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

TL;DR

This paper introduces superstatistics as a framework for describing quasi-equilibrium states in non-equilibrium systems, specifically applying it to turbulence to derive power-law spectra and improve upon existing models.

Contribution

It demonstrates that superstatistics naturally arises in systems relaxing to quasi-equilibrium states and applies this to turbulence, deriving analytical results consistent with experimental data.

Findings

Superstatistics coincides with Tsallis escort q-distributions in Gaussian noise systems.

Derived power-law spectra for turbulence velocity structure functions.

Improved the log-normal model, addressing its shortcomings and aligning better with experiments.

Abstract

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 \textit{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.