School of Physics - Theses
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ItemNon-Equilibrium Processes in Neutron Stars and Ultracold Gases( 2023-06)From the booms and busts of the economy to the schooling of fish, non-equilibrium phenomena are ubiquitous and appear at all scales. However, non-equilibrium systems have proven infamously difficult to model and understand. In this thesis we present two different of non-equilibrium systems, one classical and one quantum mechanical, and thoroughly investigate their behaviour: (i) the repeated localised mechanical failure of the crust of a spinning down neutron star, and (ii) the dynamics of quenched few-body quantum systems. As an isolated neutron star spins down the centrifugal force weakens but the gravitational force doesn't change. This results in the crust changing shape and accruing mechanical strain to the point of failure. Mechanical failure locally deforms the crust and dissipates and redistributes strain. This can result in avalanches of further failures as one region of the crust failing may prompt a neighbouring region to fail. The evolving crust is a classical far-from-equilibrium system capable of avalanche behaviour like the classic sandpile model. The statistics of crustal failure events are of much interest due to their suggested relevance to transient phenomenon such as glitches or fast radio bursts. We present a cellular automaton designed to describe the evolution of the crust over spin down and the effects of local failure. This automaton describes when and where crustal failures occur and how large they are. Additionally this automaton describes the failure-induced change in the shape of the crust. Using this automaton we find that the star needs to be born spinning over \approx 750 Hz to accumulate sufficient strain to fail at all, that the waiting-times between subsequent events are distributed as a power-law spanning seven orders of magnitude, and that the ellipticities of isolated neutron stars are in the range 10^{-13} to 10^{-12}, among many other results. It has been suggested that the mechanical failure of the crust is the cause (or result) of a variety of transient phenomena such as glitches or gamma ray bursts. This model provides predictions of the statistical behaviour of crustal failure which can be compared to the observed behaviour of these transients. Additionally, the model describes the shape of the crust and the rotational frequency at all times which allows for the wave strain of emitted gravitational waves to be calculated with implications for searches for continuous gravitational wave sources. Cold quantum gases have attracted a great deal of experimental and theoretical interest thanks to the high degree of experimental control possible over them which makes them excellent testing grounds of quantum theory. Additionally, they are excellent tools for the study of quantum thermalisation. We consider a few interacting particles initially in some equilibrium state and suddenly change (quench) the interaction strength which kicks the system away from equilibrium. Specifically, we consider systems of two and three bodies of arbitrary mass and various particle symmetries interacting via a contact interaction in an isotropic three-dimensional harmonic trap. We take particular interest in quenching between the weakly and strongly interacting regimes and the following far-from-equilibrium post-quench evolution. We describe the non-equilibrium post-quench evolution of the system by analytically and semi-analytically calculating two observables: the Ramsey signal and the particle separation. We are able to calculate these quantities for the two-body system with arbitrary particle masses for any quench in interaction strength. Additionally, we extend these calculations to three-body systems of two identical fermions and a distinct particle or three identical bosons where the quench is between the strongly and weakly interacting regimes. In the two-body case we find when quenching from weak to strong interactions the particle separation oscillates periodically between \approx0.85a_{\mu} and \approx1.15a_{\mu}, where a_{\mu} is the simple harmonic oscillator length-scale. For the same quench in the three-body case the particle separation varies depending on the specifics of the system. For the fermionic case the particle separation oscillates periodically, peaking at \approx 2.18a_{\mu} with the mass ratio of the two species determining the minimum separation. For the bosonic case the oscillation is aperiodic. Both the maximum and minimum particle separation are determined by a quantity called the three-body parameter, but particle separation generally oscillates between \approx a_{\mu} and \approx 2a_{\mu}. However, in all cases when quenching from strong to weak interactions the calculations of the particle separation do not converge. This divergence is present whatever the initial state, mass ratio, particle symmetry, etc. and is present only for this particular quench from strong to weak interactions. We investigate possible sources of this divergence and future avenues of research into its causes. Finally, we note that these theoretical predictions of Ramsey signal and particle separation are experimentally testable with current techniques.
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ItemCollective superfluid vortex dynamics and pulsar glitches( 2011)Pulsar glitches offer a way of studying the dynamics of cold, ultradense matter in systems of stellar dimensions, under extremes of density, temperature and magnetisation unattainable on Earth. This thesis aims to build a robust model of pulsar glitches, based on the superfluid vortex unpinning paradigm, which relates the physical parameters of the pulsar interior to the observed distribution of glitch sizes and waiting times (power laws and exponentials respectively). Our modelling efforts draw together knowledge about superfluid vortex dynamics and pinning, garnered from condensed matter and nuclear physics, the observational facts gathered by pulsar astronomers, and the theoretical framework of non-equilibrium stochastic systems, such as those exhibiting self-organised criticality. In each case, we emphasise the necessity of collective mechanisms in triggering avalanche-like vortex unpinning events. We begin by studying the dynamics of superfluid vortices from first principles, using numerical solutions of the Gross-Pitaevskii equation (GPE). We solve the GPE in the presence of a lattice of pinning sites, in a container that is decelerated at a constant rate, mimicking the electromagnetic spin-down torque on a pulsar. The superfluid spins down spasmodically, as vortices unpin and hop between pinning sites when the Magnus force, due to the lag between the superfluid and vortex line velocities, exceeds a threshold. Torque feedback between the superfluid and its container regulates the lag between the superfluid and crust, resulting in abrupt increases in the container angular velocity. We study how the statistics of the sizes and waiting times between spin-up events change with the mean and dispersion of pinning strengths, the electromagnetic spin-down torque, the relative number of vortices compared to pinning sites, and the ratio of the crust and superfluid moment of inertia - all parameters of interest in neutron stars. We find that mean glitch size increases with mean pinning strength and the ratio of the moments of inertia. It is independent of the relative number of pinning sites and vortices, suggesting that vortices move a characteristic distance before repinning, rather than repinning at the next available site. The mean waiting time decreases with the number of pinning sites and vortices, the ratio of the moments of inertia and the spin-down torque, and it increases with the width of the pinning strength distribution. In order to explain the broad range of observed glitch sizes using the vortex unpinning paradigm, a collective unpinning mechanism is required. Using numerical solutions of the GPE, we study how the unpinning of one vortex can cause other vortices to unpin. We identify two knock-on triggers: acoustic pulses emitted as a vortex repins, and the increased repulsive force between vortices locally, when an unpinned vortex approaches its nearest neighbours. In the second half of the thesis, we construct a suite of three large-scale stochastic models of glitches. We are inspired to prosecute this program by similarities between the statistics of archetypal self-organised critical systems, such as earthquakes and sand piles, and those of pulsar glitches. The essential features of the vortex dynamics observed in the GPE simulations are abstracted and condensed into a set of iterative rules that form the basis of automata and analytic glitch models. A cellular automaton model, in which vortices interact with nearest neighbours via the Magnus force, reveals that when all pinning sites are of the same strength, large-scale inhomogeneities in the pinned vortex distribution are necessary to produce a broad range of glitch sizes. In this case, glitch sizes and durations are power-law-distributed, and waiting times obey an exponential distribution. We find no evidence of history-dependent glitch sizes or aftershocks. A coherent noise model, based on a similar model developed to study atom hopping in glasses, in which pinning strength varies from site to site, but the pinned vortex distribution is assumed to be spatially homogeneous, exhibits power-law-distributed glitch sizes. Exponential waiting times are put in by hand, by assuming that the stress released in a glitch accumulates over exponentially-distributed time intervals. A wide range of pinning strengths is needed to find agreement with radio timing data. Mean pinning strength is found to decrease with increasing characteristic pulsar age. Finally, we construct a statistical model that tracks the vortex unpinning rate as a function of the stochastically fluctuating global lag between the superfluid and container. Monte-Carlo simulations and a jump-diffusion master equation reveal that a knock-on mechanism that is finely tuned with respect to the pinning strength, is essential to producing a broad range of glitch sizes. Estimates of the power dissipated in acoustic waves during repinning, and of the strength of the proximity effect, do not meet the fine-tuning criteria. We propose to extend this promising model to include nearest-neighbour interactions in the future, in the hope that this may lessen the need for fine tuning. The non-axisymmetric rearrangement of the superfluid velocity field during a vortex-avalanche-driven glitch is a source of gravitational radiation. We calculate the gravitational wave strain using the characteristic vortex motion observed in the GPE simulations. We set an upper bound on the wave strain of h ~ 10-23 for a glitch resulting from an unpinning avalanche of the maximum observed size. We also estimate the contribution to the stochastic gravitational wave background from the superposition of many glitches from a Galactic neutron star population. We place an upper bound on the signal-to-noise ratio of the background of ~ 10-5 for the Advanced LIGO (Laser Interferometer Gravitational-wave Observatory) detector. Detection of a gravitational wave signal from glitches can teach us about the physics of matter at nuclear densities, from the equation of state to transport coefficients like viscosity.