean-field critical behavior arises and how the functional form of the reversible thermostat affects universality.

These four questions are linked by one common objective: converting the conjectured picture — RNS as a thermostatted system undergoing a critical transition whose hydrodynamic side is statistically equivalent to ordinary turbulence — into established knowledge (rigorous or decisively numerical) about the thermodynamics of the reversible formulation of 3D Navier-Stokes turbulence. All four are part of the deliverable: the objective is met only when each of the four sub-questions has been settled, and settling a sub-question may mean either establishing or refuting the property it asks about.

References

Vishwanath Shukla, Bérengère Dubrulle, Sergey Nazarenko, Giorgio Krstulovic, Simon Thalabard, 'Phase transition in time-reversible Navier-Stokes equations', Physical Review E 100, 043104 (2019). DOI: 10.1103/PhysRevE.100.043104, https://doi.org/10.1103/PhysRevE.100.043104 (arXiv:1811.11503, https://arxiv.org/abs/1811.11503)
G. Gallavotti, 'Equivalence of dynamical ensembles and Navier-Stokes equations', Physics Letters A 223, 91-95 (1996). DOI: 10.1016/S0375-9601(96)00729-3, https://doi.org/10.1016/S0375-9601(96)00729-3
Luca Biferale, Massimo Cencini, Massimo De Pietro, Giovanni Gallavotti, Valerio Lucarini, 'Equivalence of nonequilibrium ensembles in turbulence models', Physical Review E 98, 012202 (2018). DOI: 10.1103/PhysRevE.98.012202, https://doi.org/10.1103/PhysRevE.98.012202 (arXiv:1804.06273, https://arxiv.org/abs/1804.06273)
Alice Jaccod, Sergio Chibbaro, 'Constrained Reversible System for Navier-Stokes Turbulence', Physical Review Letters 127, 194501 (2021). DOI: 10.1103/PhysRevLett.127.194501, https://doi.org/10.1103/PhysRevLett.127.194501 (arXiv:2011.09773, preprint title 'Constrained Reversible system for Navier-Stokes Turbulence: evidence for Gallavotti's equivalence conjecture', https://arxiv.org/abs/2011.09773)
Giovanni Gallavotti, 'Viscosity, Reversibillity, Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing' (title as published, including the publisher's typographical error 'Reversibillity'), Journal of Statistical Physics 185, 21 (2021). DOI: 10.1007/s10955-021-02830-1, https://doi.org/10.1007/s10955-021-02830-1 (arXiv:2102.10125, https://arxiv.org/abs/2102.10125)
Georgios Margazoglou, Luca Biferale, Massimo Cencini, Giovanni Gallavotti, Valerio Lucarini, 'Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations', Physical Review E 105, 065110 (2022). DOI: 10.1103/PhysRevE.105.065110, https://doi.org/10.1103/PhysRevE.105.065110 (arXiv:2201.00530, https://arxiv.org/abs/2201.00530)
Giovanni Gallavotti, 'Navier-Stokes and equivalence conjectures' (2022). arXiv:2211.02961, https://arxiv.org/abs/2211.02961
Guillaume Costa, Amaury Barral, Bérengère Dubrulle, 'Reversible Navier-Stokes equation on logarithmic lattices', Physical Review E 107, 065106 (2023). DOI: 10.1103/PhysRevE.107.065106, https://doi.org/10.1103/PhysRevE.107.065106 (open-access PDF: https://hal.science/hal-04122901)
Maheswar Maji, Karthik Subramaniam Eswaran, Sourangshu Ghosh, Kannabiran Seshasayanan, Vishwanath Shukla, 'Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade', Physical Review E 108, 015102 (2023). DOI: 10.1103/PhysRevE.108.015102, https://doi.org/10.1103/PhysRevE.108.015102 (arXiv:2112.12215, https://arxiv.org/abs/2112.12215)
Guillaume Costa, Amaury Barral, Adrien Lopez, Quentin Pikeroen, Bérengère Dubrulle, 'Behind the mirror: the hidden dissipative singular solutions of ideal reversible fluids on log-lattices' (2025). arXiv:2508.10659, https://arxiv.org/abs/2508.10659
Adrien Lopez, Amaury Barral, Guillaume Costa, Quentin Pikeroen, Vishwanath Shukla, Bérengère Dubrulle, 'Efficiency of turbulence', Journal of Fluid Mechanics 1019, A55 (2025). DOI: 10.1017/jfm.2025.10604, https://doi.org/10.1017/jfm.2025.10604 (arXiv:2508.05686, https://arxiv.org/abs/2508.05686)
Guillaume Costa, Bérengère Dubrulle, 'Numerical investigations around the Gallavotti-Cohen fluctuation theorem on log-lattices', Philosophical Transactions of the Royal Society A 384, 20250028 (2026). DOI: 10.1098/rsta.2025.0028, https://doi.org/10.1098/rsta.2025.0028 (arXiv:2508.17943, https://arxiv.org/abs/2508.17943)