PDF of velocity fluctuation in turbulence by a statistics based on generalized entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the Rényi entropy or the Tsallis entropy. The parameters appearing in the PDF, including the index q in the generalized measures, are determined self-consistently with the help of the observed value μ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.