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Title: Studying partially ordered systems using stochastic differential equations
Author: Rolls, Edward
ISNI:       0000 0004 8502 7041
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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This thesis looks at two different problems in the field of partially ordered systems - modelling biopolymers on multiple scales as well as modelling two-dimensional nematic liquid crystal wells - and seeks to gain insight using stochastic differential equations, through mathematical analysis and numerical simulations. In the field of biopolymers, we generalize existing models for a coarse-grained polymer to multiple resolutions, which allows for faster computations while maintaining a high level of detail in regions of interest. This is done for two bead spring models; the Rouse model, which does not account for hydrodynamic interactions, as well as the bead-spring model with hydrodynamic interactions. For both models mathematical analysis is done to find expressions for global statistics to derive scaling laws for the models. This is implemented by generalizing existing algorithms using a multi-resolution approach which utilizes multiple spatial and temporal scales. These simulations show that the scaling laws for multiple resolutions match expected analytic results through simulations. For the Rouse model, a model for the binding of a protein to DNA is introduced, along with a mechanism for changing resolutions along the filament on-the-fly. Within reasonable error, the results for the chain in multiple resolutions matches the chain modelled entirely in high-resolution. The work on nematic liquid crystals is concerned with a stochastic dynamics model of nematic liquid crystals contained within a square well, looking at their behaviour under Brownian motion using the Gay-Berne potential. In particular, we are interested in the parameter regimes which are responsible for different types of steady states to form in the system and whether steady states suggested by continuum modelling are stable.
Supervisor: Erban, Radek Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics