Asymptotic analysis of a system of algebraic equations arising in dislocation theory

The system of algebraic equations given by\ud \ud $\sum_{j=0, j \neq i}^n sgn(x_i - x_j) / |x_i - x_j|^a = 1, i = 1, 2, \ldots n, x_0 = 0,$\ud \ud appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole.\ud \ud We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n -> ∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment but, up to corrections of logarithmic order, it also leads to a differential equation.\ud \ud The continuum approximation is only valid for i not too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem

Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity

The bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg--Landau equations by the methods of formal asymptotics. The behavior of the bifurcating branch depends on the parameters d, the size of the superconducting slab, and $\kappa$, the Ginzburg--Landau parameter. The secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of $(\kappa,d)$ for which it is close to the primary bifurcation from the normal state. These values of $(\kappa,d)$ form a curve in the $\kappa d$-plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requiring a separate analysis. The results answer some of the conjectures of [A. Aftalion and W. C. Troy, Phys. D, 132 (1999), pp. 214--232]

Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions

A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the $\lambda$--\newrho model for irreversible bimolecular reactions which was introduced in [arXiv:0903.1298]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated.Comment: 16 pages, 13 figures, submitted to SIAM Appl Mat

Analysis of Brownian dynamics simulations of reversible biomolecular reactions

A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the λ-rho model for irreversible bimolecular reactions which was introduced in [11]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated

Quantifying fusion born ion populations in magnetically confined plasmas using ion cyclotron emission

Ion cyclotron emission (ICE) offers unique promise as a diagnostic of the fusion born alpha-particle population in magnetically confined plasmas. Pioneering observations from JET and TFTR found that ICE intensity $P_{ICE}$ scales approximately linearly with the measured neutron flux from fusion reactions, and with the inferred concentration, $n_\alpha/n_i$, of fusion-born alpha-particles confined within the plasma. We present fully nonlinear self-consistent kinetic simulations that reproduce this scaling for the first time. This resolves a longstanding question in the physics of fusion alpha-particle confinement and stability in MCF plasmas. It confirms the magnetoacoustic cyclotron instability (MCI) as the likely emission mechanism and greatly strengthens the basis for diagnostic exploitation of ICE in future burning plasmas

Real-Time Cavity QED with Single Atoms

The combination of cold atoms and large coherent coupling enables investigations in a new regime in cavity QED with single-atom trajectories monitored in real time with high signal-to-noise ratio. The underlying “vacuum-Rabi” splitting is clearly reflected in the frequency dependence of atomic transit signals recorded atom by atom, with evidence for mechanical light forces for intracavity photon number <1. The nonlinear optical response of one atom in a cavity is observed to be in accord with the one-atom quantum theory but at variance with semiclassical predictions