This thesis describes the development and application of models to predict the equation of mechanical state of materials from first principles, concentrating on the regime of strong shock waves. Most effort was devoted to crystalline solids, though extensions to the fluid phase and higher temperatures are proposed. Equations of state and phase diagrams were predicted for aluminium, silicon and beryllium. The method used is based on quantum mechanical treatments of the electrons in the solid and of the phonon modes. The importance of anharmonic effects (phonon-phonon interactions) was investigated, but was not included rigorously because it did not appear to contribute significantly. With fully <i>ab initio</i> methods, the equation of state and phase diagram could be predicted to a few percent in mass density, the discrepancy being caused mainly by the use of the local density approximation in predicting electron states. The accuracy of the equation of state could be improved considerably by adjusting the internal energy to reproduce the observed mass density at STP. The resulting <i>ab fere initio</i> equation of state could essentially reproduce the observed states on the shock Hugoniot to within the scatter in the experimental data. Because these equations of state are built on firm quantum mechanical and thermodynamic principles, they should predict accurate properties away from the principal Hugoniot, unlike traditional empirical equations of state. Accurate temperatures are important in the development of models of material strength (elasticity and plasticity) based on microstructural phenomena. As an illustration of the versatility of the equations of state, hydrocode simulations were made of the splitting of a shock wave in silicon, caused by the phase change. The splitting appears to be in reasonable agreement with laser-driven shock experiments.

This thesis contains various results on Vassiliev invariants: common themes running through include their polynomial nature, their functoriality, and the use of Gauβ diagrams. The first chapter examines the functoriality of Vassiliev invariants and describes how they can be defined on different types of knotty objects such as knots, framed knots and braids, and how algebraic structure naturally arises. An explicit form of the relationship between the framed and unframed knot theory is given. Chapter 2 considers the important question of whether a Vassiliev invariant can be naively obtained from a combinatorial object called a weight system. A partial answer to this is given by showing how "half" of the steps in such a transition can be performed canonically and explicitly. In Chapter 3 the first two non-trivial invariants for knots, evaluated on prime knots up to twelve crossing are examined, and some surprising graphs are obtained by plotting them. A number of results for torus knots are proved, relating unknotting number and crossing number to the first two Vassiliev invariants. The second half of the thesis is concerned primarily with Vassiliev invariants of pure braids and their connection with de Rham homotopy theory. In Chapter 4 a simple derivation is given showing the relationship between Vassiliev invariants and the lower central series of the pure braid groups. This is used to obtain closed formulae for the actual number of invariants of each type. Chapter 5 is a digression on de Rham homotopy theory and explains the geometric connections between Chen's iterated integrals, higher order Albanese manifolds, and Sullivan's 1-minimal models. A method of Chen's for obtaining integral invariants of elements of the fundamental group from a 1-minimal model is given, and in Chapter 6 this is used to find Vassiliev invariants of pure braids at low order; this extends work of M. A. Berger. Finally, a similar method using currents is employed to obtain a combinatorial formula for a type two invariant which is independent of winding numbers.

This study was motivated by the observation that most smooth bundles do not admit a smooth function that is Morse when restricted to every fibre. The complexity <I>c</I> of a critical point of a smooth map is measured by an appropriate codimension of its germ. The subset of smooth maps from a bundle to a manifold with complexity on fibres not exceeding <I>c</I> is studied. Bounds for <I>c</I> are established such that this subset is open and dense in the set of all smooth maps, where sets of smooth maps are always given the Whitney <I>C</I><SUP>∞</SUP> topology. The bounds are calculated in terms of the dimensions of the base space, the fibre and the manifold into which the bundle is mapped and are proved using the theory of finite germs and a suitable adaptation of the Thom Transversality Theorem. Recent work of Vasil'ev is used to investigate real-valued functions on compact principal <I>S</I><SUP>1</SUP>-bundles. The existence is established of a function with complexity on fibres no more than roughly half of the minimum value for <I>c</I> for the open and dense subsets mentioned above. For certain bundles with fibre of dimension one, the set of smooth real-valued functions that are Morse when restricted to every fibre is shown to be <I>C</I><SUP>0</SUP> dense but not, in general, <I>C</I><SUP>1</SUP> dense. For all <I>n</I>-sphere bundles over the circle the set is shown to be <I>C</I><SUP>0</SUP> dense. The homotopy type of the space of smooth Morse functions on the circle is derived. Arnold's determination of the fundamental group of the generalised Morse functions on the circle is included.

It is known that every scalar-type spectral operator on a Hilbert space <i>H</i> is similar to a multiplication operator on some <i>L</i>²<i> </i>space. The purpose of the main theorem in Chapter 2 of this thesis is to show that every scalar-type spectral operator on an <i>L</i>¹ space whose spectral measure has finite multiplicity is similar to a multiplication operator on the same <i>L</i>¹ space. Then we prove a similar result for scalar-type spectral operators on <i>L^p </i>(Ω, S_Ω, <i>m</i>), <i>p </i>≠<i> </i> 2, 1 < <i>p</i> < ∞, with spectral measure <i>E</i>(^.) of finite uniform multiplicity provided an extra condition is satisfied. Also, we give conditions that make a scalar-type spectral operator on <i>L</i>²(Ω, S_Ω, <i>m</i>) similar to a multiplication operator on the same <i>L</i>²(Ω, S_Ω, <i>m</i>). In 1954, Dunford proved that a bounded operator <i>T</i> on a Banach space <i>X</i> is spectral if and only if it has the canonical decomposition <i>T = S</i> +<i>Q</i>, where <i>S</i> is a scalar-type operator and <i>Q</i> is a quasinilpotent operator which commutes with <i>S</i>. In Chapter 3, we prove that any well-bounded operator <i>T</i> on a Hilbert space <i>H</i> has the form <i>T = A + Q</i>, where <i>A</i> is a self-adjoint operator and <i>Q</i> is a quasinilpotent operator such that <i>AQ - QA</i> is quasinilpotent. Then we prove that a trigonometrically well-bounded operator <i>T </i>on <i>H</i> can be decomposed as <i>T = U</i>(<i>Q + I</i>) where <i>U</i> is a unitary operator and <i>Q</i> is quasinilpotent such that <i>UQ = QU </i>is also quasinilpotent. In Chapter 4 we prove that an AC-operator with discrete spectrum on <i>H</i> can be decomposed as a sum of a normal operator <i>N</i> and a quasinilpotent <i>Q</i> such that <i>NQ - QN </i>is quasinilpotent. However, the converse of each of the last three theorems is not true in general. In the final chapter we introduce a new class of operators on <i>L</i>²([<i>a,b</i>]) which is larger than the class of well-bounded operators on <i>L</i>²([<i>a,b</i>]) and we call them operators with an <i>AC</i>₂-functional calculus. Then we give an example of an operator with an <i>AC</i>₂-functional calculus on <i>L</i>²([0,1]) which can be decomposed as a sum of a self-adjoint operator and a quasinilpotent. We also discuss the possibility of decomposing every operator <i>T</i> with an <i>AC</i>₂-functional calculus on <i>L</i>²([<i>a,b</i>]) into the sum of a self-adjoint operator <i>A</i> and a quasinilpotent operator <i>Q</i> such that <i>AQ - QA </i>is quasinilpotent.

A three-dimensional micromagnetic model is described which predicts the equilibrium magnetisation states of magnetite, one of the most common natural magnetic minerals. Solutions are presented for cubic, octahedral and irregularly shaped grains and their stability to external magnetic fields studied. Two aspects of the model make it more powerful than previous models. Firstly, a Fourier transform algorithm has been developed which reduces the number of calculations from <I>O</I>(<I>N</I><SUP>2</SUP>) to <I>O</I>(<I>N</I> log <I>N</I>) where <I>N</I> is the number of elementary magnetisations vectors. Secondly, the model is implemented on a parallel computer which reduces the computation time by a factor of approximately 1/(4<I>N<SUB>p</SUB></I>), where <I>N<SUB>p</SUB></I> = 16000 is the number of processors. Cubic and octahedral grains smaller than <I>d</I> = 0.07μm, where <I>d</I> is the width of grain, occupy uniform magnetisation states. However Néel's relaxation theory predicts that these single-domain (SD) states are only stable to thermal fluctuations in grains larger than <I>d</I> ≃ 0.045μm. Therefore SD states are only stable over a narrow size range suggesting that a significant percentage of remanence in magnetic minerals resides in grains larger than 0.07μm. Between 0.07μm and 0.1μm grains can occupy either flower states or vortex states. The magnetisation in flower states is mostly uniform but with some deflection at the corners of the grain. Vortex states are characterised by a circular magnetisation pattern which rotates around a line running through the centre of the grain. For cubic grains, vortex states are lower energy states than SD states for all grain sizes. However, for octahedral grains, vortex states are energetically favourable only in grains larger than 0.11μm. Hysteresis simulations show that vortex states can predict remanences which are ¼ of those predicted by SD theory and coercivities which are 2/3. These results are important in showing how a gradual transition in the bulk magnetic properties of magnitite, from SD states to multi-domain states can occur. Flower states are unstable in grains larger than 0.1μm and grains between 0.2μm and 1μm can occupy one of several different vortex states, each one of which is a separate local energy minimum state. The model predicts that the high values of saturation remanence and coercivity measured experimentally in grains in this size range are due to grains occupying high energy vortex states with an associated high magnetic moment. There is a gradual transition from vortex states in grains below 1μm to multi-domain states in grains larger than 1μm. During this transition vortices become localised features which can nucleate domain walls.

This thesis studies non-equilibrium dynamics in disordered "glassy" systems, focusing particularly on the response to such systems to external driving and loading. Its primary motivation is a body of experimental data suggesting that glassy dynamics underlie the mechanical properties (rheology) of a wide variety of disordered soft materials (e.g. foams, dense emulsions and dense colloidal suspensions): typically, such materials show pronounced non-linearity in their stress response to slow steady shear (often with a yield stress in the limit of zero shear) and a loss modulus which curves upwards slightly to low frequencies (in apparent violation of linear response theory). In what follows, can, when rheologically driven, broadly capture this behaviour. We also show that they predict <i>ageing </i>at small loads, in qualitative agreement with the results of recent ageing experiments. Beyond this rheological motivation, we also use the models to study more general concepts of glassy dynamics, such as the use of fluctuation-dissipation theorems (FDTs) for defining effective non-equilibrium temperatures. As a preliminary step, we extend the existing rheological formalism to include ageing materials in which time-translational invariance (TTI) is lost. Within this generalized framework, we then analyze the rheologically driven trap model - the "soft glassy rheology" (SGR) model - which considers an ensemble of elastic elements undergoing activated local yielding dynamics, with distributed yield thresholds, governed by a noise temperature <i>x</i>. (Between yields, each element follows affinely the applied shear). We review the model's exact constitutive solution and discuss its mapping, in the undriven limit, to Bouchaud's trap model. We exploit this mapping to demonstrate the existence (in linear rheology at least) of an ageing glass phase (<i>x</i> < 1), in which the relaxation time is always of order the sample age.

In this thesis we explore two different theories for modelling soft matter systems. We start by discussing density functional theory (DFT) and dynamical density functional theory (DDFT) and consider the thermodynamics underpinning these theories as well as showing how the main results may be derived from the microscopic properties of soft matter. We use this theory to set up a model for the evaporation of the solvent from a thin film of a colloidal suspension. The general background for such systems is discussed and we display some of the striking nanostructures which self-assemble during the evaporation process. We show that our theory successfully reproduces some of these patterns and deduce the various mechanisms and transport processes behind the formation of the different structures. In the second part of this thesis we discuss results for a second model; the phase field crystal (PFC) model. The model equations are discussed, showing how they may be derived from DDFT as well as discussing the general background of PFC models. We present some results for the PFC model in its most commonly used form before going on to introduce a modified PFC model. We show how the changes in the model equations are reflected in the thermodynamics of the model. We then proceed by demonstrating how this modified PFC model may be used to qualitatively describe colloidal systems. A two component generalisation of the modified PFC model is introduced and used to investigate the transition between hexagonal and square ordering in crystalline structures. We conclude by discussing the similarities and connections between the two models presented in the thesis.

No description available.

Quantum information theory is an exciting, inter-disciplinary field, combining elements of condensed matter theory, quantum mechanics and information theory. In this thesis, I shall make a modest contribution to this field by examining entanglement in many-body systems with more than two levels. In the first section, I consider the dynamics of a system of qutrits three-level quantum systems which are coupled through an SU(3)-invariant permutation Hamiltonian. Each term in this Hamil- tonian is a nearest-neighbour permutation operator, and thus this Hamiltonian may be considered a generalisation of the standard SU (2)-invariant Heisenberg Hamiltonian, in which every term (up to the addition of the identity operator) is a nearest-neighbour permutation operator for two-level system. The system considered has the topology of a cross, and thus may be considered (to a limited extent) analogous to a beam-splitter. The aim of the study is to establish a Bell singlet state between two distant parties. Building on this work, I shall go on to consider the ground state of a system made up of many-level systems coupled by the same Hamiltonian I shall show that this state is a generalisation of the two-level singlet to many levels and many systems. It thus has a high degree of symmetry. I will consider its application in entanglement distribution through measurements (localisable entanglement), and discuss how it may be physically implemented in systems of ultracold atoms, through the Hubbard model. I shall also show that in the famous valence bond solid (the ground state of the Affleck-Kennedy- Lieb-Tasaki spin chain), all the entanglement present in the state may be extracted from a single copy of the chain this is in contrast to gapless, critical chains, in which only half the total entanglement is extractable from a single copy.

Condensed matter theory is the study of systems at finite density. In this thesis we will attempt to argue that gauge-gravity dualities can give deep and meaningful insights into the behaviour of strongly coupled condensed matter systems. The first three chapters will be a review of material already available in literature. Chapter 1 will introduce holography and the AdS-CFT correspondence. Particularly, in this chapter, the technique for the extraction of diffusion constants for charge and shear stress-energy-momentum fluctuations in a field theory with a holographic dual will be demonstrated. Chapter 2 will summarise relevant literature on the relativistic fluid-gravity correspondence. In the first half of the chapter it will be shown how to calculate the transport coefficients and Navier-Stokes equations for a suitable thermal field theory. The second half of chapter 2 will then be dedicated to extracting the transport coefficients for a strongly coupled field theory dual to a Reissner-Nordstrøm AdS spacetime. In chapter 3 a scaling of the metric and gauge field found in chapter 2 will be taken such that the boundary field theory admits Galilean, as opposed to relativistic, symmetry. Consequently, the governing hydrodynamic equations will be the non-relativistic, incompressible Navier-Stokes. Chapters 4 and 5 represent novel work. In chapter 4 the transport coefficients for a particular strongly coupled thermal field theory with underlying Schrodinger symmetry will be extracted from a charged, asymptotically Schrodinger spacetime. The governing hydrodynamic equations will be compressible with non-relativistic symmetry as opposed to those found via the scaling limit of chapter 3. In chapter 5 we show how knowledge of the transport coefficients of a thermal field theory can be used as a test-bed for numerical methods to explore beyond the hydrodynamic (long wavelength and low frequency) regime. With this in mind we consider Reissner-Nordstrøm AdS4 and determine the two point correlators at arbitrary frequency and momentum. Finally in chapter 6 we summarise the work discussed in this thesis and speculate about further applications of hydrodynamic techniques to strongly coupled condensed matter theories.