| Summary: | Firstly we look at chirally imprinted elastomers. These are liquid crystalline elastomers with an internal helical structure, created by crosslinking a liquid crystal polymer in the presence of a chiral solvent. We take an existing theoretical model of the imprinting process, and add the freedom to elastically deform, and for the director array to rotate toward the helix axis. This predicts a variety of new kinds of behaviours, including new and discontinuous director rotations, and alterations to the criteria for successful imprinting. The developments in the model then allow us to make predictions about the effect of mechanically stretching such an elastomer, or of applying a constant electric field. It is found that at a critical applied field or strain, the imprinted helices may be made to undergo discontinuous unwinding. Secondly we study the problem of competitive random sequential adsorption in one dimension. This is similar to the problem of <i>Random Parking</i>, where cars of unit length sequentially attempt to park at random points along a road of length <i>x</i>. During the process, gaps smaller than unity appear between cars, in to which no further cars can be parked. This means that only a certain fraction of the road can be filled, and Alfred Renyi’s parking constant (º 0.7476...) gives this fraction for an infinite road. The problem of “competitive parking” is identical except that the length of each car is drawn from a given distribution. By generalising Renyi’s method we derive an expression for the filling fraction of an infinite line given an arbitrary distribution of finite width. Thirdly we consider the “Totally Asymmetric Exclusion Process” or TASEP. This consists of a finite one dimensional lattice where each site may either by singly occupied, or empty. Particles may hop one site to the right if a space is available, and are fed in the left end of the lattice at a rate <i>a</i> and removed from the right end at a rate <i>b</i>. The system exhibits a range of steady states which fall into three phases, governed by the input and output rates. An exact solution for the steady state occupation numbers and correlation functions exists in the literature. However, here we develop an exact path integral description of the problem where we hope has the potential to shed light on the broader class of driven systems to which the TASEP belongs. We perform a mapping from the master equation formulation of the problem onto a bosonic many body theory, and then switch to the path integral formulation in the standard way. After careful consideration of the symmetries of the system, we recover the classic mean field equations for the problem, which give the correct phase behaviour. By considering fluctuations about the mean field solutions, we hope in the future to demonstrate the existence of the phase transitions given by the exact solution.
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