Universiteit Leiden

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Gibbs-non-Gibbs transitions and vector-valued integration

Promotor: W. T. F. den Hollander, F. H. J. Redig

W.B. van Zuijlen
07 september 2016
Thesis in Leiden Repository

This thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of components that is in equilib- rium. It may happen that when the system is transformed, for example, by a stochastic dynamics that runs over a certain time interval, the evolved state is no longer a Gibbs measure. We study transitions from Gibbs tot non-Gibbs for mean-field systems and their relation to the large deviation rate function that is related to those systems. In the second part of the thesis we describe different notions of integrals for functions with values in a partially ordered vector space. We describe two extensions for integrals, called the vertical and the lateral extension. We compare combinations of them and compare them to other integrals. Another integral can be obtained for Archimedean directed ordered vector spaces, as they can be covered by Banach spaces in a natural way. This allows us to generalise the Bochner integral to function with values in such space.