Year: 2014
Type: dissertation
Abstract: This thesis deals with Markov processes in stochastic thermodynamics and fully developed turbulence. \nIn the first part of the thesis, a detailed account on the theory of Markov processes is given, forming the mathematical fundament. In the course of developing the theory of continuous Markov processes, stochastic differential equations, the Fokker-Planck equation and Wiener path integrals are introduced and embedded into the class of discontinuous Markov processes. Special attention is paid to the difficulties that arise in the case of multiplicative noise. \nReferring to the paradigm of Brownian motion, the thermodynamic quantities work, heat and entropy, and the accompanied first law and second law, are formulated on the level of individual trajectories using stochastic differential equations. \nOne of the prominent results of stochastic thermodynamics are so-called fluctuation theorems which reveal an intimate relation between entropy production and irreversibility. Using the path integral formulation, fluctuations theorems are derived in a formal setting. \nApplications of fluctuation theorems to thermodynamic systems are dependent on a reliable statistics of rare events. To access the statistics of rare events, an asymptotic method is developed. The first order asymptotics is derived for the general case of multiplicative noise, the second order asymptotics for the simpler case of additive noise. The application of the asymptotic method is demonstrated for work distributions in physically relevant models. \nThe second part of the thesis carefully introduces the concept of fully developed turbulence and gives an account on established theories. These theories can be reformulated in terms of Markov processes. An overview of these Markov representations is compiled, including a few new approaches. \nIn view of the results of stochastic thermodynamics, implications for the Markov representation are exploited. Fluctuation theorems are derived and applied to experimental data. In an experimental as well as theoretical analysis, fluctuation theorems are found to be sensitive to intermittent small-scale fluctuations. The above mentioned asymptotic method is used to assess these fluctuations. \nIn closing the thesis, the discussed approaches to fully developed turbulence are contrasted with each other using their Markov representations, and an interpretation of the respective Markov processes is offered. \n
Author: Daniel Nickelsen
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2098634820