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

A STUDY OF QUANTUM ANNEALING DEVICES FROM A CLASSICAL PERSPECTIVE

Spin glasses are experiencing a revival due to applications in quantum information theory. In particular, they are the archetypal native benchmark problem for quantum annealing machines. Furthermore, they find applications in fields as diverse as satisfiability, neural networks, and general combinatorial optimization problems. As such, developing and improving algorithms and methods to study these computationally complex systems is of paramount importance to many disciplines. This body of work attempts to attack the problem of solving combinatorial optimization problems by simulating spin glasses from three sides: classical algorithm development, suggestions for quantum annealing device design, and improving measurements in realistic physical systems with inherent noise. I begin with the introduction of a cluster algorithm based on Houdayer’s cluster algorithm for two-dimensional Ising spin-glasses that is applicable to any space dimension and speeds up thermalization by several orders of magnitude at low temperatures where previous algorithms have difficulty. I show improvement for the D-Wave chimera topology and the three-dimensional cubic lattice that increases with the size of the problem. One consequence of adding cluster moves is that for problems with degenerate solutions, ground-state sampling is improved. I demonstrate an ergodic algorithm to sample ground states through the use of simple Monte Carlo with parallel tempering and cluster moves. In addition, I present a non-ergodic algorithm to generate new solutions from a bank of known solutions. I compare these results against results from quantum annealing utilizing the D-Wave Inc. quantum annealing device. Finally, I present an algorithm for improving the recovery of ground-state solutions from problems with noise by using thermal fluctuations to infer the correct solution at the Nishimori temperature. While this method has been demonstrated analytically and numerically for trivial ferromagnetic and Gaussian distributions, a useful metric for more complex Gaussian distributions with added Gaussian noise is unavailable. We show improved recovery of numerical solutions on the chimera graph with a ferromagnetic distribution and added Gaussian noise. Next, I direct my focus to the design of future generations of quantum annealers. The first design is the two-dimensional square-lattice bimodal spin glass with next-nearest ferrromagnetic interactions proposed by Lemke and Campbell claimed to exhibit a finite-temperature spin-glass state for a particular relative strength of the next-nearest to nearest neighbor interactions. Our results from finite-temperature simulations show the system is in a paramagnetic state in the thermodynamic limit, thus not useful for quantum annealing device designs that would benefit from a spin-glass phase transition. The second design is the diluted next-nearest neighbor Ising spin-glass with Gaussian interactions in an attempt to improve the estimation of critical parameter with smaller system sizes by implementing averaging of observables over different graph dilutions. To date, this model has shown no improvement. Finally, I make suggestions for the choice of distributions of interactions that are robust to noise and present a method for using previously unaccessible continuous distributions. I begin with showing the best-case performance of quantum annealing devices. I show results for the resilience, the probability that the ground-state solution has changed due to inherent analog noise in the device, and present strategies for developing robust instance classes. The analog noise is also detrimental to interactions chosen from continuous distributions. Using Gaussian quadratures, I present a method for discretizing continuous distributions to reduce noise effects. Simulations on the D-Wave show that the average residual of the ground-state energy with the true ground-state energy is calculated and shown to be smaller in the case of the discrete distribution

Thread-safe lattice Boltzmann for high-performance computing on GPUs

We present thread-safe, highly-optimized lattice Boltzmann implementations, specifically aimed at exploiting the high memory bandwidth of GPU-based architectures. At variance with standard approaches to LB coding, the proposed strategy, based on the reconstruction of the post-collision distribution via Hermite projection, enforces data locality and avoids the onset of memory dependencies, which may arise during the propagation step, with no need to resort to more complex streaming strategies. The thread-safe lattice Boltzmann achieves peak performances, both in two and three dimensions and it allows to sensibly reduce the allocated memory ( tens of GigaBytes for order billions lattice nodes simulations) by retaining the algorithmic simplicity of standard LB computing. Our findings open attractive prospects for high-performance simulations of complex flows on GPU-based architectures

Multicomponent flow on curved surfaces: A vielbein lattice Boltzmann approach

We develop and implement a novel finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write down the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focussing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of such geometries. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries