Spinney, RE; (2012) The Use of Stochastic Methods to Explore the Thermal Equilibrium Distribution and Define Entropy Production out of Equilibrium. Doctoral thesis , UCL (University College London).

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Abstract

This thesis contains two separate bodies of research, both in terms of the period of time in which the work was done and their content, and as such is presented in two parts each of which are summarised below. The first part concerns work on entropy production in stochastic systems and describes the breakage of time reversal symmetry that arises in irreversible stochastic processes that one can associate with an entropy production contribution for a single realisation. The paradigm utilised is that of Markovian dynamics expressed using master equations and stochastic differential equations. By generalising some previously reported concepts so as to explicitly concern odd variables, some recent advances in non-equilibrium thermodynamics are refined which are then illustrated with several examples. The place of such results within the existing literature, particularly the extensive literature on fluctuation theorems, is emphasised allowing us to simultaneously demonstrate some of the widely celebrated symmetry relations to emerge from the field in recent years. The second part concerns the construction and implementation of a new Markov chain sampling algorithm called spatially local parallel tempering which improves the scaling of computational effort with system size of the well known thermal equilibrium sampling algorithm, parallel tempering. Parallel tempering accelerates thermal equilibrium sampling by performing regular sampling techniques on a composite system of replicas, each possessing a different temperature, and introducing configurational exchanges between those replicas so as to acquire configurations that would otherwise take a long time to reach. However, as the system size increases, the number of replicas required, and therefore computational effort, increases faster than linearly. To avoid this we propose local variations where this is not the case. We demonstrate these claims on several simple one dimensional models and show that the algorithms can reproduce thermodynamic accuracy in one and two dimensions.

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