Department of Applied Mathematics and Theoretical Physics (DAMTP)Carries out research of world-class excellence in a broad range of subjects across applied mathematics and theoretical physicshttps://www.repository.cam.ac.uk/handle/1810/2058712024-03-01T00:55:31Z2024-03-01T00:55:31Z9531Chaos in models of double convectionRucklidge, Alastair Michaelhttps://www.repository.cam.ac.uk/handle/1810/3649702024-02-24T01:44:14Zdc.title: Chaos in models of double convection
dc.contributor.author: Rucklidge, Alastair Michael
dc.description.abstract: 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.
Research data supporting "Data-driven classification of sheared stratified turbulence from experimental shadowgraphs"Lefauve, AdrienCouchman, Mileshttps://www.repository.cam.ac.uk/handle/1810/3645482024-02-15T01:44:39Zdc.title: Research data supporting "Data-driven classification of sheared stratified turbulence from experimental shadowgraphs"
dc.contributor.author: Lefauve, Adrien; Couchman, Miles
dc.description: For a more detailed description, refer to READ_ME.pdf
This dataset corresponds to an automated analysis of stratified turbulent flow visualisations in an inclined duct laboratory experiment. The goal is to reduce the complexity of 113 long shadowgraph movies previously shared on another repository doi.org/10.17863/CAM.104471. Our analysis takes each of the 50,155 frames and automatically extracts statistics corresponding to the morphology of density interfaces embeddd within the flow, before clustering these statistics in a low-dimensional space. The resulting clusters are physically interpretable and allow us to classify the flow instantaneously into a distinct type of turbulence.
The main dataset is the Matlab array pca_and_clustering_data.mat, providing the interface morphology 10-D vector and the ensuing clustering of each of the 50,155 frames processed in the associated article (see the methodology and variable names in Section III of the article).
In addition, we provide:
- the archive movies.zip contains six .mp4 movies showing the temporal evolution of all experiments at theta=1, 2, 3, 4, 5 and 6 degrees (including the principal components and clusters). Note that clusters initially named A, B, C, D, E in the movies correspond to L, B, O, G, U respectively in the paper.
- metadata_spreadsheet.xlsx shows all the experimental parameters and information about temporal spacing between frames.
- clustering_code.zip contains the Matlab code used to cluster the data (including the implementation of the OPTICS algorithm).
On the Factorisation of Matrix Wiener–Hopf Kernels Arising From Acoustic Scattering ProblemsAitken, Mungohttps://www.repository.cam.ac.uk/handle/1810/3641762024-02-09T01:42:25Zdc.title: On the Factorisation of Matrix Wiener–Hopf Kernels Arising From Acoustic Scattering Problems
dc.contributor.author: Aitken, Mungo
dc.description.abstract: 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.
The Many Phases of the Surface Code: Coherent Errors and Many-Body LocalisationVenn, Florianhttps://www.repository.cam.ac.uk/handle/1810/3634622024-01-19T01:41:25Zdc.title: The Many Phases of the Surface Code: Coherent Errors and Many-Body Localisation
dc.contributor.author: Venn, Florian
dc.description.abstract: 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.
On the Relationship between Canonical Quantum Gravity and the Holographic PrincipleAraujo Regado, Goncalohttps://www.repository.cam.ac.uk/handle/1810/3634592024-01-19T01:42:13Zdc.title: On the Relationship between Canonical Quantum Gravity and the Holographic Principle
dc.contributor.author: Araujo Regado, Goncalo
dc.description.abstract: 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.
Rare events and dynamics in non-equilibrium systemsKikuchi, Takaakihttps://www.repository.cam.ac.uk/handle/1810/3631972024-01-11T02:10:25Zdc.title: Rare events and dynamics in non-equilibrium systems
dc.contributor.author: Kikuchi, Takaaki
dc.description.abstract: 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.
Robustness of Fixed Points of Quantum ProcessesSalzmann, Roberthttps://www.repository.cam.ac.uk/handle/1810/3629542024-01-05T01:42:52Zdc.title: Robustness of Fixed Points of Quantum Processes
dc.contributor.author: Salzmann, Robert
dc.description.abstract: 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.
Human mobility and spatial models for infectious diseaseTang, Mariahttps://www.repository.cam.ac.uk/handle/1810/3628102024-01-03T01:42:56Zdc.title: Human mobility and spatial models for infectious disease
dc.contributor.author: Tang, Maria
dc.description.abstract: 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.
Exploring Non-Minimality in New Physics Beyond the Standard ModelBanks, Hannahhttps://www.repository.cam.ac.uk/handle/1810/3626372023-12-22T15:02:47Zdc.title: Exploring Non-Minimality in New Physics Beyond the Standard Model
dc.contributor.author: Banks, Hannah
dc.description.abstract: 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.
Shadowgraph visualisations of salt-stratified turbulence obtained in a stratified inclined duct (SID) laboratory experimentJiang, XianyangKong, GaopanLefauve, Adrienhttps://www.repository.cam.ac.uk/handle/1810/3623472023-12-16T04:29:40Zdc.title: Shadowgraph visualisations of salt-stratified turbulence obtained in a stratified inclined duct (SID) laboratory experiment
dc.contributor.author: Jiang, Xianyang; Kong, Gaopan; Lefauve, Adrien
dc.description: A collection of 113 experimental datasets corresponding to flow visualisations obtained in the G. K. Batchelor Laboratory (Department of Applied Mathematics and Theoretical Physics, University of Cambridge). The experiments were setup to create stratified turbulence generated via an exchange flow through a long tilted duct (L=2000, H=50, W=100 mm) connecting two large reservoirs (400 litres each) containing saltwater (NaCl) at different densities. The movies consist of shadowgraph visualisations, where parallel light was shone through the flow and projected onto a semi-transparent screen, where it was recorded by a video camera. The movies show the evolution of density (salinity) contrasts in the flow, including the formation and break up of density interfaces, as well as scouring and overturning behaviours. Each of the 113 movies corresponds to a distinct experiment, collectively covering a two-dimensional parameter space (Reynolds number and tilt angle) where various turbulent behaviours take place. This allows the study of a range of turbulent mixing processes responsible for transporting mass and momentum in stratified fluid systems like the oceans.
Please refer to the READ_ME file for detailed information on the folder contents and organisation.