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Turbulence as a Problem in Non-equilibrium Statistical Mechanics

An Erratum to this article was published on 24 January 2017


The transitional and well-developed regimes of turbulent shear flows exhibit a variety of remarkable scaling laws that are only now beginning to be systematically studied and understood. In the first part of this article, we summarize recent progress in understanding the friction factor of turbulent flows in rough pipes and quasi-two-dimensional soap films, showing how the data obey a two-parameter scaling law known as roughness-induced criticality, and exhibit power-law scaling of friction factor with Reynolds number that depends on the precise form of the nature of the turbulent cascade. These results hint at a non-equilibrium fluctuation-dissipation relation that applies to turbulent flows. The second part of this article concerns the lifetime statistics in smooth pipes around the transition, showing how the remarkable super-exponential scaling with Reynolds number reflects deep connections between large deviation theory, extreme value statistics, directed percolation and the onset of coexistence in predator-prey ecosystems. Both these phenomena reflect the way in which turbulence can be fruitfully approached as a problem in non-equilibrium statistical mechanics.

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NG wishes to express his gratitude to Leo P. Kadanoff for his scientific inspiration, support, collaboration and friendship over many decades. NG also wishes to thank P. Chakraborty, G. Gioia, W. Goldburg, T. Tran, H. Kellay and N. Guttenberg for collaboration on the topics in Sect. 2. We thank T.-L. Hsieh and M. Sipos for collaboration on the topics in Sect. 3. We acknowledge helpful discussions with L.P. Kadanoff, B. Hof, J. Wesfreid, P. Manneville, D. Barkley and Y. Pomeau. We thank N. Guttenberg for technical assistance with Fig. 1. This work was supported in part by the National Science Foundation through grant NSF-DMR-1044901.

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Correspondence to Nigel Goldenfeld.

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The original version of this article was revised: The equation 23 is incorrect. This has been corrected in this version.

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Goldenfeld, N., Shih, HY. Turbulence as a Problem in Non-equilibrium Statistical Mechanics. J Stat Phys 167, 575–594 (2017).

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  • Turbulence
  • Phase transitions
  • Directed percolation
  • Extreme value statistics
  • Non-equilibrium statistical mechanics
  • Fluctuation-dissipation theorem
  • Predator-prey ecosystems