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Entropy Production in Small Systems: A study of the Jarzynski inequality

October 2012 - March 2013, Supervisor: Ian Ford

Within this investigation we have considered the Jarzynski equality and its generalisations to both systems with feedback control and to non-isothermal systems. To explore and confirm all results, we have modelled the system of a single classical particle, in one dimension trapped in a harmonic potential and in thermal contact with an overdamped langevin heat bath, undergoing a cyclic work process of a step up and step down in spring constant. For feedback control systems, we considered the maximisation of the efficacy parameter within the Sagawa-Ueda generalisation of the Jarzynski equality, where the feedback for our system involves either changing or not changing the spring constant of the potential dependent on what the position of the particle is at the start of the process. We have found that changing the spring constant for a longer amount of time allows for a greater value in the efficacy parameter. Also, we have confirmed the expected result that the greater the error in the measurement that the feedback is based on, the lower the efficacy parameter becomes. Lastly for this generalisation, we considered the characteristics of the optimal value in spring constant to change to: specifically, an increase for position close to the centre of the potential, a decrease when far from the centre, and remaining the same otherwise. Numerical results from modelling these characteristics indeed show an increase in the value of the efficacy parameter. Finally, we have shown that, for the same cyclic step process, the Jarzynski equality takes on a different generalisation in terms of Tsallis q-exponentials when considering non-isothermal systems. Numerical results confirm this new equality. We have also examined the Tsallis distributions that arise in the process for non-isothermal systems, and shown how the Crooks work relation can also be generalised in this case when considering a simple single step process. Numerical modelling implies a correlation to the theoretical prediction in this case, but further numerical confirmation is required.

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Social Networks and Health

March - June 2014, Supervisors: Frances Griffiths and Thomas House

Depression forms a major contribution to the global disease burden, playing a debilitating part in the lives of millions of people across the world. It affects a significant percentage of adolescents. Depression is characterised by a set of symptoms affected by social networks. Previous work, utilising the Add Health data set, used a binary model of “depressed” versus “not depressed”, and demonstrated that being “not depressed” can transmit over a friendship network. We aim to consider greater layers of complexity to the transmission of depressive symptoms. We do this by applying two changes to the previous work. First, we consider whether people change in their level of depression at all, and in what direction, i.e. getting worse or getting better over time. Second, we consider individual symptoms. We use data from 2194 young people aged 12-19 from the Add Health dataset. We examine changes in symptom levels and the dependency of the probabilities of worsening and improving of symptoms on the number of better off and worse off friends. We achieve this with empirical data analysis and parametric inference. We find that, for almost all symptoms, having more worse off friends makes it more likely for an individual to get worse, and less likely for them to improve, and vice versa for better off friends. This suggests that whether having more friends will make you emotionally healthier is dependent on the emotional state of the friends. We also find that individuals are more likely to change in symptoms than not at all, and no bias exists towards improving or worsening. Evidence also suggests that the change in symptom level follows an exponential distribution. This suggests that the change in symptom levels between time points occurs in one direction only, i.e. people keep getting worse or keep getting better between time points, instead of following a random walk. Therefore, the effects of the friendship network happen progressively.

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Interactions Between Colloidal Knots in Liquid Crystals

June - September 2014, Supervisor: Gareth Alexander

Colloids in nematic liquid crystals interact via the effects of the distortions they create in the director field and the defects formed in response to their existence. Through these interactions, they can chain together to form crystals controllable by external electromagnetic fields. Recent experimental work has lead to the production of colloids in the shape of (p, q) torus knots. We explore the far field effects of such colloids by calculating their dipole and quadrupole moments and find that these have components related to the toroidal and knot geometry that are not present in spherical colloids. Using these dipole and quadrupole moments, we calculate the interaction potential of pairs of links to harmonic order valid at large distances. This lays the grounding for future work considering the interactions of further types of knots, the structures that may be formed from multiple knotted colloids, and what properties these structures possess and what possible uses they may have in areas such as photonics.

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