We report on results of long-time Brownian-dynamics simulations of electrostatic interactions between two rigid polyelectrolyte rods. We find that the interaction can be both repulsive, as obtained from mean-field theory, and attractive. The onset of attraction depends not only on the fixed charge density of the rod, but also on its radius. The attractive force is found to be due to the development of positional correlations between the counterions condensed on the two rods, for which we propose a simple analytical model.

We present a revision to the well known Störmer–Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behaviour of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency and time step for both underdamped and overdamped behaviour within the usual stability limit of the Verlet algorithm. Given the structure and simplicity of the method, we conclude that this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints.

Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of $T=\infty$, we identify a phase transition through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breatherlike localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.

We study driven vortex lattices in superconducting thin films. Above the critical force $F_c$ we find two dynamical phase transitions at $F_p$ and $F_t$, which could be observed in simultaneous noise measurements of the longitudinal and Hall voltage. At $F_p$ there is a transition from plastic flow to smectic flow, where the voltage noise is isotropic (Hall noise $=$ longitudinal noise) and there is a peak in the differential resistance. At $F_t$ there is a sharp transition to a frozen transverse solid, where the Hall noise falls abruptly and vortex motion is localized in the transverse direction.

A new 1D discrete nonlinear Schrödinger (NLS) Hamiltonian is introduced which includes the integrable Ablowitz-Ladik system as a limit. The symmetry properties of the system are studied. The relationship between intrinsic localized states and the soliton of the Ablowitz-Ladik NLS is discussed, including the role of discretization as a mechanism controlling collapse. It is pointed out that a staggered localized state can be viewed as a particle of a negative effective mass. It is shown that staggered localized states can exist in the discrete dark NLS. The motion of localized states and Peierls-Nabarro pinning are also studied.

Luminescence in Ce-doped materials corresponds to a transition from an excited state where the lowest Ce 5$d$ level is filled [often called the (Ce${}^{3+}$)${}^{*}$ state] to the ground state where a single 4$f$ level is filled. We have performed theoretical calculations based on density functional theory to calculate the ground-state band structure of Ce-doped materials as well as the (Ce${}^{3+}$)${}^{*}$ excited state. The excited-state calculations used a constrained occupancy approach by setting the occupation of the Ce 4$f$ states to zero and allowing the first excited state above them to be filled. These calculations were performed on a set of Ce-doped materials that are known from experiment to be scintillators or nonscintillators to relate theoretically calculable parameters to measured scintillator performance. From these studies, we developed a set of criteria based on calculated parameters that are necessary characteristics for bright Ce-activated scintillators. Applying these criteria to about 100 new materials, we developed a list of candidate materials for new bright Ce-activated scintillators. After synthesis in powder form, one of these new materials (Ba${}_2$YCl${}_7$:Ce) was found to be a bright scintillator. This approach, involving first-principles calculations of modest computing requirements, was designed as a systematic, high-throughput method to aid in the discovery of new bright scintillator materials by prioritization and down-selection on the large number of potential new materials.