2009/01/20 by Charlotte Werndl · 6 citations
Physics and Astronomy · Mathematics · Economics, Econometrics and Finance · #Statistical Mechanics and Entropy #Probability and Statistical Research #Complex Systems and Time Series Analysis
paper · doi:10.1093/bjps/axn053
From the beginning of chaos research until today, the unpredictability of chaos has been a central theme. It is widely believed and claimed by philosophers, mathematicians and physicists alike that chaos has a new implication for unpredictability, meaning that chaotic systems are unpredictable in a way that other deterministic systems are not. Hence, one might expect that the question ‘What are the new implications of chaos for unpredictability?’ has already been answered in a satisfactory way. However, this is not the case. I will critically evaluate the existing answers and argue that they do not fit the bill. Then I will approach this question by showing that chaos can be defined via mixing, which has never before been explicitly argued for. Based on this insight, I will propose that the sought-after new implication of chaos for unpredictability is the following: for predicting any event, all sufficiently past events are approximately probabilistically irrelevant. 1. Introduction2. Dynamical Systems and Unpredictability 2.1. Dynamical systems2.2. Natural invariant measures2.3. Unpredictability3. Chaos 3.1. Defining chaos3.2. Defining chaos via mixing4. Criticism of Answers in the Literature 4.1. Asymptotic unpredictability?4.2. Unpredictability due to rapid or exponential divergence?4.3. Macro-predictability and Micro-unpredictability?5. A General New Implication of Chaos for Unpredictability 5.1. Approximate probabilistic irrelevance5.2. Sufficiently past events are approximately probabilistically irrelevant for predictions6. Conclusion