State space emergence: a new formalism

Abstract

This thesis focuses on redefining the notion of emergence to a mathematically tractable concept: emergence in state spaces. In doing that, we will study the probabilistic measures of state spaces with emergence, how to control their volume growth and the differences between them and typical state spaces. This study will introduce two stylistic models, intuitively simple and practically helpful. To provide statistical tools for modelling randomness with similar emerging properties, we will introduce different probability distributions from the first principle and derive their preliminary properties. At the same time, we will see that these results are expressible in closed form, by which we can analytically study the emergence in states. Also, for practical reasons, statistical inference will be revisited for distributions’ parameter estimation. Next, we briefly study systems with emerging properties in state spaces by using information-theoretic measures. Alongside that and inspired by the ideas from this discussion, we will propose a pairing time series that combines certainty and uncertainty. In addition, we prove that the Shannon entropy and the rate entropy are well-defined in various circumstances for infinite pairing time series. And finally, we show that standard statistical mechanics methods fail to yield thermodynamical quantities for some simplistic models with emerging states. We will propose a mathematical tool rooted in the geometry of emergence states spaces from the first part of the thesis to resolve this problem.