Superstatistics as the statistics of quasi-equilibrium states: Application to fully developed turbulence

In non-equilibrium states, currents are produced by irreversible processes that take a system toward the equilibrium state, where the current vanishes. We demonstrate, in a general setting, that a superstatistics arises when the system relaxes to a (stationary) quasi-equilibrium state instead, where only the $\mathit{\text{mean}}$ current vanishes because of fluctuations. In particular, we show that a current with Gaussian white noise takes the system to a unique class of quasi-equilibrium states, where the superstatistics coincides with Tsallis escort $q$-distributions. Considering the fully developed turbulence as an example of such quasi-equilibrium states, we analytically deduce the power-law spectrum of the velocity structure functions, yielding a correction to the log-normal model which removes its shortcomings with regard to the decreasing higher order moments and the Novikov inequality, and obtain exponents that agree well with the experimental data.