The lattice Boltzmann method (LBM) have emerged as an alternative computational approach to the conventional computational fluid dynamics (CFD). Despite being computationally efficient and popular numerical method for simulation of complex fluid flow, the LBM exhibits severe instabilities in near-grid scale hydrodynamics where sharp gradients are present. Further, since the LBM often uses uniform cartesian lattices in space, the curved boundaries are usually approximated by a series of stairs that also causes computational inaccuracy in the method. An interpolation-based treatment is introduced for the curved boundaries by Mei et al. One of the recipe to stabilize the LBM is the introduction of Ehrenfests' step. The objective of this work is to investigate the efficiency of the LBM with Ehrenfests' steps for the flows around curved bluff bodies. For this purpose, we have combined the curved boundary treatment of Mei et al. and the LBM with Ehrenfests' steps and developed an efficient numerical scheme. To test the validity of our numerical scheme we have simulated the two-dimensional flow around a circular cylinder and an airfoil for a wide range of low to high Reynolds numbers (Re ≤ 30, 000). We will show that the LBM with Ehrenfests' steps can quantitatively capture the Strouhal-Reynolds number relationship and the drag coefficient without any need for explicit sub-grid scale modeling. Comparisons with the experimental and numerical results show that this model is a good candidate for the turbulence modeling of fluids around bluff bodies.

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Room temperature ionic liquids are molten salts at ambient temperature. This thesis is concerned with the structure of room temperature ionic liquids both in the bulk and in confinement. A particular theme is the rich statistical physics of such systems, which is primarily due to strong Coulomb correlations. We begin by examining the structure of ionic liquids in the bulk. One of the most important questions in understanding the structure of ionic liquids is whether ions are truly "free" and mobile (a concentrated ionic solution), or are rather bundled up as ion pairs (a dilute solution of free ions dissolved in a sea of ion pairs). We propose a mathematical model for the thermodynamics and kinetics of ion pairing in ionic liquids. Our model reveals that roughly 2/3 of ions are free, whilst those ion pairs that do exist are short- lived. We conclude that ionic liquids ought to be considered to be concentrated, rather than dilute, electrolytes. We then examine the structure of ionic liquids confined between charged surfaces. Motivated by surface force balance experiments, we use a 1D Coulomb gas model to study ionic liquids confined between charged mica surfaces. We show that the disjoining pressure- surface separation curve depends on the fugacity (the bulk cohesive energy) of the ionic liquid, and the electrostatic interaction energy of ions at closest approach. The model shows good qualitative agreement with experimental data, with all parameters independently estimated without fitting. We then turn our attention to a single layer of ionic liquid ions confined between metal surfaces, and develop a mean field model for equilibrium charge storage in a nanoporous supercapacitor. The development of nanoporous supercapacitors is hindered by the perceived tradeoff between capacitance, power delivery and reversible charging - a trilemma. Our model identifies the affinity of ions to the pore wall as a key parameter that controls charging behaviour. From this, we elucidate a region of "ionophilicity" in which the capacitance-power-hysteresis trilemma can be avoided. The non-equilibrium physics of ion transport in ionic liquids is then considered. We derive a novel system of continuum electrokinetic equations for ionic liquids that is based ?on coarse graining a simple exclusion process defined on a lattice. The resulting dynamical equations are written as a gradient flow with a degenerate mobility function. This form of the mobility function gives rise to novel charging behaviours that are qualitatively different to those known in electrolytic solutions. Finally, we consider active non-equilibrium fluids, in which energy is continuously consumed from the surrounding environment. Those systems are inherently far from thermo- dynamic equilibrium. By developing a top-down description for non-equilibrium systems with the fluctuation spectrum as the central quantity, we show that the form of the disjoin- ing force may reveal crucial elements of the microscopic physics. Our framework explains the long-ranged force observed in recent molecular dynamics simulation of active Brownian particles.

A detailed understanding of electrical transport and energy dissipation phenomena is crucial for the discovery and engineering of new, high-performance materials for application ranging from nano-electronics to thermoelectric energy conversion. The experimental investigation of charge and heat conduction in controlled conditions is often challenging and expensive, and therefore accurate and efficient theoretical and simulation approaches are pivotal to foster new advancements. In my PhD work I have developed an ab initio computational approach to predict electronic transport properties of bulk materials. This work goes beyond the state-of-the-art in the field by merging an accurate, first-principles description of the fundamental interaction between electrons and lattice vibrations, and the exact solution of the Boltzmann transport equation . This computationally challenging task is accomplished by the development of a novel, fully parallel computational infrastructure based on state-of-the-art high-performance computing techniques. This approach allows the calculation of a range of electronic transport coefficients (electrical conductivity, mobility, electronic thermal conductivity, Seebeck coefficient and Lorenz number), also accounting for the effects of non-equilibrium phonon populations due to thermal gradients. In addition, the analysis tools developed in this work provide accurate ways to examine the microscopic details of the relevant scattering mechanisms and decay channels that determine each specific transport coefficient. These developments are of fundamental importance to assess and validate a number of approximations and phenomenological models that are popular within the electronic transport and thermoelectric communities. In addition, this work provides accurate tools to test design rules to enhance the thermoelectric performance and guide the experimental synthesis and characterization of more efficient compounds. In this work, I have applied this computational framework to the investigation of many aspects of thermoelectricity in diverse classes of materials, such as metals and doped semiconductors, in a wide range of temperature and, where suitable, doping concentration. In elemental metals, predictions for resistivity and Lorenz number have been compared against experiments and previous first-principles results, confirming the effectiveness of approximate approaches to Boltzmann transport for these systems. In n-doped silicon, I have focused on a wide range of electric and thermoelectric quantities. In addition to a detailed characterization of resistivity and mobility, this work provides a unique insight on the Lorenz number, a quantity that is not directly accessible with experiments and plays an important role in thermoelectric engineering. Moreover, the analysis of the Seebeck coefficient provides a better understanding of the coupled electron and phonon dynamics in this material, also suggesting strategies towards higher thermoelectric efficiency. Finally, I have analysed the transport properties of boron-doped diamond. Here I have used phenomenological models to simulate the complex doping mechanism in place in this system. I have focused on the prediction of the best transport properties achievable in Boron-doped diamond, in order to provide a useful reference for the experiments that typically display a large uncertainty (due to the difficult assessment of impurity content in synthetic samples). The characterization of the different phonon scattering channels (also in comparison with the case of silicon) offers an insight into the origin of the extraordinary hole mobilities of this system.

This abstract contains special characters which can be better seen in the attached pdf In this project finite-size size scaling above the upper critical dimension dc is investigated. Finite-size scaling there has long been poorly understood, especially its dependency on boundary conditions. The violation of the hyperscaling relation above dc has also been one of the most visible issues. The breakdown in standard scaling is due to the dangerous irrelevant variables presented in the self-interacting term in the 4 theory, which were considered dangerous to the free energy density and associated thermodynamic functions, but not to the correlation sector. Recently, a modied nite-size scaling scheme has been proposed, which considers that the correlation length actually plays a pivotal role and is aected by dangerous variables too. This new scheme, named QFSS, considers that the correlation length, instead of having standard scaling behaviour L, scales as L. This pseudocritical exponent is connected to the critical dimension and dangerous variables. Below dc this exponent takes the value = 1, but above the upper critical dimension it is = d=dc. QFSS succeeded in reconciling the mean-eld exponents and FSS derived from the renormalisation-group for the models with short-range interactions above dc with periodic boundary conditions. If is an universal exponent, the validity of that theory should also hold for the free boundary conditions. Initial tests for such systems faced new problems. Whereas QFSS is valid at pseudocritical points where quantities such as the magnetic susceptibility experience a peak for nite systems, at critical points the standard FSS seemed to prevail, i.e., mean-eld exponents with L. Here, we show that this last picture at critical point is not correct and instead the exponents that applied there actually arise from the Gaussian fixed-point FSS where the dangerous variables are suppressed. To achieve this aim, we study Ising models with long-range interaction, which can be tuned above dc, with periodic and free boundary conditions. We also include a study of the Fourier modes which can be used as an example of scaling quantities without dangerous variables.

In this work the relaxation of a two-dimensional Bose-gas from a non-equilibrium initial condition consisting of vortices is studied. To focus on the role of the vortex excitations on the time evolution of the system, a point vortex model is used. The relaxation of the vortex gas is seen to lead to clustering of like-signed vortices that can be explained in terms of negative temperature states. The nature of the Coulomb interactions between vortices, precludes a well-defined thermodynamic limit. The large scale flow structures, therefore strongly depend on the shape of the geometry. These structures can be explained in terms of a maximum entropy principle for the vortex gas that leads to the Boltzmann-Poisson equation. For a square region the maximum entropy configuration corresponds to a monopole. This configuration results in the spontaneous acquisition of angular momentum by the ow. However, by stretching the square domain into a rectangle, the configuration which maximises the entropy switched to a dipole where like-signed vortices tend to equally occupy the two halves of the domain. In this case, the mean flow has zero angular momentum. A direct qualitative and quantitative comparison between the predictions of the mean-field theory and dynamical simulations of a point vortex model are presented. In particular, we show that vortex-antivortex annihilation results in evaporative heating of the vortex gas and the subsequent migration of the system into the negative temperature regime. Moreover, the study is extended to the dynamics of quantised vortices in the same confined geometries in a two-dimensional Bose-Einstein condensate described by the Gross-Pitaevskii equation. Despite the coexistence of phonons with vortex excitations that interact together, the above predictions continue to apply in this more realistic model of a two-dimensional superfl uid.

The Ising and $Q$-state Potts models are statistical mechanical models of spins interaction on crystal lattices. We study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena $-$ how higher level structure is affected by the detailed activity of a very large number of lower level structures. We investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. Our study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. Here we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. The partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. We extend the study of $Q$-state Potts models to a more general case called the $Z_Q$-symmetric model. We evidence the existence of multiple phase transitions for this model in case $Q=5,6,$ and discuss the possible connection to different stages of disordered state. Given sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. Our analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. We evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent $\alpha$.

This thesis investigates the usability of the lattice Boltzmann method (LBM) for the simulation of indoor air flows in real-time. It describes the work undertaken during the three years of a Ph.D. study in the School of Mechanical Engineering at the University of Leeds, England. Real-time fluid simulation, i.e. the ability to simulate a virtual system as fast as the real system would evolve, can benefit to many engineering application such as the optimisation of the ventilation system design in data centres or the simulation of pollutant transport in hospitals. And although real-time fluid simulation is an active field of research in computer graphics, these are generally focused on creating visually appealing animation rather than aiming for physical accuracy. The approach taken for this thesis is different as it starts from a physics based model, the lattice Boltzmann method, and takes advantage of the computational power of a graphics processing unit (GPU) to achieve real-time compute capability while maintaining good physical accuracy. The lattice Boltzmann method is reviewed and detailed references are given a variety of models. Particular attention is given to turbulence modelling using the Smagorinsky model in LBM for the simulation of high Reynolds number flow and the coupling of two LBM simulations to simulate thermal flows under the Boussinesq approximation. A detailed analysis of the implementation of the LBM on GPU is conducted. A special attention is given to the optimisation of the algorithm, and the program kernel is shown to achieve a performance of up to 1.5 billion lattice node updates per second, which is found to be sufficient for coarse real-time simulations. Additionally, a review of the real-time visualisation integrated within the program is presented and some of the techniques for automated code generation are introduced. The resulting software is validated against benchmark flows, using their analytical solutions whenever possible, or against other simulation results obtained using accepted method from classical computational fluid dynamics (CFD) either as published in the literature or simulated in-house. The LBM is shown to resolve the flow with similar accuracy and in less time.

Statistical mechanics is concerned with finding the macroscopic behaviour of a physical system given its microscopic characteristics. At equilibrium there is a general framework given in terms of the various statistical ensembles that describes how to calculate the macroscopic quantity that is desired. Out of equilibrium there is no such framework, leading to the treatment of microscopic models on an individual basis and the investigation of arbitrarily defined models. However, there exists a recent theory of boundary driven steady states and an associated nonequilibrium counterpart to detailed balance due to Evans. In this thesis I first review this theory of boundary driven steady states and the associated nonequilibrium counterpart to detailed balance due to Evans, before applying the theory to some toy models of driven athermal systems. These initial attempts do not reproduce the qualitative behaviour of granular systems such as jamming but are a valuable and novel starting point for a more thorough investigation of this technique. I then move on to the general theory of boundary driven systems and formulate a nonequilibrium free energy principle. The physical content of this is illustrated through a simple diffusion model. I then provide a reformulation of the principle which is more suitable for calculation and demonstrate its validity in a more complex model. Finally I investigate a particular example of a boundary driven system, a toy model of a complex fluid called the rotor model. I first use simulation to investigate the model and its phase behaviour, before using an analytical approach to do the same. This approach takes the form of a nonequilibrium real space renormalisation group calculation, and qualitatively reproduces some of the features seen in the simulations.