Parabolic Anderson Model on R^2

For my thesis project we have been studying the analysis of the parabolic Anderson model in 2 spatial dimensions on the whole plane, performed by Hairer and Labbe in early 2015. This problem is a nice example as it requires renormalization to control the singularities and weighted spaces to control the divergence at infinity. After adding the necessary logarithmic counter term and posing the problem in the correct space we are then able to prove existence and uniqueness of the solution. Our main contribution is to offer a more explicit account than was previously available, and to correct some typos in the original work. This work is of importance because the parabolic Anderson model, which models a random walk driven by a random potential, can be used to study several topics such as spectral theory and some variational problems. Moreover, this analysis is of interest because it presents a particularly clean example, in that there is no need for any complicated (though more general) renormalization procedures. Rather, we use a trick from the analysis of smooth partial differential equations to identify the diverging terms and then add an appropriate counter term.Mathematic

Ginzburg-Landau theory of the liquid-solid interface and nucleation for hard-spheres

The Ginzburg-Landau free energy functional for hard-spheres is constructed using the Fundamental Measure Theory approach to Density Functional Theory as a starting point. The functional is used to study the liquid-fcc solid planer interface and the properties of small solid clusters nucleating within a liquid. The surface tension for planer interfaces agrees well with simulation and it is found that the properties of the solid clusters are consistent with classical nucleation theory.Comment: Replacement 1. Minor changes to figure

Mechanism for the stabilization of protein clusters above the solubility curve: the role of non-ideal chemical reactions

Dense protein clusters are known to play an important role in nucleation of protein crystals from dilute solutions. While these have generally been thought to be formed from a metastable phase, the observation of similar, if not identical, clusters above the critical point for the dilute-solution/strong-solution phase transition has thrown this into doubt. Furthermore, the observed clusters are stable for relatively long times. Because protein aggregation plays an important role in some pathologies, understanding the nature of such clusters is an important problem. One mechanism for the stabilization of such structures was proposed by Pan, Vekilov and Lubchenko and was investigated using a DDFT model which confirmed the viability of the model. Here, we revisit that model and incorporate additional physics in the form of state-dependent reaction rates. We show by a combination of numerical results and general arguments that the state-dependent rates disrupt the stability mechanism. Finally, we argue that the state-depedent reactions correct unphysical aspects of the model with ideal (state-independent) reactions and that this necessarily leads to the failure of the proposed mechanism

Hydrodynamics of an inelastic gas with implications for sonochemistry

The hydrodynamics for a gas of hard-spheres which sometimes experience inelastic collisions resulting in the loss of a fixed, velocity-independent, amount of energy $\Delta$ is investigated with the goal of understanding the coupling between hydrodynamics and endothermic chemistry. The homogeneous cooling state of a uniform system and the modified Navier-Stokes equations are discussed and explicit expressions given for the pressure, cooling rates and all transport coefficients for D-dimensions. The Navier-Stokes equations are solved numerically for the case of a two-dimensional gas subject to a circular piston so as to illustrate the effects of the enegy loss on the structure of shocks found in cavitating bubbles. It is found that the maximal temperature achieved is a sensitive function of $\Delta$ with a minimum occuring near the physically important value of $\Delta \sim 12,000K \sim 1eV$Comment: 35 pages, 9 figure

Velocity correlations and the structure of nonequilibrium hard core fluids

A model for the pair distribution function of nonequilibrium hard-core fluids is proposed based on a model for the effect of velocity correlations on the structure. Good agreement is found with molecular dynamics simulations of granular fluids and of sheared elastic hard spheres. It is argued that the incorporation of velocity correlations are crucial to correctly modeling atomic scale structure in nonequilibrium fluids.Comment: Final corrections after referees' reports. To appear in PR

Atomic-scale structure of hard-core fluids under shear flow

The effect of velocity correlations on the equal-time density autocorrelation function, e.g. the pair distribution function or pdf, of a hard-sphere fluid undergoing shear flow is investigated. The pdf at contact is calculated within the Enskog approximation and is shown to be in good agreement with molecular dynamics simulations for shear rates below the shear-induced ordering transition. These calculations are used to construct a nonequilibrium generalised mean spherical approximation for the pdf at finite separations which is also found to agree well with the simulation data.Comment: 35 pages, 13 figures. To be submitted to PRE. Replacement: More data added to fig 8 and minor improvements to the tex