The self-gravitating systems are formed by particles interacting through gravity. They describe structure formation in the universe. As a consequence of the long range interaction of gravity, they are inhomogeneous even at thermal equilibrium. We study the self-gravitating systems with several kinds of particles and the self-gravitating systems in the presence of the cosmological constant $ Lambda$. We formulate the statistical mechanics and the mean field approach describing the gaseous phase. We explicitely compute the density of particles and thermodynamic quantities. The presence of $ Lambda$ extends the domain of stability of the gaseous phase. Monte Carlo simulations show that the mean field describes the gaseous phase with an excellent accuracy. Scalling law of the self-gravitating systems with several kinds of particles is found. At the critical point the fractal dimension is independant of their composition and is $1.6...$~.

We present a framework for learning in hidden Markov models with distributed state representations. Within this framework, we derive a learning algorithm based on the Expectation--Maximization (EM) procedure for maximum likelihood estimation. Analogous to the standard Baum-Welch update rules, the M-step of our algorithm is exact and can be solved analytically. However, due to the combinatorial nature of the hidden state representation, the exact E-step is intractable. A simple and tractable mean field approximation is derived. Empirical results on a set of problems suggest that both the mean field approximation and Gibbs sampling are viable alternatives to the computationally expensive exact algorithm.

A polymer is a large molecule constructed from many smaller structural units calledmonomers joined together by covalent bonds. Polymers have existed in natural formsince life began and those such as DNA, RNA, proteins and polysaccharides are someof the most important macromolecules found in plant and animal life. From the earliesttimes, the man has used many of these polymers as materials for providing clothing,decoration, tools, weapons and other requirements. However, the origins of today'spolymer industry commonly are accepted as being in the nineteenth century whenimportant discoveries were made concerning to the modification of certain naturalpolymers, as cellulose. The use of synthetic and natural polymers as stabilisers forcolloid systems (sols, dispersions, microemulsions, etc.) is becoming more importanteveryday in contemporary life. Polymer additives can be applied in preconcentrationsand dehydration of suspensions in mineral processing, purification of wastewater andeven in nutritional and pharmaceutical emulsions being their importance related to thecharacteristics of the process and the properties that they show. The present work aimsto develop appropriate numerical and analytical modelling techniques, which candescribe (considering the formation of loops and tails) the structure of a polymeric layeradsorbed on heterogeneous surfaces; this adsorbed layer is an relevant factor in theproperties showed by this kind of materials. Taking into account this, the methodologyknown as Single Chain Mean Field (SCMF) (originally used to study micellaraggregates and grafted polymers) was modified to apply on polymer adsorptionproblems. In this way, it was possible to calculate numerically properties that can beexperimentally measured, such as total monomer volume fraction profiles, loop and tailvolume fraction profiles, adsorbance or the thickness of the adsorbed layer. Thestructure of the polymeric layer was examined both for flat and spherical (colloidalparticles) surface geometries. When compared with other well establishedmethodologies for the numerical simulation of polymeric systems, this new version ofSCMF was found to be more efficient due to the improved sampling of the polymerchain configuration space.Thus, SCMF method results, in the case of the adsorption on flat surfaces, compare wellwith those obtained either with Monte Carlo simulations or with the method developedin the 80s by Scheutjens and Fleer (SCF). Due to the lack of studies focusing to polymeradsorption on colloidal particles, our results have been the first to present quantitativepredictions of the structure of the polymeric layer adsorbed on a spherical surface. Thus,we have demonstrated the dependence of the adsorbed polymer layer with the size ofthe colloidal particle as well as the characteristic lengths that influence on it. Finally, inthis work an analytical approach for the description of polymer-colloidal mixtures hasbeen developed which compares well with the numerical results obtained from theSCMF methodology. Furthermore, the analytical approach is able to predict systembehaviours, as for example the formation of gels. / Un polímero es una molécula de grandes dimensiones formada de pequeñas unidadesllamadas monómeros, los cuales se encuentran unidos por medio de enlaces covalentes.Los polímeros han existido de forma natural desde el comienzo de la vida, y aquelloscomo el DNA, RNA o las proteínas son algunos de los polímeros más importantesencontrados tanto en la vida animal como en la vegetal. Desde siempre el hombre hautilizado muchos de estos polímeros como materiales para hacer ropa, decoración,herramientas, etc. Sin embargo, el origen de la industria de polímeros que conocemoshoy en día se produjo en el siglo 19, gracias a importantes descubrimientos dentro de lamodificación de ciertos polímeros naturales, como la celulosa. El uso de polímerossintéticos y naturales como estabilizadores de sistemas coloidales (dispersiones,microemulsiones, etc.) juega en nuestros días un papel importante. Los polímerosutilizados como aditivos, pueden ser aplicados en preconcentraciones y deshidrataciónde suspensiones dentro de procesos minerales, tratamiento de aguas residuales e inclusolos podemos encontrar dentro de la industria farmacéutica y alimentaria, donde suimportancia es debida a la procesabilidad y propiedades que ellos exhiben. El trabajoque se presenta es orientado al desarrollo de técnicas de modelización, tanto analíticascomo computacionales, y su aplicación en la descripción (por medio de la formación debucles y colas) de la estructura de la capa de polímeros adsorbida en superficiesheterogéneas, siendo dicha capa de polímeros un factor importante en las propiedadesque este tipo de materiales presentan. Con este propósito, la metodología conocidacomo Single Chain Mean Field, utilizada anteriormente tanto para el estudio deagregados micelares como de polímeros anclados en superficies, ha sido modificadapara describir la adsorción de polímeros en superficies. Así se han podido calcularnuméricamente propiedades medibles experimentalmente como los perfiles de lafracción en volumen de monómeros totales, además de los pertenecientes a los bucles ycolas, adsorbancia o el espesor de la capa adsorbida, para geometrías de la superficieabsorbente tanto plana como esférica (partículas coloidales). En su comparación conotras metodologías, ya establecidas para la simulación numérica dentro de la física depolímeros, la aplicación de esta nueva versión del Single Chain Mean Field (SCMF)ha resultado ser más eficiente debido a un mejor muestreo del espacio deconfiguraciones de las cadenas poliméricas. De este modo, comparando los resultadosobtenidos a partir del SCMF, con aquellos obtenidos mediante técnicas de simulaciónMonte Carlo o la teoría desarrollada en los años 80 por Scheutjens y Fleer (SCF), se hapodido encontrar un buen acuerdo en las propiedades calculadas para el caso de laadsorción en superficies planas. Debido a la dificultad intrínseca del estudio de laadsorción en superficies curvadas, nuestros resultados son los primeros que presentanpredicciones cuantitativas sobre la estructura de la capa que se forma sobre unapartícula coloidal. Así hemos podido comprobar la dependencia de la estructura de lacapa de polímeros adsorbidos con el tamaño de la partícula sobre la que se encuentranadsorbidos además de las longitudes características de las cuales depende. Finalmente,en este trabajo se ha desarrollado, también, una teoría analítica para la descripción de lamezcla polímero-coloide. De este modo, los resultados numéricos obtenidos con elSCMF han podido ser comparados con dicha teoría, obteniendo, de nuevo, un buenacuerdo y predecir, además, comportamientos colectivos como la formación de geles.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

Nuclear spinodal instabilities are investigated in non-relativistic and relativistic stochastic mean-field approaches for charge asymmetric and charge symmetric nuclear matter. Quantum statistical effect on the growth of instabilities are calculated in non-relativistic approach. Due to quantal effects, in both symmetric and asymmetric matter, dominant unstable modes shift towards longer wavelengths and modes with wave numbers larger than the Fermi momentum are strongly suppressed. As a result of quantum statistical effects, in particular at lower temperatures, amplitude of density fluctuations grows larger than those calculated in semi-classical approximation. Relativistic calculations in the semi-classical limit are compared with the results of non-relativistic calculations based on Skyrme-type effective interactions under similar conditions. A qualitative difference appears in the unstable response of the system: the system exhibits most unstable behavior at higher baryon densities around $rho_{B}=0.4 rho_{0}$ in the relativistic approach while most unstable behavior occurs at lower baryon densities around $rho_{B}=0.2 rho_{0}$ in the non-relativistic calculations.

The nuclear matter liquid-gas phase transition is expected to be a signal of nuclear spinodal instabilities as a result of density fluctuations. Nuclear spinodal instabilities in symmetric nuclear matter are studied within a stochastic relativistic density-dependent model in semi-classical approximation. We use two parameterization for the Lagrange density, DDME1 and TW sets. The early growth of density fluctuations is investigated by employing relativistic Vlasov equation based on QHD and discussed the cluster size of the condensations from the early growth of density correlation functions. Expectations are that hot nuclear matter behaves unstable around &rho / b &asymp / &rho / 0/4 (below the saturation density) and at low temperatures. We therefore present our results at low temperature T=1 MeV and at higher temperature T=5 MeV, and also at a lower initial baryon density &rho / b = 0.2 &rho / 0 and a higher value &rho / b = 0.4 &rho / 0 where unstable behavior is within them. Calculations in density-dependent model are compared with the other calculations obtained in a relativistic non-linear model and in a Skyrme type nonivrelativistic model. Our results are consistent with them. Qualitatively similar results show that the physics of the quantities are model-independent. The size of clusterization is estimated in two ways, by using half-wavelength of the most unstable mode and from the width of correlation function at half maximum. Furthermore, the average speed of condensing fragments during the initial phase of spinodal decomposition are determined by using the current density correlation functions.

From the time of the unexpected discovery of the insulating nature of NiO by Verwey half a century ago, Oxide materials have continued to occupy the centre stage of condensed matter physics. The recent discovery of high temperature superconductivity in doped cuprates has given a new impetus to the study of the strongly correlated electron systems. Besides, the occurrence of Colossal Magneto-Resistance (CMR) in doped rare earth manganite has also created renewed interest in these rather old systems. Understanding of the rich and complex phase diagram of these materials and their sensitivity to small perturbations e.g. external magnetic field of a few Tesla, temperature, change in isotope etc. are of great theoretical interest and also these materials have many potential technological applications. A common feature of all these oxide materials is that the transition metal ions have partially filled d-shells. Unlike s and p-electrons which gives rise to hybridized Bloch states, the d-electrons retain their atomic nature in a solid. This gives rise to strong Coulomb interaction among d-electrons which may be comparable or more than its kinetic energy. The strong correlation effects are evident from the experimental fact that the undoped parent compounds are insulators rather than metals as suggested by band theory, which favours a metallic state for systems with one electron per unit cell since this gives rise to partially filled bands (and hence a metallic state). These insulators termed Mott insulators, arise solely due to strong electron-electron correlations as compared to the band insulators which arise due to complete filling of one electron bands thereby giving rise to a gap (band gap)in the excitation spectra. The delicate competition between the kinetic energy and the Coulomb energy for d-electrons is broadly responsible for the wide variety of phenomena like Mott metal-insulator transition (MIT), magnetic transitions, charge ordering, orbital ordering, ferro/antiferroelectricity, and most interestingly the observation of high Tc superconductivity in doped cuprates. In this thesis we will restrict our interest to one such class of oxide materials, namely the doped rare earth manganites. In Chapter 1 we give a brief overview of the structure and basic interactions present in the doped manganites. Also, in the same Chapter we give a brief introduction to the phenomenology of manganites, particularly its phase diagram in the doping and temperature plane and various experimental features, e.g. the wide variety of phase transitions and phenomena particularly the observation of CMR, charge ordering and incipient meso-scale phase separations etc.. Then we briefly introduce a recently proposed microscopic model which is believed to be a minimal model which, for the first time, includes the three most important interactions present in the manganites namely the following -1)coupling of the orbitally degenerate eg electrons to local lattice distortions of Jahn-Teller type which gives rise to two species of electrons. The one denoted by by ℓ is associated with Jahn-Teller effects and hence is localized whereas the other denoted by b is an extended state and propagates through the lattice. 2) The strong Hund’s couplingof ℓ and b electrons to the t2g core spin and 3) the strong Coulomb correlation between the two species of electrons. Additionally, the model includes a new doping dependent ferromagnetic exchange between the t2g core spins which can arise from “virtual double exchange” mechanism which will be discussed in great detail in Chapter 1 . Finally, we give a brief account on Dynamical Mean Field Theory (DMFT) and Numerical Renormalization Group (NRG) as an impurity solver for the single impurity problem arising under single site DMFT approximation. In Chapter 2 we study the effect of inter-site ℓ - b hybridization on the ‘ℓ - b’ model. The single impurity problem arising under DMFT approximation has close connection with the Vigman-Finkelshtein (VF)model. Then we briefly introduce the VF model and bring out its close connection with the impurity problem. We consider both the particle-hole symmetric as well as the U → ∞ particle-hole asymmetric cases. We derive various spectral functions at T = 0K and discuss the nature of fixed points under various circumstances. We explicitly show that for the particle-hole symmetric case the Hamiltonian flows from X-ray edge singularity fixed point to Free Electron fixed point under Renormalization Group transformation. This is evident from the spectral properties of the model. We write down the effective Hamiltonian at the free electron fixed point. For the particle-hole asymmetric case the model flows from X-ray edge singularity fixed point to Free Electron/Strong Coupling fixed point with additional potential scattering terms. We write down the effective Hamiltonian at this fixed point and derive various leading order deviations. We found all of them to be irrelevant in nature also most interestingly the quasi-particles describing the under lying Fermi liquid state are found to be asymptotically non-interacting. We also calculate the Fermi liquid parameter, z, by analyzing the energy level structure of a non-interacting Hamiltonian with effective renormalized parameter. Also, we consider the case of ‘self consistent bath hybridization’ without ℓ - b hybridization for Bethe lattice with infinite coordination. Low energy qualitative features are found to be same but some of the high energy features get qualitatively modified. In Chapter 3 we discuss the transport properties of doped manganites in the insulating phases and also the Hall effect in the metallic phase. In the first part of this chapter we calculate the resistivity based on the ‘ℓ - b’model and try to fit it to the semiconducting form: ρ(T )= ρ0(T /T0)−nexp[Δ(T )/kBT ] and extract the “transport gap”, Δ(T ). This gap can be characterized in terms of the “spectral gap” which can be defined for the ℓ - b model. It is found that the transport gap in the paramagnetic phase can be characterized in terms of the near constant “spectral gap” in this phase whereas the same in the ferromagnetic phase can be characterized in terms of the zero temperature spectral gap. In the last part of this chapter we calculate the Hall resistivity (ρxy) of these materials in the metallic phase. Ρxy is found to be negative and linear in applied field -quite consistent with the experimental findings but this fails to explain the positive linear Hall resistivity at low temperatures and its crossover as a function of field and temperature. We then present a reasonable explanation for this discrepancy and support it by calculating the Hall density of states for a two band “toy model” involving inter species hybridization. In Chapter 4 we calculate the optical conductivity, σ(ω), in ℓ - b model. σ(ω) arises from two independent processes. One of the processes involves ‘b’ electrons only and termed as ‘b - b channel’ and this gives rise to a Drude peak in the low frequency region. another process termed as the ‘ℓ - b channel’ involves hopping of an ℓ-electron to a neighbouring empty site and transforms into a ‘b’like state. This process gives rise to a broad mid-infrared peak. The total conductivity is the sum of contributions from these two incoherent channels. Calculated σ(ω) for metallic systems shows lot of similarities with experimental observations particularly the temperature evolution of the mid-infrared peak and the spectral weight transfer between the two peaks. But for the insulating systems the calculated optical conductivity showed trends similar to more recent experimental observations on some insulating systems (x =0.125) but contradicts with earlier experimental observations on some other insulating system (x =0.1). Finally, in the concluding chapter, we summarize results from all the chapters and also sketch some possible future directions of investigations.

The two-dimensional electronic systems (2DESs) have kept surprising physicists for the last few decades. Examples include the integer and fractional quantum Hall effects, cuprate superconductivity, and graphene. This thesis is intended to develop suitable theoretical tools which can be generalized to study new types of 2DESs with strong correlation feature. The first part of this thesis describes the investigation of heterostructures made by Mott insulators. This work is mostly motivated by the significant improvement of techniques for layer-by-layer growth of transition metal oxides in the last few years. We construct a toy model based on generalized Hubbard model complemented with long-ranged Coulomb interaction, and we study it by Hartree-Fock theory, dynamical mean-field theory, and Thomas-Fermi theory. We argue that interesting 2D strongly correlated electronic systems can be created in such heterostructures under several conditions. Since these 2D systems are formed entirely due to the gap generated by electron-electron interaction, they are not addiabatically connected to a noninteracting electron states. This feature makes these 2D systems distinguish from the ones created in semiconductor heterostructures, and they may be potential systems having non-Fermi liquid behaviors. The second part of this thesis is devoted to the study of collective excitations in high-temperature superconductors. One important achievement in this work is to develop a time-dependent mean-field theory for t-U-J-V model, an effective low energy model for cuprates. The time-dependent mean-field theory is proven to be identical to the generalized random-phase approximation (GRPA) which includes both the bubble and ladder diagrams. We propose that the famous 41 meV magnetic resonance mode observed in the inelastic neutron scattering measurements is a collective mode arising from a conjugation relation, which has been overlooked in previous work, between the antiferromagnetic fluctuation and the phase fluctuation of the d-wave superconducting order parameter near momentum ([pi, pi]). Furthermore, we find that this collective mode signals the strength of the antiferromagnetic fluctuations which are responsible for the suppression of the superfluid density in the underdoped cuprates even at zero temperature. Finally, we perform a complete analysis on an effective model with parameters fitted by experimental data of Bi2212 within the GRPA scheme and conclude that the short-range antiferromagnetic interactions which are a remnant of the parent Mott-insulator are more likely the pairing mechanism of the High-T[subscript c] cuprates. / text

This thesis is a critical examination of phenomenological nuclear mean field theories, focusing on reliable description of levels of individual particles. The approach presented here is new in the sense that it not only allows to predict the numerical values obtained with this formalism, but also yields an estimate of the probability distributions corresponding to the experimental results. We introduce the concept of 'theoretical errors' to estimate uncertainties in theoreticalmodels. We also introduce a subjective notion of 'Predictive Power' of nuclear Hamiltonians, which is analyzed in the context of the energy spectra of individual particles. The mathematical concept of 'Inverse Problem' is applied to a realistic mean-field Hamiltonian. This technique allows to predict the properties of a system from a limited number of data. To deepen our understanding of Inverse Problems, we focus on a simple mathematical problem. A function dependent on four free parameters is introduced in order to reproduce 'experimental' data. We study the behavior of the 'fitted' parameters, their correlation and the associated errors. This study helps us understand the importance of the correct formulation of the problem. It also shows the importance of including theoretical and experimental errors in the solution.