In the final discussion of their Phys. Rev. Lett. 113, 024501 (2014), Frishman and Falkovich showed that in Burgers turbulence the one-sided short-time limits of the squared velocity difference of a pair of fluid particles, ⟨du2/dt⟩t→0±\langle du^2/dt\rangle_{t\to0^\pm}, encode the coalescence of Lagrangian trajectories into shocks and the backward martingale property of the velocity, and asked whether the forward anomaly can be given a precise interpretation in terms of the spontaneous stochasticity (non-uniqueness) of the inviscid Lagrangian flow; the multidimensional compressible question traces instead to the first arXiv version of the paper, which conjectured that similar anomalies appear in compressible turbulence in any dimension, and is therefore kept below as a conjectural extension of the paper's agenda. The correspondence is currently rigorous only for Burgers (Eyink–Drivas, J. Stat. Phys. 158, 386 (2015)) and for passive scalars (Drivas–Eyink, J. Fluid Mech. 829, 153 (2017)), while for Navier–Stokes/Euler even the consistency between the martingale arrow of time and the dissipative arrow of time remains without a formal argument, and the multidimensional compressible pair anomaly has not been computed in any later work. The problem remains open.

Background

A fluid particle advected by a turbulent velocity field obeys the Lagrangian equation of motion x˙=v(x(t),t)\dot x=v(x(t),t). Frishman and Falkovich (Phys. Rev. Lett. 113, 024501 (2014)) studied the one-sided short-time limits ⟨du2/dt⟩t→0+\langle du^2/dt\rangle_{t\to0^+} and ⟨du2/dt⟩t→0−\langle du^2/dt\rangle_{t\to0^-}, where uu is the velocity difference of a pair of fluid particles (markers) initially separated by a fixed distance, in one-dimensional Burgers turbulence — the inviscid limit of the Burgers equation, whose solutions develop discontinuities called shocks at which velocity gradients and energy dissipation remain finite as the viscosity ν\nu vanishes (the dissipation anomaly). At a large Reynolds number (the ratio of inertial to viscous forces) the velocity is effectively rough in time, so the expectation of the squared increment du2du^2 over a short time dtdt need not scale as dt2dt^2 as it would in a smooth field, and ⟨du2/dt⟩\langle du^2/dt\rangle can have a finite one-sided limit; the classical relation ⟨du2/dt⟩=−4εˉ\langle du^2/dt\rangle=-4\bar\varepsilon, with εˉ\bar\varepsilon the mean dissipation rate, implicitly takes t→0t\to0 before ν→0\nu\to0. Frishman and Falkovich showed that in Burgers turbulence the two limits do not commute: taking ν→0\nu\to0 first, the backward-in-time limit vanishes while the forward-in-time limit is nonzero (−6εˉ-6\bar\varepsilon in the paper's notation), because marker particles fall into shocks forward in time, whereas particles marked outside the zero-measure shocks at t=0t=0 cannot have come from them backward in time; a similar anomaly holds for the single-particle energy change dv2/dtdv^2/dt. Comparing a movie of Burgers turbulence run forward with one run backward, this is a Lagrangian manifestation of time irreversibility. Spontaneous stochasticity refers to the non-uniqueness of Lagrangian trajectories that persists in the inviscid limit: when the velocity is non-Lipschitz (as at Burgers shocks, where trajectories coalesce), two particles starting at the same point can separate, and this non-uniqueness is generated by the flow itself rather than by molecular noise. The martingale property is the statement that the velocity evaluated along a (suitably defined) Lagrangian trajectory has no drift when viewed backward in time, a property that forces the backward-in-time short-time limit to vanish. In their final discussion the authors proposed interpreting the forward/backward-in-time anomalies as due to trajectory uniqueness/non-uniqueness — two trajectories entering the same shock forward in time mean non-uniqueness, hence spontaneous stochasticity backward in time, producing the forward anomaly — and hypothesized that the symmetric anomaly of incompressible flow, for which the two limits commute and ⟨du2/dt⟩=−4εˉ\langle du^2/dt\rangle=-4\bar\varepsilon from either side, reflects an equality of the degrees of trajectory non-uniqueness forward and backward in time. The first arXiv version of the paper conjectured that similar anomalies appear in compressible turbulence in any dimension; the published version, however, only remarks that for real fluid particles of weakly compressible one-dimensional flow the shocks sweep past particles at the sound speed, the jump in the Lagrangian dissipation rate disappears, and the correctly ordered anomaly vanishes altogether, and does not pose the multidimensional finite-Mach question as a claim — that extension is therefore formulated below as a conjectural extension of the paper's own agenda. The problem should also be distinguished from the case of passive scalars (a contaminant advected by the flow without reacting back on it): Drivas and Eyink (J. Fluid Mech. 829, 153 (2017)) proved that anomalous dissipation of a passive scalar is equivalent to backward spontaneous stochasticity, and identified the pressure-gradient term as the obstruction to the analogous statement for kinetic energy, which is what makes the Navier–Stokes case genuinely open.

Problem Statement

The problem is the agenda posed in the final discussion of Frishman and Falkovich (Phys. Rev. Lett. 113, 024501 (2014)) together with the open questions of Eyink and Drivas (J. Stat. Phys. 158, 386 (2015)), with the compressible extension kept at its conjectural status. The parts are unified by the single question of whether anomalous dissipation and spontaneous stochasticity are two faces of the same phenomenon, and by the shared requirement that the inviscid limit be taken with the source's time-ordering respected (viscosity ν→0\nu\to0 first, then the one-sided short-time limit t→0±t\to0^\pm). (1) In Burgers turbulence the correspondence is now largely rigorous; what remains open there is the uniqueness question of Eyink and Drivas (J. Stat. Phys. 158, 386 (2015)): do the stochastic Lagrangian processes arising from the inviscid limit — backward Markov processes for which the velocity is a backward martingale — uniquely characterize the entropy solution (the weak solution selected by the admissibility condition) of Burgers? (2) For the incompressible Navier–Stokes and Euler equations, determine whether the arrow of time specified by the backward martingale property is the same as the arrow specified by anomalous dissipation of energy — in the precise sense posed by Eyink and Drivas, whether weak Euler solutions (if any) satisfying the backward martingale property for circulations must have kinetic energies always decreasing in time, for which they state that there is not even a formal physicists' argument — and, following the interpretive agenda of Frishman and Falkovich, establish (or refute) the equivalence between the Lagrangian time-irreversibility anomaly ⟨du2/dt⟩t→0±\langle du^2/dt\rangle_{t\to0^\pm} (the one-sided short-time rate of change of the squared velocity difference of a pair of fluid particles, with an analogous single-particle statement for dv2/dtdv^2/dt) and spontaneous stochasticity (non-uniqueness of inviscid Lagrangian trajectories). This part includes the paper's own hypothesis that the symmetric anomaly of incompressible flow — for which the two limits commute and ⟨du2/dt⟩=−4εˉ\langle du^2/dt\rangle=-4\bar\varepsilon from either side — reflects an equality of the degrees of trajectory non-uniqueness forward and backward in time, which requires first formalizing that degree. (3) As a conjectural extension of the paper's agenda, for multidimensional compressible turbulence at finite Mach number (the ratio of flow speed to sound speed), compute or rigorously characterize the pair short-time anomaly in the correctly ordered inviscid limit — viscosity ν→0\nu\to0 at fixed finite Mach number, then t→0±t\to0^\pm — and determine whether it is in correspondence with trajectory non-uniqueness there; note that the paper itself observed that for weakly compressible one-dimensional flow the correctly ordered anomaly vanishes because shocks sweep past finite-sound-speed particles without coalescence. A complete answer to any of the three parts — the Burgers uniqueness converse, the Navier–Stokes martingale-versus-dissipation-arrow equivalence and the equivalence between the Lagrangian time-irreversibility anomaly and spontaneous stochasticity, including the incompressible symmetry hypothesis, or the multidimensional compressible pair anomaly with its stochasticity correspondence — advances the problem.

References

  1. A. Frishman and G. Falkovich, "New Type of Anomaly in Turbulence", Phys. Rev. Lett. 113, 024501 (2014). DOI: 10.1103/PhysRevLett.113.024501; https://doi.org/10.1103/PhysRevLett.113.024501 (also arXiv:1401.6141; https://arxiv.org/abs/1401.6141).
  2. A. Frishman and G. Falkovich, "New type of anomaly in turbulence", arXiv:1401.6141v1 (2014); https://arxiv.org/abs/1401.6141v1. First arXiv version, whose final section conjectures that similar anomalies appear in compressible turbulence in any dimension.
  3. G. L. Eyink and T. D. Drivas, "Spontaneous Stochasticity and Anomalous Dissipation for Burgers Equation", J. Stat. Phys. 158, 386–432 (2015). DOI: 10.1007/s10955-014-1135-3; https://doi.org/10.1007/s10955-014-1135-3 (also arXiv:1401.5541; https://arxiv.org/abs/1401.5541).
  4. T. D. Drivas and G. L. Eyink, "A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls", J. Fluid Mech. 829, 153–189 (2017). DOI: 10.1017/jfm.2017.567; https://doi.org/10.1017/jfm.2017.567 (also arXiv:1606.00729; https://arxiv.org/abs/1606.00729).
  5. T. D. Drivas, "Turbulent Cascade Direction and Lagrangian Time-Asymmetry", J. Nonlinear Sci. 29, 65–88 (2018). DOI: 10.1007/s00332-018-9476-8; https://doi.org/10.1007/s00332-018-9476-8 (also arXiv:1802.02289; https://arxiv.org/abs/1802.02289).
  6. J. Jucha, H. Xu, A. Pumir and E. Bodenschatz, "Time-reversal-symmetry Breaking in Turbulence", Phys. Rev. Lett. 113, 054501 (2014). DOI: 10.1103/PhysRevLett.113.054501; https://doi.org/10.1103/PhysRevLett.113.054501.
  7. T. Grafke, A. Frishman and G. Falkovich, "Time irreversibility of the statistics of a single particle in compressible turbulence", Phys. Rev. E 91, 043022 (2015). DOI: 10.1103/PhysRevE.91.043022; https://doi.org/10.1103/PhysRevE.91.043022 (also arXiv:1408.5580; https://arxiv.org/abs/1408.5580).
  8. Q. Zhang and Z. Xiao, "On the time irreversibility of compressible turbulence reflected by particles of various inertias", Phys. Fluids 33, 036113 (2021). DOI: 10.1063/5.0041210; https://doi.org/10.1063/5.0041210.
  9. S. De, D. Mitra and R. Pandit, "Uncovering the multifractality of Lagrangian pair dispersion in shock-dominated turbulence", Phys. Rev. Research 6, L022032 (2024). DOI: 10.1103/PhysRevResearch.6.L022032; https://doi.org/10.1103/PhysRevResearch.6.L022032 (also arXiv:2311.06836; https://arxiv.org/abs/2311.06836).
  10. S. De, D. Mitra and R. Pandit, "Intermittency and nonuniversality of pair dispersion in isothermal compressible turbulence", Phys. Rev. Research 8, L012037 (2026). DOI: 10.1103/kr1z-rf6b; https://doi.org/10.1103/kr1z-rf6b (also arXiv:2504.17292; https://arxiv.org/abs/2504.17292).
  11. N. Valade, S. Thalabard and J. Bec, "Anomalous Dissipation and Spontaneous Stochasticity in Deterministic Surface Quasi-Geostrophic Flow", Ann. Henri Poincaré 25, 1261–1283 (2024). DOI: 10.1007/s00023-023-01284-3; https://doi.org/10.1007/s00023-023-01284-3 (also arXiv:2210.12366; https://arxiv.org/abs/2210.12366).
  12. G. L. Eyink and N. Goldenfeld, "Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence", arXiv:2512.24469 (2025); https://arxiv.org/abs/2512.24469. Contribution to the Philosophical Transactions of the Royal Society A themed issue "Frontiers of Turbulence and Statistical Physics Meet".
  13. V. Valadão, E. Aurell, G. Boffetta, M. Cencini, S. Musacchio and A. Vulpiani, "The real butterfly effect: from the pop culture to mathematics and physics", arXiv:2607.24715 (2026); https://arxiv.org/abs/2607.24715.