Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials

In this work we prove global stability for the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $\gamma + 2s \ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma + 2s < 0$, our solutions decay polynomially fast in time with any rate. These results are constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma + 2s \ge 0$, as conjectured in Mouhot-Strain (2007).Comment: This file has not changed, but this work has been combined with (arXiv:0912.0888v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v

The Vlasov-Poisson-Landau System in Rx3\R^3_x

For the Landau-Poisson system with Coulomb interaction in $\R^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.Comment: 50 page

Optimal time decay of the non cut-off Boltzmann equation in the whole space

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space \threed_x with \DgE. We use the existence theory of global in time nearby Maxwellian solutions from \cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption \cite{MR677262,MR2847536}. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}}) in the L^2_\vel(L^r_x)-norm for any $2\leq r\leq \infty$.Comment: 31 pages, final version to appear in KR

A variable delay integrated receiver for differential phase-shift keying optical transmission systems

An integrated variable delay receiver for DPSK optical transmission systems is presented. The device is realized in silicon-on-insulator technology and can be used to detect DPSK signals at any bit-rates between 10 and 15 Gbit/s

Hilbert Expansion from the Boltzmann equation to relativistic Fluids

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations.Comment: 50 page

Global Newtonian limit for the Relativistic Boltzmann Equation near Vacuum

We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time "mild" solutions are obtained uniformly in the speed of light parameter $c \ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon \in(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.Comment: 35 page

Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of (Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves the open question of global existence for the soft potentials.Comment: 64 page

Optimal time-domain combination of the two calibrated output quadratures of GEO 600

GEO 600 is an interferometric gravitational wave detector with a 600 m arm-length and which uses a dual-recycled optical configuration to give enhanced sensitivity over certain frequencies in the detection band. Due to the dual-recycling, GEO 600 has two main output signals, both of which potentially contain gravitational wave signals. These two outputs are calibrated to strain using a time-domain method. In order to simplify the analysis of the GEO 600 data set, it is desirable to combine these two calibrated outputs to form a single strain signal that has optimal signal-to-noise ratio across the detection band. This paper describes a time-domain method for doing this combination. The method presented is similar to one developed for optimally combining the outputs of two colocated gravitational wave detectors. In the scheme presented in this paper, some simplifications are made to allow its implementation using time-domain methods

Hybrid integration of diamond membranes with GaN waveguides

The nitrogen vacancy (NV) is a photostable emitter in diamond which is optically accessible at room temperature and a potential candidate for quantum information processing as a spin register. The challenge facing research today is the efficient collection and manipulation of the NV’s emissions, such as by enhancing the zero phonon line transitions for a coherent spin-photon interface.Integrating diamond with other photonic materials would allow for resonant coupling of the defect centre to optical devices on large area photonic integrated circuits (PICs). Emitted photons collected by bus waveguides could then be guided elsewhere on chip for entanglement or measurements.This work focuses on integrating ultra-thin diamond membranes with GaN waveguide and resonator devices. Mode simulations (see Fig. 1(a)) show that light can be coupled significantly into and out of the membranes by this method. Membranes of < 200 nm have been fabricated using Ar-Cl2 etch recipes that cumulatively smooth the diamond over time; an r.m.s roughness value of 0.19 nm has been achieved.1 The smooth surface and ultra-low thickness allow a good conformation and strong bonding of the membrane to other materials. This should allow for the integration of diamond membranes with photonic integrated circuits as shown schematically in Fig. (b).Free standing ultra-thin diamond membranes can also be used in tuneable open access cavities – where a low mode volume and high Q factor are desired; 2 or as templates for fabricating diamond optical devices on non-native substrates.

Diabetes, cardiovascular disease and the microcirculation.

Cardiovascular disease (CVD) is the leading cause of mortality in people with type 2 diabetes mellitus (T2DM), yet a significant proportion of the disease burden cannot be accounted for by conventional cardiovascular risk factors. Hypertension occurs in majority of people with T2DM, which is substantially more frequent than would be anticipated based on general population samples. The impact of hypertension is considerably higher in people with diabetes than it is in the general population, suggesting either an increased sensitivity to its effect or a confounding underlying aetiopathogenic mechanism of hypertension associated with CVD within diabetes. In this contribution, we aim to review the changes observed in the vascular tree in people with T2DM compared to the general population, the effects of established anti-diabetes drugs on microvascular outcomes, and explore the hypotheses to account for common causalities of the increased prevalence of CVD and hypertension in people with T2DM.This article is freely available via Open Access. Click on the Additional Link above to access the full-text via the publisher's site