Markov processes linking stochastic thermodynamics and turbulent cascades

etc15 Tracking Number 302

An elementary example of a Markov process (MP) is Brownian motion. The work done and the entropy produced for single trajectories of the Brownian particles are random quantities. Statistical properties of such fluctuating quantities are central in the field of stochastic thermodynamics \cite{seifert_ov}. Prominent results of stochastic thermodynamics are so-called fluctuation theorems (FTs) which express the balance between production and consumption of entropy \cite{seifert_ft}. Turbulent cascades of eddies are assumed to be the predominant mechanism of turbulence fixing the statistical properties of fully developed (boundary-free) turbulent flows. These properties typically adress the two-point statistics of the flow field and hold universally for any kind of turbulence generation \cite{frisch}. Various models of turbulence aim at reproducing the observed universal properties. An intriguing phenomenon of developed turbulence are violent small-scale fluctuations in flow velocity that exceed any Gaussian prediction, commonly referred to as small-scale intermittency \cite{frisch}. The correct reproduction of small-scale intermittency by models of turbulence is of particular importance in turbulence research \cite{frisch,sreenivasan}. In analogy to Brownian motion, we show how the assumption of the Markov property leads to a MP for the turbulent cascade that is equivalent to the seminal K62 model \cite{k62}. In addition to the K62 model, we demonstrate how many other models of turbulence can be written as a MP, including scaling laws, multiplicative cascades, multifractal models and field-theoretic approaches. Based on the various MPs, we discuss the production of entropy and the corresponding FTs. In particular, an experimental analysis indicates that entropy consumption is linked to small-scale intermittency and, as a consequence, the corresponding FT probes the correct modeling of small-scale intermittency of the underlying model of turbulence \cite{nickelsen}. Using the FT as a citerion, we demonstrate in another experimental study that the three-point statistics of a developed turbulent flow field is universal only for the same kind of turbulence generation \cite{reinke}.