1. Statistical Mechanics (S.M.) on Turbulence* Sunghwan (Sunny) Jung Harry L. Swinney Physics Dept. University of Texas at Austin *Supported by ONR.

2. Contents Revise Castaing’s method Introduce another method under transformation Stochastic Model from statistical mechanics Revise Kolmogorov 1962 (K62) in terms of statistical mechanics

3. Couette-Taylor Exp. At moderate rotation rate In turbulence regime Data Out

4. Observed quantities Extensive variable Intensive variable Velocity Difference Energy dissipation rate

5. Statistical Universality Coarse-Grained Quantity Physical Quantity Temporal information

6. Separation(r) Dependence r ~ LGaussian Dist.Delta function r << LGaussian Dist.Log-normal Dist. Castaing’s model We can rewrite its as

7. Cascade to the smaller scale(r) r ~ L r << L

8. Transform Castaing Model Gaussian Dist.Log-normal Dist. Transform

9. Statistical Universality Coarse-Grained Quantity Physical Quantity Temporal information

10. Probability of beta where

11. Conditioned Probability

12. Separation(r) Dependence d ~ NNon-GaussianDelta function d << NGaussian Dist.Log-normal Dist. Where N is the total number of data sets. We can rewrite it as

13. Cascade to large coarse-grain cell d << N d = L

14. Compared the predicted PDF

15. Lebesgue Measure x x Changes from Delta function to Log-normal Dist. Gaussian Dist. K62 :

16. S.M. Interpretation on K62 Taylor Expansion Probability of velocity differences Thermodynamic variable If we assume that

17. Conclusion Castaing’s method and Beck-Cohen’s method are the same under the transformation. Beck-Cohen’s method represents a cascade from a small coarse-grain to a large one. We revised Kolmogorov’s 1962 theory in terms of the thermodynamic fluctuation of physical variables.

18. Conditional PDF (Stolovitzky et. al, PRL, 69)

19. Thanks all