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.

Test of Lorentz symmetry with a 3 He,129 Xe clock-comparison experiment

In case of violation of CPT- and Lorentz Symmetry, the minimal Standard Model Extension (SME) of Kostelecky and coworkers predicts sidereal modulations of atomic transition frequencies as the Earth rotates relative to a Lorentz-violating background field. One method to search for these modulations is the so-called clock-comparison experiment, where the frequencies of co-located clocks are compared as they rotate with respect to the fixed stars. In this work an experiment is presented where polarized 3He and 129Xe gas samples in a glass cell serve as clocks, whose nuclear spin precession frequencies are detected with the help of highly sensitive SQUID sensors inside a magnetically shielded room. The unique feature of this experiment is the fact that the spins are precessing freely, with transverse relaxation times of up to 4.4 h for 129Xe and 14.1 h for 3He. To be sensitive to Lorentz-violating effects, the influence of external magnetic fields is canceled via the weighted difference of the 3He and 129Xe frequencies or phases. The Lorentz-violating SME parameters for the neutron are determined out of a fit on the phase difference data of 7 spin precession measurements of 12 to 16 hours length. The result of the fit gives an upper limit for the equatorial component of the neutron parameter b_n of 3.7×10^(−32) GeV at the 95% confidence level. This value is not limited by the signal-to-noise ratio, but by the strong correlations between the fit parameters. To reduce the correlations and therewith improve the sensitivity of future experiments, it will be necessary to change the time structure of the weighted phase difference, which can be realized by increasing the 129Xe relaxation time.