Analysis of fully developed turbulence by the non-extensive Tsallis statistics

An analysis of fully developed turbulence is developedbased on the assumption that the underlyingstatistics of the system is of Tsallis ensemble.The distribution function for the probability density function ofthe local dissipation is derived by taking extremal ofTsallis entropy under the two constraints, i.e., one isthe normalization of probability and the other is to fixthe intermittency exponent being constant.The multifractal spectrum $f_{\rm T}(\alpha)$corresponding to the distribution functionis determined self-consistently in the sense thatall parameters can be obtained through the observed valueof the intermittency exponent.It is shown that the scaling exponents $\zeta_m$of velocity structure function derived with the help ofthe multifractal spectrum explains experimental data very well.It is also revealed that the asymptotic expressionof $\zeta_m$ for $m \gg 1$ has a log term,which was not recognized before.I hope that the present approach will provide us witha deeper understanding of statistics underlyingthe system of fully developed turbulence, and with a reasonableinterpretation of somewhat funny statistics based onTsallis {\it non-extensive} entropy.