In Memoriam, Solomon Marcus

This book commemorates Solomon Marcus’s fifth death anniversary with a selection of articles in mathematics, theoretical computer science, and physics written by authors who work in Marcus’s research fields, some of whom have been influenced by his results and/or have collaborated with him

Between order and disorder: ultracold atoms in a quasicrystalline optical lattice

Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as self-similarity, the inheritance of topological properties from higher dimensions, and the presence of non-trivial structure on all lengthscales. However, quasicrystalline materials are notoriously hard to synthesise, as defects and impurities may greatly alter their final microscopic composition. The field of quantum simulation with ultracold atoms offers a solution to this problem, namely the use of optical lattices – standing waves of light. Optical lattices are free of impurities and therefore ideally suited to study quasicrystals, enabling unprecedented access to observables that are unattainable in condensed matter systems. This study presents the first experimental realisation of a two-dimensional quasicrystalline potential for ultracold atoms, based on an eightfold symmetric optical lattice. Features pertaining to both ordered and disordered phases are observed, from sharp diffraction peaks in the matter-wave interference pattern, to a disorder-induced localised phase emerging at a critical lattice depth V_loc ~ 1.78 E_rec. The localised phase seems to be resilient against moderate interactions, which would make this the first experimental realisation of a 2D Bose glass.EPSR

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

Complex dynamics of solid-fluid systems

The focus of this thesis was the investigation of the complex dynamics of solid-fluid systems. These systems are of great industrial importance, such as in methane clathrate formation in sub-sea pipelines. As well as being crucial to furthering our understanding of various natural phenomena, such as the rate of rain droplet formation in clouds. We began by considering the problem of the orbits tracked by ellipsoids immersed in viscous and inviscid environments. This investigation was carried out by a combination of analytical and numerical techniques: direct numerical simulations of resolved full-coupled solid-fluid systems, analysis the Kirchhoff-Clebsch equations for the case of inviscid flows, and characterising dynamics through advanced techniques such as recurrence quantification analysis. We demonstrate that the ellipsoid tracks a chaotic orbit not only in an inviscid environment but also when submerged in a viscous fluid, under specific conditions. Under inviscid environments, an ellipsoid subject to arbitrary initial conditions of linear and angular momentum demonstrates chaotic orbits when all the three axes of the ellipsoid are unequal, in agreement with the Kozlov and Onishchenko’s theorem of non-integrability of Kirchhoff’s equations and also with Aref and Jones’s potential flow solution. We then extended our methodology to understand the dynamics of a single ellipsoid tumbling in a viscous environment with the presence of both passive and viscosity coupled tracers in addition to the chaotic dynamics predicted by the Kirchhoff-Clebsch equations. Our results show that the bodies move along from viscosity gradients towards minima of the viscous stress. These bodies might become trapped in unstable minima. However, more work is needed to understand the long-term mixing of viscosity coupled tracers. Our direct numerical solver was also extended to include contact models for solid-solid interactions in the simulation domain. The validation of the contact models was presented. Finally, we expand, the theoretical framework of the Kirchhoff-Clebsch equations to account for the presence of multiple bodies. This extension was done by using Hamiltonian mechanics to extend the derivation proposed by Lamb. We present our preliminary result of simulating two solids systems using the extended Kirchhoff-Clebsch equations. The rel- ative orientations of the two solids were found to regularly switch from being correlated to anti-correlated in an otherwise chaotic system. Further work is required to understand the mechanism behind this behaviour

Cooperative surmounting of bottlenecks

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated high-dimensional phase space are the source for the directed long-range transport

Methods for Analysis of Nonlinear Thermoacoustic Systems

This thesis examines the nonlinear behaviour of thermoacoustic systems by using approaches from the field of nonlinear dynamics. The underlying behaviour of a nonlinear system is determined by two things: first, by the type and form of the attractors in phase space, and second, by the mechanism that the system transitions from one attractor to another. For a thermoacoustic system, both of these things must be understood in order to define a safe operating region in parameter space, where no high-amplitude oscillations exist. Triggering in thermoacoustics is examined in a simple model of a horizontal Rijke tube. A triggering mechanism is presented whereby the system transitions from a stable fixed point to a stable limit cycle, via an unstable limit cycle. The practical stability of the Rijke tube was investigated when the system is forced by stochastic noise. Low levels of noise result in triggering much before the linear stability limit. Stochastic stability maps are introduced to visualise the practical stability of a thermoacoustic system. The triggering mechanism and stochastic dependence of the Rijke tube match extremely well with results from an experimental combustor. The most common attractors in thermoacoustic systems are fixed points and limit cycles. In order to define the nonlinear behaviour of a thermoacoustic system, it is therefore important to find the regions of parameter space where limit cycles exist. Two methods of finding limit cycles in large thermoacoustic sytems are presented: matrix-free continuation methods and gradient methods. Continuation methods find limit cycles numerically in the time domain, with no additional assumptions other than those used to form the governing equations. Once the limit cycles are found, these continuation methods track them as the operating condition of the system changes. Most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory. In the literature, there are therefore only a few applications of continuation methods to thermoacoustics, all with low-order models. Matrix-free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics, but are as yet unused in thermoacoustics. These matrix-free methods are shown to converge quickly to limit cycles by implicitly using a ‘reduced order model’ property. This is because the methods preferentially use the influential bulk motions of the system, whilst ignoring the features that are quickly dissipated in time. The matrix-free methods are demonstrated on a model of a ducted 2D diffusion flame, and the safe operating region is calculated as a function of the Peclet number and the heat release parameter. Both subcritical and supercritical Hopf bifurcations are found. Physical information about the flame-acoustic interaction is found from the limit cycles and Floquet modes. Invariant subspace preconditioning, higher order prediction techniques, and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces. Two types of shooting are compared, and two types of matrix-free evaluation are compared. The matrix-free methods are also demonstrated on a model of a ducted axisymmetric premixed flame, using a kinematic G-equation solver. The methods find limit cycles, period-2 limit cycles, fold bifurcations, period-doubling bifurcations and Neimark-Sacker bifurcations as a function of two parameters: the location of the flame in the duct, and the aspect ratio of the steady flame. The model is seen to display rich nonlinear behaviour and regions of multistability are found. Gradient methods can also efficiently calculate the limit cycles of large systems. A scalar cost function is defined that describes the proximity of a state to a limit cycle. The gradient of the cost function is used in an optimisation routine to iteratively converge to a limit cycle (or fixed point). The gradient of the cost function is found with a forwards-backwards process: first, the direct equations are marched forwards in time, second, the adjoint equations are marched backwards in time. The adjoint equations are derived by partially differentiating the direct governing equations. The gradient method is demonstrated on a model of a horizontal Rijke tube. This thesis describes novel nonlinear analysis techniques that can be applied to coupled systems with both advanced acoustic models and advanced flame models. The techniques can characterise the rich nonlinear behaviour of thermoacoustic models with a level of detail that was not previously possible.EPSRC Doctoral Training Partnershi

Nonlinear Dynamics In Musical Interactions

This thesis examines nonlinear dynamical processes in musical tools, identifying certain roles that they play in creative interactions with existing tools, and investigates the roles they might play in digital tools. Nonlinear dynamical processes are fundamental in the everyday physical world. They lie at the core of many acoustic instruments, playing a particularly significant role in bowed and blown instruments. Two major studies are presented that approach these issues from different perspectives. Firstly a set of comparative studies explore the ways in which musicians engage with systems that do and do not incorporate nonlinear dynamical processes. Secondly, interviews with a range of musicians engaged in contemporary musical practices — particularly free improvisation — are used to investigate the role of nonlinear dynamical processes in instrumental interactions in relation to unpredictability and creative exploration. Evidence is presented demonstrating that nonlinear dynamical processes can be drawn on as resources for exploration over long time periods. An approach to creative interaction that explicitly draws on the properties of nonlinear dynamical processes is uncovered and connected to material-oriented notions of creative processes. Nonlinear dynamics are shown to facilitate a productive ‘‘sweet spot’’ between unpredictability and complexity on the one hand, and detailed, sensitive, deterministic control, coupled with the potential to repeat and develop particular actions on the other. The importance of timing in interactions with nonlinear dynamical processes is highlighted as being significant in creating explorable interactions, particularly close to critical thresholds. A distinction is raised between instantaneous unpredictabilities that emerge from the interaction with the tool (interactional ), and unpredictabilities that result from the unexpected implications of the conjunction of otherwise anticipated elements (combinatorial). While the usefulness of the latter in creative interactions is frequently acknowledged in HCI research, the former is often excluded, or seen as a hinderance or obstruction. Engagements with nonlinear dynamical processes in existing musical instruments and practices provide clear evidence of the utility of both nonlinear dynamics, and interactional surprises more generally, suggesting that they can be of use in other domains where creative exploration is a concern

Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum

A quantum version of transition state theory based on a quantum normal form (QNF) expansion about a saddle-centre-...-centre equilibrium point is presented. A general algorithm is provided which allows one to explictly compute QNF to any desired order. This leads to an efficient procedure to compute quantum reaction rates and the associated Gamov-Siegert resonances. In the classical limit the QNF reduces to the classical normal form which leads to the recently developed phase space realisation of Wigner's transition state theory. It is shown that the phase space structures that govern the classical reaction d ynamicsform a skeleton for the quantum scattering and resonance wavefunctions which can also be computed from the QNF. Several examples are worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008) R1-R11