Firman So. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [159]-160). / Text in English; abstracts in English and Chinese. / Firman So. / Abstract --- p.i / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.vii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Derivation of the Fokker-Planck Equation --- p.4 / Chapter 2.1 --- Brownian Motion --- p.4 / Chapter 2.2 --- Non-Linear Langevin Equation --- p.7 / Chapter 2.3 --- Conditional Probability Density --- p.9 / Chapter 2.4 --- Kramers-Moyal Expansion --- p.11 / Chapter 2.5 --- Fokker-Planck Equation --- p.13 / Chapter Chapter 3. --- Method & Solution of the One-variable Fokker-Planck Equation with Time-Independent Coefficients --- p.15 / Chapter 3.1 --- Stationary Solution --- p.16 / Chapter 3.2 --- Ornstein-Ulhenbeck Process: An Exactly Solvable Fokker-Planck Equation --- p.17 / Chapter 3.3 --- Eigenfunction Expansion --- p.19 / Chapter 3.4 --- Ornstein-Ulhenbeck process by Eigenfunction Expansion --- p.29 / Chapter 3.5 --- Eigenfunctions and Eigenvalues of Inverted Potentials --- p.30 / Chapter 3.6 --- Kramers' Escape Rate --- p.32 / Chapter Chapter 4. --- Diffusion in Potential Wells --- p.36 / Chapter 4.1 --- Symmetric Double-Well Potential --- p.36 / Chapter 4.2 --- Asymmetric Bistable Potential --- p.61 / Chapter Chapter 5. --- Stochastic Resonance --- p.100 / Chapter 5.1 --- Introduction --- p.100 / Chapter 5.2 --- Probability Density........................... --- p.101 / Chapter 5.3 --- Power Spectrum of the Autocorrelation Function of x --- p.113 / Chapter 5.4 --- Stochastic Resonance --- p.120 / Chapter Chapter 6. --- Colored Noise --- p.124 / Chapter 6.1 --- Introduction --- p.124 / Chapter 6.2 --- Approximation Schemes for the Colored Noise Problem --- p.125 / Chapter 6.3 --- Stationary Probability Density of the Colored Noise Driven Bistable System --- p.132 / Chapter 6.4 --- Escape Rate in the Presence of Colored Noise --- p.140 / Chapter Chapter 7. --- Conclusion --- p.146 / Appendix A --- p.149 / Chapter A.1 --- State-Dependent Diagonalization Method --- p.149 / Chapter A.2 --- Infinite-Square-Well Basis Diagnalization --- p.153 / Chapter A.3 --- Solving the Fokker-Planck equation --- p.156 / References

No description available.

by Tam Chi Yung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 66-67). / by Tam Chi Yung. / Contents --- p.i / List of Figures --- p.iii / Acknowledgement --- p.iv / Abstract --- p.v / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Open systems and quasinormal modes --- p.1 / Chapter 1.2 --- Gravitational waves --- p.3 / Chapter Chapter 2. --- Green's Function Formalism --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Constructing the Green's function --- p.7 / Chapter 2.3 --- The norm --- p.9 / Chapter 2.4 --- Completeness --- p.11 / Chapter Chapter 3. --- Potentials With No Tail --- p.13 / Chapter 3.1 --- Introduction --- p.13 / Chapter 3.2 --- Completeness --- p.14 / Chapter 3.2.1 --- Proof --- p.14 / Chapter 3.2.2 --- Examples --- p.16 / Chapter 3.3 --- The two-component approach --- p.20 / Chapter 3.3.1 --- Formalism --- p.21 / Chapter 3.3.2 --- Comparison of different expansion schemes --- p.23 / Chapter 3.3.3 --- Linear Space --- p.29 / Chapter 3.4 --- Perturbation theory --- p.31 / Chapter 3.4.1 --- Formalism --- p.31 / Chapter 3.4.2 --- Examples --- p.33 / Chapter 3.5 --- Conclusion --- p.35 / Chapter Chapter 4. --- Potentials With Exponential Tails --- p.36 / Chapter 4.1 --- Introduction --- p.36 / Chapter 4.2 --- Single exponential tail --- p.37 / Chapter 4.3 --- Asymptotics of QNM's --- p.40 / Chapter 4.4 --- The Born series --- p.43 / Chapter 4.5 --- Poschl-Teller potential --- p.44 / Chapter 4.5.1 --- Analytic solutions --- p.44 / Chapter 4.5.2 --- The norm --- p.46 / Chapter 4.6 --- The problem of cut-off --- p.48 / Chapter 4.7 --- An effective numerical scheme --- p.49 / Chapter 4.8 --- Conclusion --- p.53 / Chapter Chapter 5. --- Logarithmic Perturbation --- p.54 / Chapter 5.1 --- Introduction --- p.54 / Chapter 5.2 --- Formalism --- p.54 / Chapter 5.3 --- Examples --- p.57 / Chapter 5.4 --- Conclusion --- p.59 / Chapter Chapter 6. --- Conclusion --- p.60 / Appendix A. Asymptotic behaviour of the Green's function --- p.61 / Appendix B. Derivation of the equation (4.16) --- p.63 / Appendix C. Different definitions of the norm --- p.64 / Bibliography --- p.66

We prove two results; first, we show that a boundary value problem for the semilinear wave equation with smooth, asymptotically linear nonlinearity and sinusoidal smooth forcing along a characteristic cannot have a continuous solution. Thereafter, we show that if the sinusoidal forcing is not isolated to a characteristic of the wave equation, then the problem has a continuous solution.

In this thesis, we propose a new microstructural model to describe the rheology of entangled linear polymers. In order to reduce the number of non-linear adjustable parameters, we develop a model capable of predicting both the linear and the non-linear response, using a single set of material parameters. In a first step, a linear differential formulation of the thermal constraint release mechanism is introduced and validated against experimental results for linear polystyrene melts. In a second step, we extend the linear model to the non-linear regime by generalizing the state variables to conformation tensors and accounting for the relevant non-linear relaxation phenomena. The numerical predictions of the resulting model are then compared to experimental data for entangled polymer melts and solutions in different flow regimes. Finally, we show, on a simple reptation model, how the single generator bracket formalism of non-equilibrium thermodynamics can be used for the phenomenological improvement of microstructural constitutive models.

No description available.

Aurorae often break down into elongated filaments parallel to the geomagnetic field lines (B) with cylindrically symmetric structures. The object of this thesis is to study the ion distribution function and transport properties in response to the sudden introduction of a radial electric field (E) in such a cylindrical geometry. Both collision-free and collisional situations are considered.<p> The thesis starts by solving a collision-free problem where the electric field is constant in time but increases linearly with radius, while the initial ion density is uniform in space. The attendant Boltzmann equation is solved by tracking the ions back in time, thereby using the temporal link between the initial position and velocity of an ion and its position and velocity at an arbitrary time and place. Complete analytical solutions show that the ion distribution function is a pulsating Maxwellian in time, and all transport parameters (e.g., bulk speed, temperature, etc.) oscillate in time but independent of radius. If the ion-neutral collisions are taken into account by employing a simple relaxation model, analytical solutions are also obtained. In this case, the ion distribution function can be driven to horseshoe shapes which are symmetric with respect to the ExB direction. The bulk parameters evolve in a transition period of the order of one collision time as they go from oscillating to the non-oscillating steady state.<p> In more realistic electric field structures which are spatially inhomogeneous but still constant in time, a generalized semi-numerical code is developed under collision-free conditions. This code uses a backmapping approach to calculate the ion velocity distribution and bulk parameters. With arbitrarily selected electric field rofiles, calculations reveal various shapes of ion velocity distribution functions (e.g., tear-drop, core-halo, ear-donut, etc). The associated transport properties are also obtained and discussed.<p> Under both collision-free and collisional conditions, the effect of the density inhomogeneities at the initial time is studied in an electric field which is proportional to radius and constant in time. With two profiles of the initial ion density for the collision-free case, and one profile for the collisional case, complete analytical solutions are obtained. The results reveal that the distribution function and the bulk properties are now strongly dependent on radial position.<p> If the radial electric field is unable to stay constant with time but modulated by in-coming charged particles, a fluid formalism is used to study the excitation of several plasma waves under different kinds of initial conditions. These identified waves include the ion cyclotron oscillation, the ion and electron upper-hybrid oscillations, and the lower-hybrid oscillation.<p> The results of this thesis are expected to be applicable to high-resolution observations. Future work should also include the mirror effect and the formation of conics in velocity space. Finally, the velocity distributions obtained in this thesis could trigger various plasma instabilities, and this topic should also be looked at in the future.

In this thesis we prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H^{s}_{x}(\mathbb{R})$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

In this thesis we prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H^{s}_{x}(\mathbb{R})$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

Aurorae often break down into elongated filaments parallel to the geomagnetic field lines (B) with cylindrically symmetric structures. The object of this thesis is to study the ion distribution function and transport properties in response to the sudden introduction of a radial electric field (E) in such a cylindrical geometry. Both collision-free and collisional situations are considered.<p> The thesis starts by solving a collision-free problem where the electric field is constant in time but increases linearly with radius, while the initial ion density is uniform in space. The attendant Boltzmann equation is solved by tracking the ions back in time, thereby using the temporal link between the initial position and velocity of an ion and its position and velocity at an arbitrary time and place. Complete analytical solutions show that the ion distribution function is a pulsating Maxwellian in time, and all transport parameters (e.g., bulk speed, temperature, etc.) oscillate in time but independent of radius. If the ion-neutral collisions are taken into account by employing a simple relaxation model, analytical solutions are also obtained. In this case, the ion distribution function can be driven to horseshoe shapes which are symmetric with respect to the ExB direction. The bulk parameters evolve in a transition period of the order of one collision time as they go from oscillating to the non-oscillating steady state.<p> In more realistic electric field structures which are spatially inhomogeneous but still constant in time, a generalized semi-numerical code is developed under collision-free conditions. This code uses a backmapping approach to calculate the ion velocity distribution and bulk parameters. With arbitrarily selected electric field rofiles, calculations reveal various shapes of ion velocity distribution functions (e.g., tear-drop, core-halo, ear-donut, etc). The associated transport properties are also obtained and discussed.<p> Under both collision-free and collisional conditions, the effect of the density inhomogeneities at the initial time is studied in an electric field which is proportional to radius and constant in time. With two profiles of the initial ion density for the collision-free case, and one profile for the collisional case, complete analytical solutions are obtained. The results reveal that the distribution function and the bulk properties are now strongly dependent on radial position.<p> If the radial electric field is unable to stay constant with time but modulated by in-coming charged particles, a fluid formalism is used to study the excitation of several plasma waves under different kinds of initial conditions. These identified waves include the ion cyclotron oscillation, the ion and electron upper-hybrid oscillations, and the lower-hybrid oscillation.<p> The results of this thesis are expected to be applicable to high-resolution observations. Future work should also include the mirror effect and the formation of conics in velocity space. Finally, the velocity distributions obtained in this thesis could trigger various plasma instabilities, and this topic should also be looked at in the future.