Well-balanced and asymptotic preserving schemes for kinetic models

In this paper, we propose a general framework for designing numerical schemes that have both well-balanced (WB) and asymptotic preserving (AP) properties, for various kinds of kinetic models. We are interested in two different parameter regimes, 1) When the ratio between the mean free path and the characteristic macroscopic length $\epsilon$ tends to zero, the density can be described by (advection) diffusion type (linear or nonlinear) macroscopic models; 2) When $\epsilon$ = O(1), the models behave like hyperbolic equations with source terms and we are interested in their steady states. We apply the framework to three different kinetic models: neutron transport equation and its diffusion limit, the transport equation for chemotaxis and its Keller-Segel limit, and grey radiative transfer equation and its nonlinear diffusion limit. Numerical examples are given to demonstrate the properties of the schemes

Full-counting statistics of charge and spin transport in the transient regime: A nonequilibrium Green's function approach

We report the investigation of full-counting statistics (FCS) of transferred charge and spin in the transient regime where the connection between central scattering region (quantum dot) and leads are turned on at $t=0$. A general theoretical formulation for the generating function (GF) is presented using a nonequilibrium Green's function approach for the quantum dot system. In particular, we give a detailed derivation on how to use the method of path integral together with nonequilibrium Green's function technique to obtain the GF of FCS in electron transport systems based on the two-time quantum measurement scheme. The correct long-time limit of the formalism, the Levitov-Lesovik's formula, is obtained. This formalism can be generalized to account for spin transport for the system with noncollinear spin as well as spin-orbit interaction. As an example, we have calculated the GF of spin-polarized transferred charge, transferred spin, as well as the spin transferred torque for a magnetic tunneling junction in the transient regime. The GF is compactly expressed by a functional determinant represented by Green's function and self-energy in the time domain. With this formalism, FCS in spintronics in the transient regime can be studied. We also extend this formalism to the quantum point contact system. For numerical results, we calculate the GF and various cumulants of a double quantum dot system connected by two leads in transient regime. The signature of universal oscillation of FCS is identified. On top of the global oscillation, local oscillations are found in various cumulants as a result of the Rabi oscillation. Finally, the influence of the temperature is also examined

Novel biodegradable polyesters derived from carbohydrates

This thesis concerns the synthesis of new biodegradable polyesters from carbohydrates. All the new products have been characterised using nuclear magnetic resonance (NMR) spectroscopy and mass spectrometry; some have been studied using X-ray crystallography. In Chapter 1, which is the introduction, some of the leading biodegradable polymers are introduced, in particular attention is paid to polyesters which are synthesised from carbohydrates. In addition, the standard polymerisation methods for the polyester synthesis - ring opening polymerisation and step polymerisation - are described. Chapter 2 details the synthesis and polymerisation of acetic acid 5-acetoxy-6-oxo-tetrahydro-pyran-2-yl methyl ester. Three different initiating systems have been used for ring opening polymerisations; the kinetics and the polymer product of the polymerisation are discussed. L-Lactide was used in copolymerisations with acetic acid 5-acetoxy-6-oxo-tetrahydro-pyran-2-yl methyl ester; the copolymers show distinct thermal properties and accelerated degradation rates compared with poly(L-Lactide) (PLLA). Chapter 3 involves applying functionalised carbohydrates, with one free hydroxyl group, as the co-initiator for the ring opening polymerisation (ROP) of L-lactide. The polymerisations were well controlled with linear relationships between the percentage lactide conversion and the polymer’s molecular weight. The carbohydrates form the functionalised end group of the PLLA, which, in turn, improve the hydrophilicity of the resulting PLLA. These end-group functionalised polylactides have been used as the matrix for human-osteoblast-derived osteosarcoma cells (SaOS-2 cells) culturing, and these studies proved that the carbohydrate end groups were non toxic. Chapter 4 describes the synthesis of a series of other carbohydrate lactones derived from D-glucono-1,5-lactone, D-xylose and 2-deoxy-Dribose; the lactones include 2,3,4-tri-O-benzyl-D-xylonolactone, 6-methyl-2-oxotetrahydro- 2H-pyran-3-yl acetate, (S)-6-(benzyloxymethyl)-tetrahydropyran-2-one. The ROP of these lactones resulted in formation of low molecular weight oligomers, except for 2,3,4-tri-O-benzyl-D-xylonolactone, which can not be polymerised. Chapter 5 describes the overall conclusions resulting from the experiments described in the thesis. Chapter 6 is the experimental section and thus provides a detailed description of the synthesis of all the compounds prepared in the thesis and their analytical data. The Appendices include two papers that have already been published, describing some of the work in the thesis; important additional NMR spectra and MALDI-ToF spectra; and the complete X-ray crystallography data

A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and Keller-Segel limit

A pathway-based mean-field theory (PBMFT) was recently proposed for E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we derived a new moment system of PBMFT by using the moment closure technique in kinetic theory under the assumption that the methylation level is locally concentrated. The new system is hyperbolic with linear convection terms. Under certain assumptions, the new system can recover the original model. Especially the assumption on the methylation difference made there can be understood explicitly in this new moment system. We obtain the Keller-Segel limit by taking into account the different physical time scales of tumbling, adaptation and the experimental observations. We also present numerical evidence to show the quantitative agreement of the moment system with the individual based E. coli chemotaxis simulator.Comment: 21 pages, 3 figure

Statistical Thermodynamics of General Minimal Diffusion Processes: Constuction, Invariant Density, Reversibility and Entropy Production

The solution to nonlinear Fokker-Planck equation is constructed in terms of the minimal Markov semigroup generated by the equation. The semigroup is obtained by a purely functional analytical method via Hille-Yosida theorem. The existence of the positive invariant measure with density is established and a weak form of Foguel alternative proven. We show the equivalence among self-adjoint of the elliptic operator, time-reversibility, and zero entropy production rate of the stationary diffusion process. A thermodynamic theory for diffusion processes emerges.Comment: 23 page

Sensitivity of Capital Stock and Multifactor Productivity Estimates to Depreciation Assumptions: A Canada-U.S. Comparison

This article provides consistent estimates for capital stock and multifactor productivity (MFP) for Canada and the United States across major industries for the 1987-2007 period. For this purpose, capital stock estimates are developed for Canadian and U.S. industries using the same asset depreciation rates (either from the U.S. Bureau of Economic Analysis or from Statistics Canada) for the two countries. The results show that on an hours worked basis Canadian industries invest more in total capital than their U.S. counterparts. This situation reflects much greater investment in structures, with less in machinery and equipment (including information and communications technologies). The results imply that all of the Canada-U.S. labour productivity gap arises from the multifactor productivity gap.multifactor productivity, capital stock, depreciation rate, labour productivity