A Newtonian diffusion process describing a stochastic dynamical system is a diffusion process satisfying a Nelson-Newton law. In this thesis we primarily study Newtonian diffusions and in particular Nelson's stochastic mechanics. In Chapter 1 we describe the concept of a Newtonian diffusion process and how this leads to a corresponding Schrodinger equation. We also give some motivating discussions for the work contained in the ensuing chapters. In Chapter 2 we apply Newtonian diffusion theory to the problem of giving a mathematical model for the condensation of planets out of a protosolar nebula. We consider two examples. Firstly the case of a cloud nebula with its mass concentrated near the origin and secondly the case of a cloud nebula with spherically uniform density. Newtonian diffusions are derived from a particular stationary state solution of the corresponding Schrodinger equation and a model constructed in which the planets condense out of the protosolar nebula describing circular orbits (Prop. 2.2.2, 2.2.3, and 2.2.4). In Chapter 3 we generalize the results of Chapter 2 to include a wider class of potentials corresponding to different cloud densities. To this end we establish some results concerning the log-concavity of radial Schrodinger wave-functions (Prop. 3.3.1 and 3.3.3). In Chapter 4 we study the Newtonian diffusions arising from Nelson's stochastic mechanics of the hydrogen atom. In particular we describe some computer simulations of the diffusions corresponding to the first few (low energy) states and generate some interesting coded pictures. The accuracy of our simulations is also analysed via tests based on the ergodic theorem. Finally, in Chapter 5 we continue our study of stochastic processes and their computer simulation. We study Brownian motion on hypersurfaces and present a useful characterization in terms of local parametric coordinates. Using this characterization we discuss some computer simulations of Brownian motion on surfaces in R<SUP>3</SUP>.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:636752 |
| Date | January 1991 |
| Creators | Durran, R. M. |
| Publisher | Swansea University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0126 seconds