Analysis of Fluctuations in the Early Universe Using Fokker-Planck Formalism

The thesis begins with a review of basic elements of general theory of relativity (GTR) which forms the basis for the theoretical interpretation of the observations in cosmology. The first chapter also discusses the standard model in cosmology, namely the Friedmann model, its predictions and problems. We have also made a brief discussion on fractals and inflation of the early universe in the first chapter. In the second chapter we discuss the formulation of a new approach to cosmology namely a stochastic approach. In this model, the dynam ics of the early universe is described by a set of non-deterministic, Langevin type equations and we derive the solutions using the Fokker—Planck formalism. Here we demonstrate how the problems with the standard model, can be eliminated by introducing the idea of stochastic fluctuations in the early universe. Many recent observations indicate that the present universe may be approximated by a many component fluid and we assume that only the total energy density is conserved. This, in turn, leads to energy transfer between different components of the cosmic fluid and fluctuations in such energy transfer can certainly induce fluctuations in the mean to factor in the equation of state p = wp, resulting in a fluctuating expansion rate for the universe. The third chapter discusses the stochastic evolution of the cosmological parameters in the early universe, using the new approach. The penultimate chapter is about the refinements to be made in the present model, by means of a new deterministic model The concluding chapter presents a discussion on other problems with the conventional cosmology, like fractal correlation of galactic distribution. The author attempts an explanation for this problem using the stochastic approach.

Studies on the dynamics and rheology of dilute suspensions of Brownian particles in simple shear flow under the action of an external force

We present a novel approach to computing the orientation moments and rheological properties of a dilute suspension of spheroids in a simple shear flow at arbitrary Peclct number based on a generalised Langevin equation method. This method differs from the diffusion equation method which is commonly used to model similar systems in that the actual equations of motion for the orientations of the individual particles are used in the computations, instead of a solution of the diffusion equation of the system. It also differs from the method of 'Brownian dynamics simulations' in that the equations used for the simulations are deterministic differential equations even in the presence of noise, and not stochastic differential equations as in Brownian dynamics simulations. One advantage of the present approach over the Fokker-Planck equation formalism is that it employs a common strategy that can be applied across a wide range of shear and diffusion parameters. Also, since deterministic differential equations are easier to simulate than stochastic differential equations, the Langevin equation method presented in this work is more efficient and less computationally intensive than Brownian dynamics simulations.We derive the Langevin equations governing the orientations of the particles in the suspension and evolve a procedure for obtaining the equation of motion for any orientation moment. A computational technique is described for simulating the orientation moments dynamically from a set of time-averaged Langevin equations, which can be used to obtain the moments when the governing equations are harder to solve analytically. The results obtained using this method are in good agreement with those available in the literature.The above computational method is also used to investigate the effect of rotational Brownian motion on the rheology of the suspension under the action of an external force field. The force field is assumed to be either constant or periodic. In the case of con- I stant external fields earlier results in the literature are reproduced, while for the case of periodic forcing certain parametric regimes corresponding to weak Brownian diffusion are identified where the rheological parameters evolve chaotically and settle onto a low dimensional attractor. The response of the system to variations in the magnitude and orientation of the force field and strength of diffusion is also analyzed through numerical experiments. It is also demonstrated that the aperiodic behaviour exhibited by the system could not have been picked up by the diffusion equation approach as presently used in the literature.The main contributions of this work include the preparation of the basic framework for applying the Langevin method to standard flow problems, quantification of rotary Brownian effects by using the new method, the paired-moment scheme for computing the moments and its use in solving an otherwise intractable problem especially in the limit of small Brownian motion where the problem becomes singular, and a demonstration of how systems governed by a Fokker-Planck equation can be explored for possible chaotic behaviour.

Studies on Pseudoscalar Meson Bound States and Semileptonic Decays in a Relativistic Potential Model

In this thesis quark-antiquark bound states are considered using a relativistic two-body equation for Dirac particles. The mass spectrum of mesons includes bound states involving two heavy quarks or one heavy and one light quark. In order to analyse these states within a unified formalism, it is desirable to have a two-fermion equation that limits to one body Dirac equation with a static interaction for the light quark when the other particle's mass tends to infinity. A suitable two-body equation has been developed by Mandelzweig and Wallace. This equation is solved in momentum space and is used to describe the complete spectrum of mesons. The potential used in this work contains a short range one-gluon exchange interaction and a long range linear confining and constant potential terms. This model is used to investigate the decay processes of heavy mesons. Semileptonic decays are more tractable since there is no final state interactions between the leptons and hadrons that would otherwise complicate the situation. Studies on B and D meson decays are helpful to understand the nonperturbative strong interactions of heavy mesons, which in turn is useful to extract the details of weak interaction process. Calculation of form factors of these semileptonic decays of pseudo scalar mesons are also presented.

Density of states in high-Tc superconductors

The density of states and the low temperature specific heat of higb-Tc superconductors are calculated in a functional integral formalism using the slave boson technique. The manybody calculation in a saddle point approximation shows that the Iow energy sector is dominated by 3 single band. The calculated values of density of states are in good agreement with experimental results.

Studies on Quasinormal Modes and Late-time Tails in Black Hole Spacetimes

This thesis Entitled Studies on Quasinormal modes and Late-time tails black hole spacetimes. In this thesis, the signature of these new theories are probed on the evolution of field perturbations on the black hole spacetimes in the theory. Chapter 1 gives a general introduction to black holes and its perturbation formalism. Various concepts in the area covered by the thesis are also elucidated in this chapter. Chapter 2 describes the evolution of massive, charged scalar field perturbations around a Reissner-Nordstrom black hole surrounded by a static and spherically symmetric quintessence. Chapter 3 comprises the evolution of massless scalar, electromagnetic and gravitational fields around spherically symmetric black hole whose asymptotes are defined by the quintessence, with special interest on the late-time behavior. Chapter 4 examines the evolution of Dirac field around a Schwarzschild black hole surrounded by quintessence. Detailed numerical simulations are done to analyze the nature of field on different surfaces of constant radius . Chapter 5is dedicated to the study of the evolution of massless fields around the black hole geometry in the HL gravity.

Analytical Studies on Diffusion and lntermittency in Chaotic Maps

The study of simple chaotic maps for non-equilibrium processes in statistical physics has been one of the central themes in the theory of chaotic dynamical systems. Recently, many works have been carried out on deterministic diffusion in spatially extended one-dimensional maps This can be related to real physical systems such as Josephson junctions in the presence of microwave radiation and parametrically driven oscillators. Transport due to chaos is an important problem in Hamiltonian dynamics also. A recent approach is to evaluate the exact diffusion coefficient in terms of the periodic orbits of the system in the form of cycle expansions. But the fact is that the chaotic motion in such spatially extended maps has two complementary aspects- - diffusion and interrnittency. These are related to the time evolution of the probability density function which is approximately Gaussian by central limit theorem. It is noticed that the characteristic function method introduced by Fujisaka and his co-workers is a very powerful tool for analysing both these aspects of chaotic motion. The theory based on characteristic function actually provides a thermodynamic formalism for chaotic systems It can be applied to other types of chaos-induced diffusion also, such as the one arising in statistics of trajectory separation. It was noted that there is a close connection between cycle expansion technique and characteristic function method. It was found that this connection can be exploited to enhance the applicability of the cycle expansion technique. In this way, we found that cycle expansion can be used to analyse the probability density function in chaotic maps. In our research studies we have successfully applied the characteristic function method and cycle expansion technique for analysing some chaotic maps. We introduced in this connection, two classes of chaotic maps with variable shape by generalizing two types of maps well known in literature.