Daniel W. Hook

International collaboration clusters in Africa

Recent discussion about the increase in international research collaboration suggests a comprehensive global network centred around a group of core countries and driven by generic socio-economic factors where the global system influences all national and institutional outcomes. In counterpoint, we demonstrate that the collaboration pattern for countries in Africa is far from universal. Instead, it exhibits layers of internal clusters and external links that are explained not by monotypic global influences but by regional geography and, perhaps even more strongly, by history, culture and language. Analysis of these bottom-up, subjective, human factors is required in order to provide the fuller explanation useful for policy and management purposes.

Quantum effects in classical systems having complex energy

On the basis of extensive numerical studies it is argued that there are strong analogies between the probabilistic behavior of quantum systems defined by Hermitian Hamiltonians and the deterministic behavior of classical mechanical systems extended into the complex domain. Three models are examined: the quartic double-well potential V(x) = x4 ? 5x2, the cubic potential , and the periodic potential V(x) = ?cos?x. For the quartic potential a wave packet that is initially localized in one side of the double-well can tunnel to the other side. Complex solutions to the classical equations of motion exhibit a remarkably analogous behavior. Furthermore, classical solutions come in two varieties, which resemble the even-parity and odd-parity quantum-mechanical bound states. For the cubic potential, a quantum wave packet that is initially in the quadratic portion of the potential near the origin will tunnel through the barrier and give rise to a probability current that flows out to infinity. The complex solutions to the corresponding classical equations of motion exhibit strongly analogous behavior. For the periodic potential a quantum particle whose energy lies between ?1 and 1 can tunnel repeatedly between adjacent classically allowed regions and thus execute a localized random walk as it hops from region to region. Moreover, if the energy of the quantum particle lies in a conduction band, then the particle delocalizes and drifts freely through the periodic potential. A classical particle having complex energy executes a qualitatively analogous local random walk, and there exists a narrow energy band for which the classical particle becomes delocalized and moves freely through the potential.

Information geometry in vapour?liquid equilibrium

Using the square-root map p-->\sqrt{p} a probability density function p can be represented as a point of the unit sphere S in the Hilbert space of square-integrable functions. If the density function depends smoothly on a set of parameters, the image of the map forms a Riemannian submanifold M in S. The metric on M induced by the ambient spherical geometry of S is the Fisher information matrix. Statistical properties of the system modelled by a parametric density function p can then be expressed in terms of information geometry. An elementary introduction to information geometry is presented, followed by a precise geometric characterisation of the family of Gaussian density functions. When the parametric density function describes the equilibrium state of a physical system, certain physical characteristics can be identified with geometric features of the associated information manifold M. Applying this idea, the properties of vapour-liquid phase transitions are elucidated in geometrical terms. For an ideal gas, phase transitions are absent and the geometry of M is flat. In this case, the solutions to the geodesic equations yield the adiabatic equations of state. For a van der Waals gas, the associated geometry of M is highly nontrivial. The scalar curvature of M diverges along the spinodal boundary which envelopes the unphysical region in the phase diagram. The curvature is thus closely related to the stability of the system.

Complex trajectories of a simple pendulum

The motion of a classical pendulum in a gravitational field of strength g is explored. The complex trajectories as well as the real ones are determined. If g is taken to be imaginary, the Hamiltonian that describes the pendulum becomes PT-symmetric. The classical motion for this PT-symmetric Hamiltonian is examined in detail. The complex motion of this pendulum in the presence of an external periodic forcing term is also studied.

Complexified dynamical systems

Many dynamical systems, such as the Lotka–Volterra predator–prey model and the Euler equations for the free rotation of a rigid body, are symmetric. The standard and well-known real solutions to such dynamical systems constitute an infinitessimal subclass of the full set of complex solutions. This paper examines a subset of the complex solutions that contains the real solutions, namely those having symmetry. The condition of symmetry selects out complex solutions that are periodic.

On optimum Hamiltonians for state transformations

For a prescribed pair of quantum states |ψI and |ψF we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation |ψI → |ψF in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.

Complex correspondence principle.

Quantum mechanics and classical mechanics are distinctly different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum and classical mechanics into the complex domain. These complex extensions continue to exhibit a correspondence, and this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship requires the use of asymptotics beyond all orders.

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviours of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.

Quantum Phase Transitions Without Thermodynamic Limits

A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterized by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures.

Probability Density in the Complex Plane

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density (z) in the complex plane. It is shown that there exist innitely many complex contours C of innite length on which (z)dz is real and positive. Furthermore, the probability integral R C (z)dz is nite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.