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>.

In part I we study quantum modified photon trajectories in a Schwarzschild blackhole spacetime. The photon vacuum polarization effect in curved spacetime leads to birefringence, i.e. the photon velocity becomes <i>c</i>±<i>δc </i>depending on its polarization. This velocity shift then results in modified photon trajectories. We find that photon trajectories are shifted by equal and opposite amounts for the two photon polarizations, as expected by the sum rule. Therefore, the critical circular orbit at <i>u</i> =1/3<i>M</i> in Schwarzschild spacetime, is split depending on polarization as <i>u</i> = 1/3<i>M</i> ± <i>Aδ</i>(<i>M</i>) (to first order in <i>A</i>), where <i>A</i> is a constant found to be ~ 10^-32 for a solar mass blackhole. Then using general quantum modified trajectory equations we find that photons projected into the blackhole for a critical impact parameter tend to the critical orbit associated with that polarization. We then use an impact parameter that is lower than the critical one. In this case the photons tend to the event horizon in coordinate time, and according to the affine parameter the photons fall into the singularity. This means even with the quantum corrections the event horizon behaves in the classic way, as expected from the horizon theorem. We also construct a quantum modified Schwarzschild metric, which encompasses the quantum polarization corrections. This is then used to derive the photons general quantum modified equations of motion, as before. We also show that when this modified metric is used with wave vectors for radically projected photons we obtain the classic equations of motion, as expected, because radial velocities are not modified by the quantum polarization correction. In Part II we use the 2+1d Nambu-Jona-Lasino (NJL) model to study the superfluid behaviour of two-dimensional quark matter. In previous work it was suggested that the high density phase of the 2+1d NJL model could be a relativistic gapless thin film BCS superfluid. In this work we find that as we raise the baryon chemical potential (<i>µ</i>) the baryon supercurrent jumps from a non-superfluid (zero) phase to a superfluid (non-zero) phase. This strongly first order transition is seen to occur at <i>µ </i>= 0.65, which was shown to be the point of chiral symmetry restoration. Therefore, we prove that at high density the 2+ 1<i>d</i> NJL model is in a superfluid phase. We then go on to study the dynamics of the superfluid phase, represented by the helicity modulus (<i>Ү</i>), which is the constant of proportionality between the supercurrent and the gradient of the diquark state function. We find that below the temperature associated with lattice size <i>L_t = </i>4, the system is in a non-superfluid phase, and above <i>L_t</i> = 24 the system is in a superfluid phase. We also find a possible 2^nd order transition at <i>L_t</i> ≈ 6, which corresponds to the critical point as described by Kosterlitz and Thouless’ theory of 2<i>D</i> critical systems with <i>U</i>(1) global symmetry – such as the <i>XY </i>model.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.