In this dissertation, we study velocity distributions in granular gases. For granular systems at low density, kinetic theory reduces to the Boltzmann equation which is based on the assumption of molecular chaos. At large velocity scales, stationary solutions with power-law tails, f( v) ∼ v–σ, have been derived from the Boltzmann equation for spatially homogeneous granular systems [6]. The behavior of power-law tail is complete generic, holding for arbitrary dimension, arbitrary collision rules, and general collision rates. We find the non-Maxwellian steady states using event-driven molecular dynamics simulations. Firstly, power-law steady states are observed in driven systems where energy is injected rarely at large velocity scale V . The range of power-law tail shrinks when we increase the heating-dissipation ratio [special characters omitted], where NI and NC are number of injections and number of collisions, respectively. Then a crossover from a power-law to a stretched exponential distribution is developed when the heating-dissipation ratio [special characters omitted] is close to 1. It is the energy cascade from a few energetic particles to the overwhelming majority of slowly moving particles that causes the non-Maxwellian velocity distributions. Steady states with power-law tail are robust as long as the injection velocity scale V is essentially separated from the typical velocity scale v0. These steady states are shown to exist for a wide range of number densities, different combinations of injection velocities and injection rates. The injection velocity scale V, the typical velocity scale v0, and the injection rate per particle are related by energy balance. This energy balance relation is confirmed by data collapse of velocity distributions for various choices of parameters.

In this work, we have developed a self-consistent field theory (SCFT) for polyelectrolytic systems and studied four important problems of contemporary interest: microphase separation in the melts of charged-neutral diblock copolymers, confinement effects on flexible polyelectrolytes, counterion adsorption on single flexible polyelectrolyte chain and the origin of translocation barriers in polyelectrolytic systems. Using the theory, we have been able to capture the effects of the degree of ionization, salt concentration, electrostatic and the excluded volume interaction strengths, degree of polymerization, role of architecture and solvent quality on these polyelectrolytic systems. Within saddle-point approximation, the polyelectrolyte chain configuration is described as a walk in the presence of fields coming from the excluded volume interactions and the other effects such as incompressibility in addition to the electrostatic potential. The electrostatic potential, on the other hand, is obtained from Poisson-Boltzmann like equation. So, in contrast to the SCFT for neutral polymers, there are two coupled non-linear equations namely modified diffusion equation describing the walk in the fields and the Poisson-Boltzmann equation, which have to be solved self-consistently. In this work, we have developed various numerical schemes to solve these coupled non-linear sets of equations. Furthermore, comparison of the SCFT results with a previous developed variational theory for polyelectrolytes has been carried out. Also, systematic expansions around the saddle-point results have been carried out to capture the effects of the density fluctuations of the small ions in the systems.

<p> Originally proposed in high energy physics as particles, which are their own anti-particles, Majorana fermions have never been observed in experiments. However, possible signatures of their condensed matter analog, zero energy, charge neutral, quasiparticle excitations, known as Majorana zero modes (MZMs), are beginning to emerge in experimental data. The primary method of engineering topological superconductors capable of supporting MZMs is through proximity-coupled semiconductor nanowires with strong Rashba spin-orbit coupling and an applied magnetic field. Recent tunneling transport experiments involving these materials, known as semiconductor-superconductor heterostructures, were capable for the first time of measuring quantized zero bias conductance plateaus, which are robust over a range of control parameters, long believed to be the smoking gun signature of the existence of MZMs. The possibility of observing Majorana zero modes has garnered great excitement within the field due to the fact that MZMs are predicted to obey non-Abelian quantum statistics and therefore are the leading candidates for the creation of qubits, the building blocks of a topological quantum computer. In this work, we first give a brief introduction to Majorana zero modes and topological quantum computing (TQC). We emphasize the importance that having a true topologically protected state, which is not dependent on local degrees of freedom, has with regard to non-Abelian braiding calculations. We then introduce the concept of partially separated Andreev bound states (ps-ABSs) as zero energy states whose constituent Majorana bound states (MBSs) are spatially separated on the order of the Majorana decay length. Next, through numerical calculation, we show that the robust 2<i> e²/h</i> zero bias conductance plateaus recently measured and claimed by many in the community to be evidence of having observed MZMs for the first time, can be identically created due to the existence of ps-ABSs. We use these results to claim that all localized tunneling experiments, which have been until now the main way researchers have tried to measure MZMs, have ceased to be useful. Finally, we outline a two-terminal tunneling experiment, which we believe to be relatively straight forward to implement and fully capable of distinguishing between ps-ABSs and true topologically protected MZMs.</p><p>

<p> In this dissertation, we investigate the approach of pure SU(2) lattice gauge theory to its continuum limit using the deconfinement temperature, six gradient scales, and six cooling scales. We find that cooling scales exhibit similarly good scaling behavior as gradient scales, while being computationally more efficient. In addition, we estimate systematic error in continuum limit extrapolations of scale ratios by comparing standard scaling to asymptotic scaling. Finally we study topological observables in pure SU(2) using cooling to smooth the gauge fields, and investigate the sensitivity of cooling scales to topological charge. We find that large numbers of cooling sweeps lead to metastable charge sectors, without destroying physical instantons, provided the lattice spacing is fine enough and the volume is large enough. Continuum limit estimates of the topological susceptibility are obtained, of which we favor &chi;^1/4/<i>T_c</i> = 0.643(12). Differences between cooling scales in different topological sectors turn out to be too small to be detectable within our statistical error.</p><p>

<p> Large vacuum fluctuations of a quantum stress tensor can be described by the asymptotic behavior of its probability distribution. Here we focus on stress tensor operators which have been averaged with a sampling function in time. The Minkowski vacuum state is not an eigenstate of the time-averaged operator, but can be expanded in terms of its eigenstates. We calculate the probability distribution and the cumulative probability distribution for obtaining a given value in a measurement of the time-averaged operator taken in the vacuum state. In these calculations, we use the normal ordered square of the time derivative of a massless scalar field in Minkowski spacetime as an example of a stress tensor operator. We analyze the rate of decrease of the tail of the probability distribution for different temporal sampling functions, such as compactly supported functions and the Lorentzian function. We find that the tails decrease relatively slowly, as exponentials of fractional powers, in agreement with previous work using the moments of the distribution. Our results lead additional support to the conclusion that large vacuum stress tensor fluctuations are more probable than large thermal fluctuations, and may have observable effects.</p><p>

<p> Elastic lepton scattering off of a nucleon has proven to be an efficient tool to study the structure of the hadron. In particular, the spatial distributions of the nucleon's charge and magnetization can be accessed through measurements of its electric (<i>G_E</i>) and magnetic (<i>G_M</i>) form factors. These form factors can be extracted from unpolarized cross sections measurements by using the Rosenbluth separation technique. At the current level of accuracy, a determination of <i>G_E</i> and <i>G_M</i> from an analysis of elastic lepton-nucleon scattering data requires effects beyond the leading-order (Born) approximation to be taken into account. </p><p> In this work, I study higher-order QED corrections to elastic lepton-nucleon scattering. First of all, I perform a model-independent calculation of conventional charge-dependent contributions in unpolarized lepton-proton scattering without making use of ultra-relativistic approximations. Second, in a connection to the future MUSE experiment in Switzerland, I estimate helicity-flip meson exchanges that make a difference in a comparison of ultra-relativistic vs non-ultra-relativistic lepton-proton scattering. Finally, I present a model calculation of the target-normal single-spin asymmetry in elastic electron-nucleon scattering. Such an asymmetry gives us a direct tool to studies of the imaginary part of the two-photon exchange amplitude.</p><p>

We work with two different mechanisms to generate geometric frustration on thin elastic sheets; isotropic differential growth and folding. We describe how controlled growth and prescribing folding patterns are useful tools for designing three-dimensional objects from information printed in two dimensions. The first mechanism is inspired by the possibility to control shapes by swelling polymer films, where we propose a solution for the problem of shape formation by asking the question, “what 2D metric should be prescribed to achieve a given 3D shape?”', namely the reverse problem. We choose two different types of initial configurations of sheets, disk-like with one boundary and annular with two boundaries. We demonstrate our technique by choosing four examples of 3D axisymmetric shapes and finding the respective swelling factors to achieve the desired shape. Second, we present a mechanical model for a single curved fold that explains both the buckled shape of a closed fold and its mechanical stiffness. The buckling arises from the geometrical frustration between the prescribed crease angle and the bending energy of the sheet away from the crease. This frustration increases as the sheet's area increases. Stiff folds result in creases with constant space curvature while softer folds inherit the broken symmetry of the buckled shape. We extend the application of our numerical model to show the potential to study multiple fold structures.

In this dissertation the parity violating neutron deuteron scattering amplitudes are calculated using pionless effective field theory to leading order. The five low energy parity violating constants present in pionless effective field theory are estimated by matching onto the ``best" values for the parameters of the model by Desplanques, Donoghue, and Holstein (DDH). Using these estimates and the calculated amplitudes, predictions for the spin rotation of a neutron through a deuteron target are given with a value of 1.8 × 10-8 rad cm-1. Also given are the longitudinal analyzing power in neutron deuteron scattering with a polarized neutron yielding 2.2 × 10-8, and a polarized deuteron giving 4.0 × 10-8. These observables are discussed in the broader context of hadronic parity violation and as possible future experiments to determine the values of the five low energy parity violating constant present in pionless effective theory.

The phase behavior of chiral block copolymers (BCPs*), namely, BCPs with at least one of the constituent block is formed by chiral monomers, is studied both experimentally and theoretically. Specifically, the formation of a unique morphology with helical sense, the H* phase, where the chiral block forms nanohelices hexagonally embedded in the matrix of achiral block, is investigated. Such unique morphology was first observed in the cast film of polystyrene- b-poly(L-lactide) (PS-b-PLLA) from a neutral solvent dichloromethane at room temperature with all the nanohelices being left-handed, which would switch to right-handed if the PLLA block changes to PDLA. Further studies revealed that such morphology only forms when the chiral PLLA block possesses certain volume fraction (from 0.32 to 0.36), and the molecular weight exceeds certain critical value (around 20,000 to 25,000 g/mol). Achiral phases such as lamellae, gyroid, cylinder, and sphere will form if the above criteria are not satisfied. Even though the unique H* phase has been extensively studied and utilized for many applications, many fundamental and important questions remain unanswered for such BCP* system. Specifically, how does the molecular level chirality transfer from the several-angstrom scale of the lactide monomer to the tens-of-nanometer size scale of the H* domain morphology? Why is the chirality transfer not automatic for this BCP* system? Is H* phase a thermodynamic stable or metastable phase? Are there other novel phases other than the H* phase that could form within the BCP* system? We aimed at providing answers to the abovementioned questions regarding the formation of chiral H* phase, which is no longer limited to the PS- b-PLLA/PDLA system. We divided our studies into both experimental and theoretical parts. In the experiments, we studied the effect of solvent casting conditions, including solvent removal rate and polymer-solvent interactions, on the formation of the H* phase in PS-b-PLLA/PDLA BCPs*. In addition, we monitored the morphological evolution during solvent casting using time-resolved x-ray scattering technique. We found that good solubility towards both PS and PLLA/PDLA blocks are required for the formation of the H* phase, and microphase separation has to happen prior to crystallization of chiral block. Most importantly, we found that crystalline ordering is not necessary for the H* phase formation. This result led us to propose melt-state twisted molecular packing as the underlying driving force for such helical phase to form, and began our work on the theory for BCPs*. First we built the theoretical tool by incorporating the orientational segmental interactions into the self-consistent field theory (SCFT) for BCPs. As a demonstration, we constructed the phase diagrams for one-dimensional (1D) and two-dimensional (2D) phases, for achiral BCPs with different orientational stiffness. We found that orientational stiffness could serve as another parameter to introduce asymmetry into BCP systems, in addition to conformational and architectural asymmetry. This model was further applied to study the phase behavior of BCPs*, and two phase diagrams were constructed. Another chiral phase, wavy lamellae (L* phase), was observed for BCPs*. The H* phase was found to be a thermodynamic stable phase, as long as the segregation strength χN and chiral strength q0 exceed certain critical values. Energetically favorable cholesteric texture was observed for the chiral segment packing inside the H* phase, which is believed to drive such unusual morphology to form. A simple geometrical argument based on bending of cylindrical microdomain and twisted packing of the bended microdomain can be given to explain the nonlinear chiral sensitivity of BCP* morphology, which further explains the non-automatic feature of chirality transfer in such system.

We investigate the properties of the Kitaev honeycomb model with both site dilution and exchange randomness. Embarking on this work, we review disorder in some strongly correlated electron systems, including spin-½ and spin-1 Heisenberg antiferromagnetic chains, two dimensional Heisenberg antiferromagnets, the cuprates and graphene. We outline some aspects of resonating valence bond phases, valence bond solids, spin liquids and quantum computation that are pertinent to an understanding of the Kitaev model. The properties of the Kitaev model without disorder are discussed and it is found to be a critical spin liquid, with algebraic correlations in two spin operators sigma^{alpha}_{i}sigma^{alpha}_{j}, where i and j,/em> are either end of a link of type alpha = x, y or z on the honeycomb lattice. The Kitaev model is exactly solvable and we show that this remains so in the presence of site dilution and exchange randomness. We find that vacancies bind a flux. In the gapped phase, a vacancy forms an effective paramagnetic moment. With two or more vacancies we describe the interaction of their effective moments and show that a finite density of vacancies leads to a divergent macroscopic susceptibility at small fields. In the gapless phase the effective moment has a susceptibility that is, to leading order at small fields, chi(h)~log(1/h). Interaction between the moments from two vacancies on opposite sublattices cuts off this divergence in susceptibility at a large but finite constant. Two vacancies on the same sublattice behave quite differently and we find the combined susceptibility is parametrically larger than that of an isolated vacancy, chi(h)sim [h(log(1/h))^{3/2}]^{-1}. We also investigate the effects of slowly varying, quenched disorder in exchange coupling. We demonstrate that this does not qualitatively affect the susceptibility but show that the heat capacity C ~ T^{2/z}, where z is a measure of the disorder and increases from one with increasing disorder strength.