The statistical properties of fully developed hydrodynamic turbulence
can be well understood using techniques from a generalized version of statistical mechanics,
so-called nonextensive statistical mechanics and superstatistics.
Look at my PRL (2007) on Lagrangian turbulence, and more recent papers in PRE and EPL on quantum turbulence (the turbulent state of superfluid Helium).
The stochastic models that are relevant for this are based on the so-called superstatistics approach.

The following picture shows probability densities of velocity differences as measured on various scales in a
turbulent Taylor-Couette flow, and the theoretical predictions of a model based on nonextensive statistical mechanics.
There is excellent agreement. Other theoretical models for probability densities in turbulent flows do not yield anything of similar precision. Turbulent velocity differences u

The measurements were made by Greg Lewis and Harry Swinney at the
Center for Nonlinear Dynamics, University of Texas at Austin.

More details can be found in the following references:

  • C. Beck, G.S. Lewis, and H.L. Swinney, Measuring non-extensitivity parameters in a turbulent Couette-Taylor flow, Phys. Rev 63E , 035303(R) (2001)
    ps file available

  • C. Beck, Dynamical foundations of nonextensive statistical mechanics, Phys. Rev. Lett. 87 , 180601 (2001)
    ps file available

  • Here is a link to a popular science article on my turbulence work.

    An article in the SCIENCE magazin (23 August 2002) also mentions this.