Theses - Applied Mathematics and Theoretical Physics

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  • ItemOpen Access
    Chaos in models of double convection
    Rucklidge, Alastair Michael
    This dissertation concentrates on the derivation and analysis of low-order sets of ordinary differential equations (ODEs) that accurately describe the behaviour of a fluid in convective motion. A second-order set of ODEs is presented and analysed, and then related to a particular double convection problem (compressible convection in a vertical magnetic field); the low-order model proves to be useful in interpreting the behaviour of the full system. Equations describing several types of double convection (convection in a magnetic field, convection in a rotating layer of fluid and convection in a solute gradient) are reduced to low-order sets of ODEs that are asymptotically exact descriptions of the partial differential equations (PDEs) from which they were derived. The ODE model for incompressible convection in a vertical magnetic field is analysed in detail, and a rich variety of periodic orbits and chaotic behaviour is found. A numerical study of the full set of PDEs for this case confirms that the low-order model provides an asymptotically correct description of the full problem; in particular, the PDEs have the chaotic solutions predicted by the low-order model.
  • ItemOpen Access
    On the Factorisation of Matrix Wiener–Hopf Kernels Arising From Acoustic Scattering Problems
    Aitken, Mungo
    The research undertaken in this thesis is in the broad area of diffraction theory. We consider three separate and distinct problems of acoustic scattering with rectangular geometries, which have a common underlying mathematical structure. The geometries are: the infinite wedge, the waveguide with a barrier, and the semi-infinite plate of finite thickness. It turns out that these problems may be formulated as matrix Wiener–Hopf problems with the special property that their matrix kernels $\mathsf K$ may be formulated as $\mathsf K = \mathsf M^{-1} \mathsf J \mathsf M$, where $\mathsf J^2 = \mathsf I$, the identity matrix. This special property makes the problems amenable to factorisation which enables an exact solution to be derived, at least in theory. In practice, in two of the cases, we end up with an infinite system of equations which must be truncated to allow for practical computation of coefficients. However, these coefficients are rapidly convergent aided by the use of a novel technique termed the `corner singularity method', in which the integration contour of an integral is shifted upwards in the complex plane to pick up a contribution from the infinite 'tail'. This work has applications in industrial and marine acoustics, and bears promise of fruitful extension to elastodynamics and other areas of wave theory.
  • ItemControlled Access
    The Many Phases of the Surface Code: Coherent Errors and Many-Body Localisation
    Venn, Florian
    This thesis investigates the far-from-ground-state physics of the surface code, in particular its quantum error correction applications and formulations. We contribute to this field via two lines of research: we study the behaviour of the surface code under coherent errors, which create superpositions of excited states, and we probe topological many body localization (MBL) which protects topological order for all eigenstates. In the first strand, we develop an interpretation of the error correction threshold for coherent error rotations as a phase transition. For this, we first generalize a numerical method for the simulation of coherent errors in surface codes on square lattices to work with surface codes on general planar graphs. This method is based on a mapping to a free fermion model which allows calculating the expectation values using fermion linear optics. Using this method, we show that the connectivity of the graph can shift the error correcting performance between resilience against *X*- and *Z*-rotations. Building on this work, we further explore the relationship between coherent errors in surface codes and free fermion models. We develop a formalism to map the surface code under coherent errors to a complex Ising model and from there to a Majorana fermion scattering model. We analyze its conductivity and find that for rotations below the error correction threshold the resulting model is an insulator, and it becomes a metal above the threshold. By estimating the position of this phase transition, we obtain the achievable error correction threshold for coherent errors. The second line of research is focused on the disordered and perturbed toric code. We implement a recently proposed method that numerically approximates the local integrals of motion that are present in (topological) MBL phases using sets of stabilizers that are dressed by optimized quantum circuits. First, we apply this method to the disordered Kitaev chain as a benchmark. Then, we proceed by adapting it to the toric code. We show how it can be used to distinguish topological and trivial MBL and how it can be combined with exact diagonalization to obtain an approximate phase diagram.
  • ItemOpen Access
    On the Relationship between Canonical Quantum Gravity and the Holographic Principle
    Araujo Regado, Goncalo
    This thesis explores the connection between two approaches to the problem of quantum gravity. On the one hand, we have the canonical approach which imposes the gauge constraints on the physical states. This leads to the notoriously hard problem of solving the Wheeler-deWitt (WdW) equation. On the other hand, we have the holographic principle, which defines the gravitational path integral in terms of the partition function of a non-gravitational CFT living on the boundary, leading to the flourishing field of the AdS/CFT correspondence. The connection between the two becomes clear after a reformulation of the holographic principle in which the emergent dimension is time instead of space. For that we need to consider Euclidean field theories living on a slice of space. They are defined starting from the usual type of holographic CFTs followed by a special type of deformation called the $T^2$ deformation. Such partition functions solve the WdW equation, thus providing canonical quantum states of the bulk theory. The deformation flow is uniquely fixed by the bulk gauge constraints and it has several exotic properties. This formulation extends the AdS/CFT framework naturally to other quantum gravity scenarios. We explain the what, how and why of the $T^2$ deformation in quantum gravity by studying general solutions to the WdW equation. This leads naturally to an explicit map between field theory states living on the boundary of space and quantum gravity states living on the bulk of space. This is a manifestation of the holographic principle, hiding inside the WdW equation. We also propose a reconstruction of the boundary state from bulk data. We conjecture about an isomorphism between the quantum gravity and field theory Hilbert spaces. The dynamics of the boundary state with respect to boundary time is shown to induce a time evolution of the quantum gravity state. We discuss, at several points in the thesis, how the bulk theory manages to keep being unitary, despite the lack of unitarity of the deformed field theory. Along the way, we also propose a more general version of the holographic principle in the language of equating bulk and boundary path integrals. We discuss at length the application of this formalism to quantum cosmology. This requires us to consider complexified deformations. Crucially, we are forced to consider superpositions of field theory branches in order to describe the bulk. This leads to several discussions about the structure of quantum gravity and its hypothetical UV completion. In particular, we discuss the phenomenon of spontaneous CPT breaking for the UV completion of the $T^2$-deformed theory along its RG flow. The partition function is computed explicitly in minisuperspace, touching base with previously known solutions to the WdW equation applied to this restricted toy model. We then go on to conjecture that the choice of lapse contour in the gravitational path integral is intimately related to the superposition of field theory branches and, therefore, to the different UV completions for the holographic dual. All these features point in the direction of the long-standing conjecture that there is a unique quantum state of the universe.
  • ItemOpen Access
    Rare events and dynamics in non-equilibrium systems
    Kikuchi, Takaaki
    The matter of this thesis is divided in two parts, both of which are substantially different from the other, but nevertheless belong to disciplines that lie within the purview soft matter physics. In the first part, we study the infinite-dimensional probability space of stochastic differential equations. In particular, we study the transition path ensemble (TPE), the set of transition paths between meta-stable states of Ito diffusions. In the limit of vanishing diffusivity, the Freidlin-Wentzell action characterises the asymptotics of the path-probability distribution over the TPE. We develop spectral Ritz methods to efficiently find minimisers of this action, and to construct quasipotentials of steady-state distributions, and we test our algorithm on a number of benchmark systems. To study the TPE in the finite temperature regime, we develop an MCMC algorithm to sample the infinite-dimensional space of transition paths, which we call the *teleporter MCMC*. The algorithm was designed to efficiently sample the TPEs of Ito diffusions with multiple competing transition channels, avoiding the issue of slow-mixing common to MCMC schemes. We concluded this part of the thesis by applying our MCMC method to study the temperature-dependence of the TPE. Using two model systems, we show that the dominant transition channel does not in general coincide with the most probable path of the path distribution, even in a low-to-intermediate temperature regime. In the second part of this thesis we develop a general theory of the geometric mechanics of a broad class of microstructured continuum systems. Specifically, we consider systems with configuration spaces that are either Lie groups, or homogeneous spaces. We demonstrate that this theory, which we call a generalised geometric Cosserat theory (GGCT), can be seen as a unifying framework with which to study classical Cosserat systems, and numerous non-classical variations. As a paradigmatic example we first study the Cosserat rod model, we identify its configuration space as a curve in $SE(3)$, the Lie group of translations and rotations on Euclidean space, and use the Lie algebra-Lie group correspondence to relate its configuration to curves in the Lie algebra. Using the Euler-Poincaré theorem we then proceeded to formulate the dynamics of the Cosserat rod on the dual Lie algebra. The resulting kinodynamical - kinematic and dynamic - theory of the Cosserat rod is defined completely on the trivialisation of the tangent bundle of $SE(3)$, the Lie algebra $\mathfrak{se(3)}$. We then constructed the GGCT by extrapolating these above steps to systems with generalised configuration spaces. In the final chapter of this thesis, we constructed geometric numerical integrators designed to preserve the qualitative features of the system geometry.
  • ItemOpen Access
    Robustness of Fixed Points of Quantum Processes
    Salzmann, Robert
    The thesis combines two independent lines of research, both of which lie in the general area of the theory of robustness of fixed points (or invariant states) of quantum processes. In the first part of the thesis, we address the following question: Given a quantum channel and a quantum state which is almost a fixed point of the channel, can we find a new channel and a new state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? This question can be asked under many interesting constraints in which the original channel and state are assumed to have certain structures which the new channel and state are supposed to satisfy as well. We answer this question in the affirmative under fairly general assumptions on afore-mentioned structures through a compactness argument. We then concentrate on specific structures of states and channels and establish explicit bounds on the approximation errors between the original- and new states and channels respectively. We find a particularly desirable form of these approximation errors for a variety of interesting examples. These include the structure of general quantum states and general quantum channels, unitary channels, mixed unitary channels and unital channels, as well as the structure of classical states and classical channels. On the other hand, for the setup of bipartite quantum systems for which the considered channels are demanded to act locally, we are able to lower bound the possible approximation errors. Here, we show that these approximation errors necessarily scale in terms of the dimension of the quantum system in an undesirable manner. We apply our results to the robustness question of quantum Markov chains (QMC) and establish the following: For a tripartite quantum state we show the existence of a dimension-dependent upper bound on the distance to the set of QMCs, which decays as the conditional mutual information of the state vanishes. In the second part of the thesis we prove the so-called quantum Zeno- and strong damping limits for infinite-dimensional open quantum systems. In the former case, which we refer to as the quantum Zeno regime, the dynamics of the open quantum system is governed by a quantum dynamical semigroup, which is repeatedly and frequently interrupted by the action of a quantum operation. The quantum operation is considered to be mixing, in the sense that if applied multiple times it converges to its fixed point space. We then analyse the effective dynamics of the overall process in the limit of the application frequency of the quantum operation going to infinity. The strong damping regime can be considered as a continuous variant of the quantum Zeno regime. Here, the discrete and frequent action of the quantum operation is replaced by an additional term in the generator of the dynamical semigroup, whose individual dynamics is mixing, in the sense that it converges to its fixed point space in the infinite time limit. We analyse the overall dynamics in the limit of infinite interaction strength. All previous proofs of quantum Zeno limits in the literature relied on an assumption given by a certain spectral condition. We give a full characterisation of quantum operations which are mixing in the uniform topology under this assumption. Then, using a novel perturbation technique, we are able to go beyond this assumption and prove quantum Zeno- and strong damping limits in an unified way, if the mixing happens in the strong sense, i.e. pointwise for a given state. Here, we see that the effective processes converge to the fixed point spaces, on which they are governed by an effective quantum Zeno dynamics. The result is quantitative and gives a bound on the speed of convergence of the quantum Zeno- and strong damping limits, given a bound on the speed of convergence of the mixing process.
  • ItemOpen Access
    Human mobility and spatial models for infectious disease
    Tang, Maria; Tang, Maria Lan [0000-0002-9671-8302]
    Human mobility is an important determinant for the spatial spread of human infectious diseases such as influenza but obtaining human mobility datasets has historically been difficult. This thesis investigates two ways to represent human mobility in spatial metapopulation models for the spread of influenza in the US and UK – using gravity models with data-based distance metrics and using survey mobility data from the BBC Pandemic project and the 2011 UK census. Our metapopulation models describe the spread of influenza on a network of geographically segregated subpopulations that make up the whole population. Interactions between subpopulations are characterised by the human mobility proxies, while homogeneous mixing is assumed within subpopulations. The choice of subpopulations can therefore potentially have a large influence on the model output, and so this thesis also considers how this choice of spatial scale for the aggregation of the human mobility data and for the model can affect the epidemic dynamics produced. Chapter 2 investigates the use of data-based distance metrics in a gravity model framework fit to influenza spread in the US. Given that people do not move via straight lines, we consider driving distance by road and driving time as alternative distance metrics to great-circle distance. Gravity models are fit to outbreak onset dates in the US for the 2009 A/H1N1pdm influenza pandemic and the 2003/04 and 2007/08 influenza seasons, derived from influenza-like-illness medical claims timeseries at the scale of 3-digit ZIP codes (ZIPs). Driving distance and time are found to give better gravity model fits than great-circle distance to this data and simulations highlight spatial differences in the spread predicted by the different distance metrics. Chapter 3 explores the effect that spatial scale of the data and model has on the results in the previous chapter and considers two spatial scales in addition to ZIPs: sectional centre facilities (SCFs) and states. We compare the results from using different scales for obtaining outbreak onset dates from the influenza-like-illness timeseries, model fitting to the outbreak onset dates, and simulating from the model parameters. The better modelling performance of driving distance and driving time compared to great-circle distance persisted at the SCF level but not at the state level. Chapter 4 describes the England mobility data from the BBC Pandemic citizen science project that recorded location data of participants via a mobile phone app in 2017-2018. Compared to the most widely used open-source England human mobility data in the last decade, the 2011 census commuter workflow matrices, the BBC location data is more recent and records the movement of a wider range of people and trips but is relatively sparser. To compare the two datasets, we aggregate the BBC data into origin-destination matrices and fit competing destination models, an extension of the gravity model, to both BBC and census mobility data at three spatial scales: local authority districts (LADs), upper tier local authorities (UTLAs) and regions. Model preference was similar between datasets and scales, but parameter estimates differed. Chapter 5 uses the fitted mobility matrices in the previous chapter in a compartmental metapopulation model for influenza disease spread in England to compare simulated output from using the BBC and census mobility datasets. The resulting simulated epidemic dynamics are evaluated at the three scales (LADs, UTLAs, regions). Additionally, Chapter 6 presents a retrospective analysis of another source of survey data – for coughs, colds, and influenza-like illness in the University of Cambridge from 2007-2008. This self-reported data from university students and staff is one of the most detailed datasets of infectious respiratory disease in UK universities pre-COVID-19. Although a simple survey that comes with biases, it provides insights into risk factors for infectious disease in the relatively closed environment of a university and suggests ways in which future surveys could be carried out.
  • ItemOpen Access
    Exploring Non-Minimality in New Physics Beyond the Standard Model
    Banks, Hannah
    The need to extend the Standard Model of particle physics is now well established with a multitude of observations heralding the existence of new physics beyond the realms of our present understanding. A plethora of new theoretical possibilities have been proposed to this end, each with vastly different microphysical realisations and in turn, phenomenological signatures. The notion of minimality has traditionally been appealed to as a guiding force in the organisation of our experimental explorations of this space to date, with a handful of simple benchmark scenarios receiving the lion's share of attention. With all dedicated searches for new physics as-yet returning null results however, it is becoming increasingly apparent that a more thorough survey of the diverse landscape of prospective theoretical models is required. This thesis considers a number of different ways in which we might introduce complexity into our searches for new physics beyond the Standard Model in order to probe previously unchartered theoretical territory. We begin in the arena of flavour physics where we re-interpret LHC search data to place exclusion bounds on a specific extension of the Standard Model which, in order to address both the hierarchy of the fermion masses and anomalies observed in meson decay processes, is non-trivial in its flavour structure. The latter part of this thesis then focuses on new physics relating to the dark sector. We begin by developing an entirely general analysis framework with which to structure searches for scalar operator `fifth forces' that may arise between Standard Model particles due to the exchange of new light states. By encapsulating the phenomenology of an extremely broad range of theoretical possibilities in terms of a single real, positive-definite spectral density function, we demonstrate that this approach enables exotic scenarios which go beyond the simplest possibility of tree-level scalar exchange to be considered with ease. We also show how this prescription provides the scaffolding to probe speculative violations of quantum field theoretic principles such as unitarity, causality and locality. Continuing along the lines of generalising searches for new light physics, we next apply ourselves to the phenomenon of neutrino oscillations. Here, we introduce a new, flexible language in which a diverse range of new physics effects on neutrino propagation, such as the existence of additional light neutrino species, are described by a single spectral function. We further demonstrate that the relevant phenomenology of a host of complex theoretical models can be conveniently approximated by way of a simple mass spectrum which comprises three `broadened' states. By allowing for a model-independent analysis of neutrino oscillation data, we illustrate how this phenomenological ansatz enables deviations from the canonical three-neutrino scenario to be probed in a systematic and general fashion. We finally turn to a specific possible manifestation of complexity in the dark sector - namely the formation of exotic compact objects. Provided such structures form binary systems, they may generate unique, identifiable signals at near future gravitational wave observatories sensitive to sub-Hz frequencies. We show that studying the gravitational wave background generated by the mergers of such objects may not only provide an indication of their existence but offer a unique opportunity to probe their properties and in turn, the dark sector states from which they are composed.
  • ItemOpen Access
    Contact and coalescence of viscous drops
    Beaty, Edward; Beaty, Edward [0000-0001-6995-8645]
    When two fluid drops come into contact, surface tension quickly pulls the drops together into a single larger drop. This coalescence process is an example of a singular flow resulting from a topological transition, in this case between the disjoint and connected drops. In this thesis, we consider theoretically the flow in two viscous drops with inertia either side of this topological transition. The deformation of drops prior to contact sets the initial shape of the drops for the subsequent coalescence. We consider a mechanism for contact between nearby, effectively stationary drops: when such drops are in sufficiently close proximity, van der Waals attraction between the drops overcomes surface tension and deforms the surfaces into contact. We solve for the viscous dynamics of this deformation both with and without inertia and find in each case a self-similar solution. The self-similar surface evolution determines the initial surface profile, and therefore the strength of the singularity in surface curvature, for the subsequent coalescence. At sufficiently early times, coalescence is described geometrically by a small fluid bridge over which the touching drops are in contact. At the edge of the fluid bridge the surface is tightly curved. The corresponding surface tension drives an expansion of the fluid bridge and consequently coalescence. We solve for the dynamics of viscous coalescence for general initial surface profiles set by a contact process. The strength of the singularity in surface curvature at contact determines the initial rate of coalescence at leading order. For drops with both viscosity and inertia, the early-time coalescence dynamics transitions between several regimes that are determined by the relative strengths of viscosity and inertia on the different length scales of the problem. We identify regimes in which the momentum imparted on the fluid by surface tension is confined to a viscous wake over the fluid bridge. Entrainment into the wake alters the drop profile ahead of the fluid bridge and subsequently alters the rate of coalescence.
  • ItemOpen Access
    Advances in Reinforcement Learning for Decision Support
    Jarrett, Daniel
    On the level of decision support, most algorithmic problems encountered in machine learning are instances of pure prediction or pure automation tasks. This dissertation takes a holistic view of decision support, and begins by identifying four important problem classes that lie between the two extremes: exploration, mediation, interpretation, and generation---specifically with an eye towards the role of reinforcement learning in helping humans 'close the loop' in sequential decision-making. In particular, we focus on the problems of: exploring new environments without guidance, interpreting observed behavior from data, generating synthetic time series, and mediating between humans and machines. For each of these, we proffer novel mathematical formalisms, propose algorithmic solutions, and present empirical illustration of their utility. In the first instance, we refine our notion of curiosity-driven exploration to separate epistemic knowledge from aleatoric variation in hindsight, propose an algorithmic framework that yields a simple and scalable generalization of curiosity that is robust to all types of stochasticity, and demonstrate state-of-the-art results in a popular benchmark. In the second instance, we formalize a unifying perspective on inverse decision modeling that generalizes existing work on imitation learning and reward learning while opening up a broader class of research problems in behavior representation, and instantiate an example for learning interpretable representations of boundedly rational decision-making. In the third instance, we propose a probabilistic generative model of time-series data that optimizes a local transition policy by reinforcement from a global energy model learned by contrastive estimation, draw a rich analogy between synthetic generation and sequential imitation, and verify that it yields useful samples on real-world datasets. In the fourth instance, we formalize the sequential problem of online decision mediation with abstentive feedback, propose an effective solution that seeks to trade off immediate loss terms against future improvements in generalization error, and illustrate its efficacy relative to applicable benchmark algorithms on a variety of metrics. Like so, this dissertation contributes and advances a broader perspective on machine learning for augmenting decision-making processes.
  • ItemEmbargo
    Radiative Transitions in Charmonium
    Delaney, James
    In this thesis, we analyse radiative transitions in charmonium using Lattice QCD, to provide insight into the internal structure and photocoupling of charmonium states. Distillation is used to allow the efficient computation of a high number of correlation functions, leading to the construction of highly optimized operators ideal as an input for the calculation of matrix elements. We explore techniques which leverage data over many time slices to increase statistics as well as a robust method of determining multiple form factors with generic kinematics. From these matrix elements, form factors are determined over a wide range of *Q2* and extrapolations are performed to the zero virtuality point. Quantities computed are in reasonable agreement with other lattice studies and experimental values, where available. First determinations of radiative partial widths of exotic charmonia using dynamical quarks are calculated.
  • ItemOpen Access
    Inference and Entropy in Free-Surface Flows
    Young, Benjamin
    In spite of the ubiquity of free-surface flows across both nature and industry, theoretical and numerical modelling remains challenging. Continuum flows, whether Newtonian or Non-Newtonian, are governed by the Navier–Stokes equations, whilst the particle-particle interactions within a granular flow can be modelled by simple collision mechanics. However, the computational effort required to solve these equations can often render numerical modelling intractable as modellers must either resolve all scales of the velocity field or model every particle-particle interaction. This issue is compounded by the fact that in many cases, modellers are not interested in the fine detail of the flow, but instead wish to study the bulk features of the flow. These bulk features include: the development of a free-surface due to topographic changes or the spatial distribution of the coarse-grained stress field within a granular flow, to name the two examples explored in this thesis. A common approach to model bulk features, without resolving the smallest scale features, is to perform a filtering operation on the governing Navier–Stokes equations or equations of collision mechanics. Unfortunately, the resulting equations often rely on a heuristic closure model to capture the interaction between the sub-filter scale features and the bulk features. In this thesis we explore two methods for deriving closure laws for filtered models – information entropy/energy dissipation maximisation and rheological inference – and apply these to two free-surface flows. In our first problem, we theoretically and numerically examine the interaction between a Blasius boundary layer and free-surface within a shallow Newtonian fluid. Depth-averaging the Navier Stokes equations yields an infinite system of equations that describes how the shape-factors (depth-wise moments of the stream-wise velocity profile) of the flow evolve. We apply an entropy maximisation method that: (i) produces a first-order accurate closure model for our equations; (ii) predicts analytical, steady-state solutions to the free-surface flow and (iii) provides valuable, new insight into the relationship between the rheology and information entropy of a flow. In our second problem, we experimentally and numerically investigate the stress/deformation-rate relationship (granular rheology) in the statistically steady flow of a refractive-index-matched, granular suspension within a rotating drum. Coarse-graining the governing equations yields the Cauchy momentum equations, which describe the relationship between coarse-grained velocity, pressure and stress. We develop and apply an inference algorithm to estimate the latent coarse-grained stress and pressure fields from granular velocimetry data and examine the relationship between inferred stress and shear rate within the context of existing rheology models. We demonstrate that our inferred data follows many of the previously observed stress/shear-rate trends and discuss where our data deviates away from theory. Finally, we posit how our method could be used to develop and validate more accurate models of granular rheology.
  • ItemOpen Access
    Cauchy Slice Holography and the Semiclassical Approximation
    Shafi, Mohammed Rifath Khan
    We investigate a new approach to holography in asymptotically AdS spacetimes, in which time rather than space is the emergent dimension. By making a sufficiently large $T^2$-deformation of a Euclidean CFT, we define a holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More generally, for an arbitrary Hamiltonian constraint equation that closes, we show how to obtain it by an irrelevant deformation from a CFT with suitable anomalies.) The partition function of this theory defines a natural map between the bulk canonical quantum gravity theory Hilbert space, and the Hilbert space of the usual (undeformed) boundary CFT. We argue for the equivalence of the ADM and CFT Hamiltonians. We also explain how bulk unitarity emerges naturally, even though the boundary theory is not reflection-positive. This allows us to reformulate the holographic principle in the language of Wheeler-DeWitt canonical quantum gravity. Along the way, we outline a procedure for obtaining a bulk Hilbert space from the gravitational path integral with Dirichlet boundary conditions. Following previous conjectures, we postulate that this finite-cutoff gravitational path integral agrees with the $T^2$-deformed theory living on an arbitrary boundary manifold---at least near the semiclassical regime. However, the $T^2$-deformed theory may be easier to UV complete, in which case it would be natural to take it as the definition of nonperturbative quantum gravity. We then explore the semiclassical approximation to canonical quantum gravity and how a classical background emerges from the Wheeler-DeWitt (WDW) states. By employing the Wigner functional analysis, we derive the backreacted Einstein-Hamilton-Jacobi equation as an approximation to the WDW equation, along with the requisite validity conditions. We then apply this understanding to both AdS/CFT and dS/CFT correspondences, to explain how the bulk is encoded in the correlation functions of the $T^2$-deformed theory. We then explain an appropriate description for scenarios in which gravity behaves quantum mechanically in certain regions of spacetime and explain its relation to subregion holography. We derive the validity conditions for gravity to be semiclassical near any co-dimension 1 time-like surface and employ these conditions to explore the black hole information paradox. Our analysis suggests that for evaporating black holes, there might be a violation of semiclassical gravity in the near-horizon region close to the Page time, although this is contingent upon certain assumptions. This also provides insights into the fate of information trapped within evaporating black holes. We then explore this issue from the perspectives of both external and infalling observers. We then explain how to employ this new approach to study the retrieval of information from evaporating black holes, presenting a comprehensive approach to tackle this complex issue in quantum gravity.
  • ItemOpen Access
    Neural Network Training and Inversion with a Bregman Learning Framework
    Wang, Xiaoyu
    Deep Neural Networks (DNNs) are powerful computing systems that have revolutionised a wide range of research domains and have achieved remarkable success in various realworld applications over the past decade. Despite their significant recent advancements, training DNNs still remains a challenging task due to the non-convex and (potentially) non-smooth nature of the objective function. Back-propagation in combination with gradient-based minimisation approaches has been the predominant strategy for training DNNs for decades. Yet the popular error backpropagation algorithm is susceptible to potential drawbacks and limitations, for example its non-parallelisablity and biological implausibility, and vanishing or exploding gradients issues, etc. Inverting DNNs to infer likely inputs of the system from given outputs, is the other side of the same coin. Early ideas of DNNs inversion trace back to the 1990s, but research interests in more generic network inversion problems have been rekindled and primarily driven due to the rapid advancements in generative modelling in recent years. While several approaches for the inversion of DNNs have been proposed, the stability of the inversion is an often neglected crucial aspect. The neural network inversion problem is ill-posed as the solution does not depend continuously on the input datum hence can be highly sensitive to perturbations. The core theme of this thesis is at the training of DNNs. Built up on distributed optimisation approaches, this work contributes to both the learning problems and the inversion problems of DNNs. In particular, we propose a lifted Bregman learning framework that goes beyond the classical back-propagation approach, and aims to address unresolved and overlooked issues in training and the inversion of DNNs. More specifically, we propose a family of loss (penalty) functions that are based on a tailored Bregman distance. We provide detailed mathematical analysis on the derived Bregman learning framework and propose a whole range of deterministic and stochastic optimisation strategies to enable solving the learning problem. Bringing techniques and tools from Inverse Problems and Regularisation Theory, we provide theoretical guarantees as well as computational optimisation strategies for the stable, model-based inversion of neural networks.
  • ItemOpen Access
    Identifying Optimal Mixing Strategies and Structures through Sobolev Norms
    Heffernan, Conor
    The mixing of fluids is a hugely important physical phenomenon for understanding processes such as, but not limited to, climate change and improving industrial efficiency. A particular question of interest is how one may optimise the underlying mixing underpinning these processes. Recent advances have shown that optimisation of certain mixing measures yield a higher quality mixing strategy while using less energy than those triggering turbulent flows. These multiscale measures, equivalent to Sobolev norms of negative index, have been employed successfully for this purpose. This thesis investigates these norms and their use in nonlinear optimal mixing problems. In Chapter 3, we compare the optimisation of the classical variance measure of mixing with that of the multiscale norms. These measures are found to be computationally efficient as they identify optimal perturbations which continue mixing of the passive scalar very well compared with variance-based strategies that were optimised over a longer horizon. These multiscale measures also lead to the identification of vertical coherent structures which correspond to effective mixing within the system. In Chapter 4, we study the effect of the underlying geometry of the passive scalar on mixing optimisation. This is investigated through two different problems of interest. In the first problem, we perform a comparative study between a rectilinear stripe geometry and a circular disc geometry. In this study, we find that the stirring of the scalar varies considerably depending on which geometry is prescribed. In the second problem, we restrict our attention to the disc geometry but add more discs to the domain to investigate how the mixing dynamics scale up. We find that while the qualitative behaviour of the solutions remains stable, but through non-local interactions the vortex structure exhibits some variability as the Péclet number is increased. Finally, in Chapter 5, we study a mixing problem with a prescribed stream function chosen according to observations from the studies performed in this thesis. We find that a closed form solution emerges for a special choice of a parameter of particular interest in the literature. We then derive through perturbation methods a system which describes mixing of a passive scalar based on free parameters which set the shape of the initial vortices.
  • ItemOpen Access
    Dynamically induced uncertainty in stratified turbulent mixing models
    Lewin, Samuel; Lewin, Samuel [0000-0002-2602-4751]
    The interaction of turbulence with buoyancy effects due to gravity is of central importance in a wide range of environmental and industrial flows. On both global and regional scales in the ocean, it is necessary to model small-scale turbulent processes and the way in which they enhance mixing in order to form an accurate picture of the vertical transport of heat, nutrients and anthropogenic pollutants. This applies to parameterisation schemes in coarsely resolved large-scale numerical simulations, as well as the processing and interpretation of spatio-temporally sparse observational field measurements. Most studies of stratified turbulence modelling support the conclusion that models of mixing should depend on parameters of the flow derived from diagnosable turbulent statistics in addition to (or in some cases, instead of) those associated with the ambient background state and the physical properties of the fluid. However, there is no consensus on precisely what these parameters should be, how many are necessary, and what functional form these dependencies should take. Using data from a variety of idealised direct numerical simulations (DNS), and with careful consideration of the underlying fluid dynamics, this thesis serves to highlight the compound importance of the many varieties of uncertainty that can arise during the different stages of model application, from selecting and pre-processing suitable inputs to then transforming these inputs into estimates for relevant mixing properties. For the purposes of illustration, we frame our discussion in terms of two classical and commonly used models for calculating local values of turbulence-enhanced vertical diffusivity in the ocean, described in chapter 1. We first investigate mixing processes driven by Kelvin-Helmholtz instability, which acts to overturn density interfaces in a stratified, vertically sheared flow. Chapters 2 and 3 illustrate two distinct ways in which geophysically realistic departures from an idealised model background flow contribute towards generating uncertainty in various important mixing diagnostics. In particular, chapter 2 tackles issues surrounding the `suitable' definition of a local mixing region and associated background density profile in the presence of inverted local density gradients, as well as considering the relevance of different approximate overturning length scales for estimating the mixing efficiency of the flow. Chapter 3 utilises a time-dependent forcing scheme to highlight the modification of mixing behaviour by a competition between shear and convective processes driving turbulence. Chapters 4 and 5 are concerned with turbulence in a strongly stratified environment that exhibits the formation of a characteristic vertically ‘layered’ structure in the velocity and density fields. The results are motivated by considering the inferences that can be made from typical observational datasets consisting of a limited number of measured velocity and density gradients in the flow. Chapter 4 looks at the transient pathways to strongly stratified turbulence in a freely evolving purely horizontal shear flow, discussing the potential links to parameterisations based on individual overturning events. Looking to the future, chapter 5 considers the emerging relevance of machine learning tools for problems in stratified turbulence. We introduce a novel data-driven method to investigate and improve upon some of the existing uncertainties due to small-scale anisotropy that arise when making estimates of energy dissipation rates in flows with low levels of turbulence intensity, as measured by the buoyancy Reynolds number *Re*b.
  • ItemOpen Access
    Mathematical Studies on the Asymptotic Behaviour of Gravitational Radiation in General Relativity
    Kehrberger, Leonhard; Kehrberger, Leonhard [0000-0003-4485-8351]
    This thesis contains various constructions that attempt to answer and clarify the long-standing question of how to model the asymptotic behaviour of gravitational radiation in astrophysical processes within the theory of general relativity. It is argued that the correct mathematical setup to address this question is the *scattering problem* (as opposed to the Cauchy problem), where data are posed in the infinite past. The choice for these data is informed by making certain heuristic connections to the Newtonian theory (Post-Newtonian theory). It is then proved that there exists a unique solution that attains these data in the limit, and the asymptotic properties of this solution are analysed. In this way, it is found that the constructed solutions, which have clear physical interpretations afforded by the connection to the Newtonian theory, neither admit a smooth past null infinity nor a smooth future null infinity (they violate peeling near both), and moreover decay slower towards spatial infinity than typically assumed in the literature. In other words, our constructions show that many of the commonly-used concepts to model isolated physical systems are, in fact, not suitable for this purpose. Furthermore, it is shown that the non-smoothness of future null infinity, is, in a certain sense, exactly conserved and even determines the asymptotic behaviour of gravitational radiation at late times. Our constructions either concern the nonlinear Einstein-Scalar field equations under the assumption of spherical symmetry, or the system of linearised gravity around Schwarzschild with no symmetry assumptions, but the methods employed are capable of generalisation to the nonlinear Einstein vacuum equations.
  • ItemEmbargo
    Nonlinear Dynamics of Warped Astrophysical Discs
    Fairbairn, Callum William; Fairbairn, Callum [0000-0002-9285-2184]
    The classical perspective of astrophysical discs models them as being flat, circular and coplanar. However, a wealth of observational evidence has highlighted the ubiquity of warped geometries in a variety of contexts. This has motivated theoretical efforts to explain the origins of such distortions and understand their ensuing dynamical evolution. Despite marked progress, a complete theory remains elusive, particularly in the observationally relevant regime of large warp amplitudes wherein nonlinear effects and hydrodynamic instabilities significantly modify the warp evolution -- both of which we aim to elucidate in this work. In the first part of this thesis, we construct a novel analytical model which proffers a flexible hydrodynamic framework for studying nonlinear warps. Crucially, this model allows us to investigate inviscid, Keplerian discs for which resonances between the orbital and epicyclic frequencies demand a separate treatment compared with previous theoretical frameworks. We solve the ideal, compressible fluid equations for a non-selfgravitating elliptical cylinder within a local shearing sheet. By restricting attention to flow fields which are a linear function of the coordinates, we capture the lowest order global motions (tilting, shearing and breathing modes) and reduce the dynamics to a set of coupled ordinary differential equations. The connection between tilting tori and warped discs is demonstrated, before the linear modes are confirmed in numerical grid based simulations. In the second part of this thesis we exploit this model to uncover two distinct nonlinear regimes as the warp amplitude is increased. Initially, we find a smooth modulation theory that describes warp evolution in terms of the averaged Lagrangian of the oscillatory vertical motions of the disc. Upon the warp amplitude exceeding a critical value, which scales as the square root of the aspect ratio of our ring, the disc enters into a bouncing regime with extreme vertical compressions twice per orbit. We develop an impulsive theory that predicts special retrograde and prograde precessing warped solutions. Such solutions emphasize the essential activation of nonlinear vertical oscillations within the disc and may have important implications for energy and warp dissipation. In the third part of this thesis, we investigate the growth, saturation and feedback of a parametric instability, which feeds off the free-energy associated with the warped geometry. Using a Lagrangian-Godunov code, we perform simulations of our tilting ring and identify several locally growing modes, as predicted by a three-mode coupling analysis of inertial waves. After finding decent agreement with the theoretical growth rates, we proceed to understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wave-like turbulence. The associated Reynolds stresses can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Finally, in the fourth part of this thesis we consider a simplified model for warps in young protostellar discs, where small dust grains are coupled to the gas. By using a two-fluid approach we find the internal shearing flows associated with the warp and then perform a stability analysis in the terminal velocity approximation. This reveals the suppression of parametric growth by the dusty damping of inertial waves.
  • ItemOpen Access
    The Cosmological Implications of Unitarity
    Goodhew, Harry
    The unitarity of time evolution is a vitally important constituent of our descriptions of fundamental interactions via quantum field theory. The implications of unitarity for scattering amplitudes are well understood, for example through the optical theorem and cutting rules. In contrast, the implications for interactions in curved spacetimes are only now beginning to be investigated. Understanding such curved spacetimes is vital to deepening our knowledge of the early universe where, within the leading paradigm, we expect an approximately de Sitter expansion. In this thesis we show that unitarity implies a set of relations among the coefficients of the wavefunction of the universe in such an expanding epoch, which we name the cosmological optical theorem. We also show that this result can equivalently be understood as a single cut rule that relates diagrams at different orders in perturbation theory. The validity of this relationship is explored within the cosmological context, and we present several explicit checks of this relation. Finally, we demonstrate the power of this result by using it to derive the graviton four-point wavefunction coefficient. This requires some additional recent developments in the cosmological bootstrap program exploiting locality and the relationship to flat space amplitudes which are reviewed.
  • ItemOpen Access
    On the evolution & equilibration of submesoscale fronts
    Wienkers, Aaron
    Submesoscale fronts with large horizontal density gradients and O(1) Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability — a form of convective–inertial instability which occurs when the potential vorticity is of opposite sign to the Coriolis parameter. Symmetric instability is characterised by growing slantwise convection cells nearly aligned with isopycnals and which encourage vertical transport of important biogeochemical tracers in addition to geostrophic momentum. This momentum transport destabilises the balanced thermal wind and can prompt geostrophic adjustment and re-stratification which often leaves remnant inertial oscillations. This thesis sets out to model the impacts of symmetric instability on the structure, evolution, and equilibration of the broad range of submesoscale fronts. We develop a theory for the linear growth and weakly-nonlinear saturation of symmetric instability in the Eady model. This idealised front configuration is motivated by nearly-uniform symmetrically-unstable frontal zones observed during the SUNRISE field campaign in the Gulf of Mexico. Both the fraction of the balanced thermal wind mixed down by symmetric instability and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of symmetric instability we call 'slantwise inertial instability' by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in 'slantwise convection.' Using a set of direct numerical simulations, we further investigate the non-linear consequences of these flavours of symmetric instability together with the effect of lateral constraints imposed on the growing modes within finite-width fronts by varying the balanced Rossby number. While weak fronts develop narrow frontlets and excite small-amplitude vertically-sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. Furthermore, fronts with a super-critical balanced Rossby number equilibrate to a nearly self-similar profile dependent only on the deformation radius, but for small enough Rossby number, self-similar frontlets form and sub-divide the frontal region. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state. These emergent front properties are incorporated into a scaling model framework to describe the ultimate state of the equilibrated front, and broader implications of these results are discussed in the context of current parameterisations of symmetric instability affecting the upper ocean mixed layer and ultimately climate & earth system models.