Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Exchange of nitrogen dioxide (NO 2) between plants and the atmosphere under laboratory and field conditions

Nitrogen is an essential nutrient. It is for human, animal and plants a constituent element of proteins and nucleic acids. Although the majority of the Earth’s atmosphere consists of elemental nitrogen (N2, 78 %) only a few microorganisms can use it directly. To be useful for higher plants and animals elemental nitrogen must be converted to a reactive oxidized form. This conversion happens within the nitrogen cycle by free-living microorganisms, symbiotic living Rhizobium bacteria or by lightning. Humans are able to synthesize reactive nitrogen through the Haber-Bosch process since the beginning of the 20th century. As a result food security of the world population could be improved noticeably. On the other side the increased nitrogen input results in acidification and eutrophication of ecosystems and in loss of biodiversity. Negative health effects arose for humans such as fine particulate matter and summer smog. Furthermore, reactive nitrogen plays a decisive role at atmospheric chemistry and global cycles of pollutants and nutritive substances.rnNitrogen monoxide (NO) and nitrogen dioxide (NO2) belong to the reactive trace gases and are grouped under the generic term NOx. They are important components of atmospheric oxidative processes and influence the lifetime of various less reactive greenhouse gases. NO and NO2 are generated amongst others at combustion process by oxidation of atmospheric nitrogen as well as by biological processes within soil. In atmosphere NO is converted very quickly into NO2. NO2 is than oxidized to nitrate (NO3-) and to nitric acid (HNO3), which bounds to aerosol particles. The bounded nitrate is finally washed out from atmosphere by dry and wet deposition. Catalytic reactions of NOx are an important part of atmospheric chemistry forming or decomposing tropospheric ozone (O3). In atmosphere NO, NO2 and O3 are in photosta¬tionary equilibrium, therefore it is referred as NO-NO2-O3 triad. At regions with elevated NO concentrations reactions with air pollutions can form NO2, altering equilibrium of ozone formation.rnThe essential nutrient nitrogen is taken up by plants mainly by dissolved NO3- entering the roots. Atmospheric nitrogen is oxidized to NO3- within soil via bacteria by nitrogen fixation or ammonium formation and nitrification. Additionally atmospheric NO2 uptake occurs directly by stomata. Inside the apoplast NO2 is disproportionated to nitrate and nitrite (NO2-), which can enter the plant metabolic processes. The enzymes nitrate and nitrite reductase convert nitrate and nitrite to ammonium (NH4+). NO2 gas exchange is controlled by pressure gradients inside the leaves, the stomatal aperture and leaf resistances. Plant stomatal regulation is affected by climate factors like light intensity, temperature and water vapor pressure deficit. rnThis thesis wants to contribute to the comprehension of the effects of vegetation in the atmospheric NO2 cycle and to discuss the NO2 compensation point concentration (mcomp,NO2). Therefore, NO2 exchange between the atmosphere and spruce (Picea abies) on leaf level was detected by a dynamic plant chamber system under labo¬ratory and field conditions. Measurements took place during the EGER project (June-July 2008). Additionally NO2 data collected during the ECHO project (July 2003) on oak (Quercus robur) were analyzed. The used measuring system allowed simultaneously determina¬tion of NO, NO2, O3, CO2 and H2O exchange rates. Calculations of NO, NO2 and O3 fluxes based on generally small differences (∆mi) measured between inlet and outlet of the chamber. Consequently a high accuracy and specificity of the analyzer is necessary. To achieve these requirements a highly specific NO/NO2 analyzer was used and the whole measurement system was optimized to an enduring measurement precision.rnData analysis resulted in a significant mcomp,NO2 only if statistical significance of ∆mi was detected. Consequently, significance of ∆mi was used as a data quality criterion. Photo-chemical reactions of the NO-NO2-O3 triad in the dynamic plant chamber’s volume must be considered for the determination of NO, NO2, O3 exchange rates, other¬wise deposition velocity (vdep,NO2) and mcomp,NO2 will be overestimated. No significant mcomp,NO2 for spruce could be determined under laboratory conditions, but under field conditions mcomp,NO2 could be identified between 0.17 and 0.65 ppb and vdep,NO2 between 0.07 and 0.42 mm s-1. Analyzing field data of oak, no NO2 compensation point concentration could be determined, vdep,NO2 ranged between 0.6 and 2.71 mm s-1. There is increasing indication that forests are mainly a sink for NO2 and potential NO2 emissions are low. Only when assuming high NO soil emissions, more NO2 can be formed by reaction with O3 than plants are able to take up. Under these circumstance forests can be a source for NO2.

Canonical group quantization and boundary conditions

In the present thesis, we study quantization of classical systems with non-trivial phase spaces using the group-theoretical quantization technique proposed by Isham. Our main goal is a better understanding of global and topological aspects of quantum theory. In practice, the group-theoretical approach enables direct quantization of systems subject to constraints and boundary conditions in a natural and physically transparent manner -- cases for which the canonical quantization method of Dirac fails. First, we provide a clarification of the quantization formalism. In contrast to prior treatments, we introduce a sharp distinction between the two group structures that are involved and explain their physical meaning. The benefit is a consistent and conceptually much clearer construction of the Canonical Group. In particular, we shed light upon the 'pathological' case for which the Canonical Group must be defined via a central Lie algebra extension and emphasise the role of the central extension in general. In addition, we study direct quantization of a particle restricted to a half-line with 'hard wall' boundary condition. Despite the apparent simplicity of this example, we show that a naive quantization attempt based on the cotangent bundle over the half-line as classical phase space leads to an incomplete quantum theory; the reflection which is a characteristic aspect of the 'hard wall' is not reproduced. Instead, we propose a different phase space that realises the necessary boundary condition as a topological feature and demonstrate that quantization yields a suitable quantum theory for the half-line model. The insights gained in the present special case improve our understanding of the relation between classical and quantum theory and illustrate how contact interactions may be incorporated.