The weather dynamics are significantly determined by the motion of the atmosphere and the ocean. This motion is often turbulent, characterized by fluctuations of the flow velocity over wide spatial and temporal scales. This fact, besides limited observability and inaccurate models, impedes the predictability of quantities such as the velocity of winds, which are relevant for the everyday life. One is always interested in improving such predictions - by employing better models or obtaining more information about the system.
The Crooks fluctuation theorem is a relation from nonequilibrium thermodynamics, which has its typical applications in nanoscale systems. It quantifies the distribution of imposed work in a process, where the system is pushed out of thermal equilibrium. This distribution is broadened due to the fluctuations of the microscopic degrees of freedom in the system.
The fluctuations of the velocity field in turbulent flow suggest the derivation of an analogy of Crooks' theorem for this macroscopic system. The knowledge about the validity of such a relation is additional information, which one in reverse could use to improve predictions about the system. In this thesis both issues are addressed: the derivation of the theorem, and the improvement of predictions.
We illustrate the application of Crooks' theorem to hydrodynamic flow within a model of a two-dimensional inviscid and incompressible fluid field, when pushed out of dynamical equilibrium. The flow on a rectangular domain is approximated by the two-dimensional vorticity equation with spectral truncation. In this setting, the equilibrium statistics of the flow can be described through a canonical ensemble with two conserved quantities, kinetic energy and enstrophy. To perturb the system out of equilibrium, we change the shape of the domain according to a protocol, which changes the kinetic energy but leaves the enstrophy constant. This is interpreted as doing work to the system. Evolving along a forward and its corresponding backward process, we find that the distributions of the work performed in these processes satisfy the Crooks relation with parameters derived from the canonical ensembles.
We address the issue of prediction in this thesis in a concrete setting: There are examples where the distributions of a variable in the forward and the backward process collapse into one, hence Crooks' theorem relates the distribution of one variable with itself. For a finite data set drawn from such a distribution, we are interested in an estimate of this variable to exceed a certain threshold. We demonstrate that, using the knowledge about Crooks' relation, forecast schemes can be proposed which improve compared to a pure frequency estimate on the data set. The findings are illustrated in three examples, studies of parameters such as exceedance threshold and data set size are presented.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:14-qucosa-156748 |
| Date | 06 January 2015 |
| Creators | Gundermann, Julia |
| Contributors | Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, Prof. Dr. Holger Kantz, Prof. Dr. Holger Kantz, Prof. Dr. Günter Radons |
| Publisher | Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
Page generated in 0.0026 seconds