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Turbulence and stochastic processes. (English) Zbl 1142.76398

Vulpiani, Angelo (ed.) et al., The Kolmogorov legacy in physics. Transl. from the 2003 French original. Berlin: Springer (ISBN 3-540-20307-9/hbk). Lecture Notes in Physics 636, 173-186 (2003).
Summary: In 1931 the monograph “Analytical Methods in Probability Theory” appeared [Math. Ann. 104, 415–458 (1931; Zbl 0001.14902 and JFM 57.0613.03)], in which A. N. Kolmogorov laid the foundations for the modern theory of Markov processes. According to Gnedenko: ‘In the history of probability theory it is difficult to find other works that changed the established points of view and basic trends in research work in such a decisive way’. Ten years later, his article on fully developed turbulence provided the framework within which most, if not all, of the subsequent theoretical investigations have been conducted [A. N. Kolmogorov, Proc. R. Soc. Lond., Ser. A 434, No. 1890, 9–13 (1991; Zbl 1142.76389)] (see e.g. the review by L. Biferale, G. Boffetta and B. Castaing in this volume [Lect. Notes Phys. 636, 149–172 (2003; Zbl 1142.76388)]. Remarkably, the greatest advances made in the last few years towards a thorough understanding of turbulence developed from the successful marriage between the theory of stochastic processes and the phenomenology of turbulent transport of scalar fields.
In this article we summarize these recent developments which expose the direct link between the intermittency of transported fields and the statistical properties of particle trajectories advected by the turbulent flow (see also [B. I. Shraiman and E. D. Siggia, Nature 405, 639–649 (2000)] and, for a more thorough review [G. E. Fal’kovich, K. Gawẹdzki and M. Vergassola, Rev. Mod. Phys. 73, No. 4, 913–975 (2001)]). We also discuss the perspectives of the Lagrangian approach beyond passive scalars, especially for the modeling of hydrodynamic turbulence.
For the entire collection see [Zbl 1138.37002].

MSC:

76F55 Statistical turbulence modeling
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
76M35 Stochastic analysis applied to problems in fluid mechanics
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