Journal of Statistical Physics

, Volume 157, Issue 2, pp 205–218 | Cite as

Non-Equilibrium Statistical Mechanics of Turbulence

  • David Ruelle


The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system (Ruelle, PNAS 109:20344–20346, 2012). Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents \(\tau _p\) and \(\zeta _p\) associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note (Ruelle, 2012), is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov (J. Fluid Mech. 13:82–85, 1962) and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents \(\tau _p\) and \(\zeta _p\).


Turbulence Non-equilibrium Statistical mechanics 



I am indebted to Christian Beck and Victor Yakhot for useful interaction during the preparation of this paper, and to Giovanni Gallavotti for earlier discussions; VY communicated reference [13] which is essential here. This work was initiated while I visited the Isaac Newton Institute in Cambridge, UK, at the end of 2013.


  1. 1.
    Anselmet, F., Gagne, Y., Hopfinger, E.J., Antonia, R.A.: High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63–89 (1984)ADSCrossRefGoogle Scholar
  2. 2.
    Beck, C.: Chaotic cascade model for turbulent velocity distributions. Phys. Rev. E 49, 3641–3652 (1994)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benzi, R., Paladin, G., Parisi, G., Vulpiani, A.: On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17, 3521–3531 (1984)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Castaing, B.: The temperature of turbulent flows. J. Phys. II France 6, 105–114 (1996)CrossRefGoogle Scholar
  5. 5.
    Frisch, U., Parisi, G.: On the singularity structure of fully developed turbulence. In: Ghil, M., Benzi, R., Parisi, G. (ed.) Turbulence and Predictability in Geophysical Fluid Dynamics, pp. 84–88. North-Holland, Amsterdam (1985)Google Scholar
  6. 6.
    Gallavotti, G.: Foundations of Fluid Mechanics. Springer-Verlag, Berlin, 2005 (see Section 6.3)Google Scholar
  7. 7.
    Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301–305 (1941)ADSGoogle Scholar
  8. 8.
    Kolmogorov, A.N.: On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540 (1941)zbMATHGoogle Scholar
  9. 9.
    Kolmogorov, A.N.: Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (1941)ADSzbMATHGoogle Scholar
  10. 10.
    Kolmogorov, A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85 (1962)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Meneveau, C., Sreenivasan, K.R.: Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427 (1987)ADSCrossRefGoogle Scholar
  12. 12.
    Ruelle, D.: Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics. PNAS 109, 20344–20346 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Schumacher, J., Scheel, J., Krasnov, D., Donzis, D., Sreenivasan, K., Yakhot, V.: Small-scale universality in turbulence. Preprint (2014)Google Scholar
  14. 14.
    Stresing, R., Peinke, J.: Towards a stochastic multi-point description of turbulence. New J. Phys. 12, 103046+14 (2010)Google Scholar
  15. 15.
    Yakhot, V.: Pressure-velocity correlations and scaling exponents in turbulence. J. Fluid Mech. 495, 135–143 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematics DepartmentRutgers UniversityPiscatawayUSA
  2. 2.IHESBures sur YvetteFrance

Personalised recommendations