Charlotte Werndl

What Are the New Implications of Chaos for Unpredictability

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 Introduction Dynamical Systems and Unpredictability 2.1. Dynamical systems2.2. Natural invariant measures2.3. Unpredictability Dynamical systems Natural invariant measures Unpredictability Chaos 3.1. Defining chaos3.2. Defining chaos via mixing Defining chaos Defining chaos via mixing 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? Asymptotic unpredictability? Unpredictability due to rapid or exponential divergence? Macro-predictability and Micro-unpredictability? A General New Implication of Chaos for Unpredictability 5.1. Approximate probabilistic irrelevance5.2. Sufficiently past events are approximately probabilistically irrelevant for predictions Approximate probabilistic irrelevance Sufficiently past events are approximately probabilistically irrelevant for predictions Conclusion

Climate Models, Calibration and Confirmation

We argue that concerns about double-counting—using the same evidence both to calibrate or tune climate models and also to confirm or verify that the models are adequate—deserve more careful scrutiny in climate modelling circles. It is widely held that double-counting is bad and that separate data must be used for calibration and confirmation. We show that this is far from obviously true, and that climate scientists may be confusing their targets. Our analysis turns on a Bayesian/relative-likelihood approach to incremental confirmation. According to this approach, double-counting is entirely proper. We go on to discuss plausible difficulties with calibrating climate models, and we distinguish more and less ambitious notions of confirmation. Strong claims of confirmation may not, in many cases, be warranted, but it would be a mistake to regard double-counting as the culprit. 1 Introduction 2 Remarks about Models and Adequacy-for-Purpose 3 Evidence for Calibration Can Also Yield Comparative Confirmation   3.1 Double-counting I   3.2 Double-counting II 4 Climate Science Examples: Comparative Confirmation in Practice   4.1 Confirmation due to better and worse best fits   4.2 Confirmation due to more and less plausible forcings values 5 Old Evidence 6 Doubts about the Relevance of Past Data 7 Non-comparative Confirmation and Catch-Alls 8 Climate Science Example: Non-comparative Confirmation and Catch-Alls in Practice 9 Concluding Remarks 1 Introduction 2 Remarks about Models and Adequacy-for-Purpose 3 Evidence for Calibration Can Also Yield Comparative Confirmation   3.1 Double-counting I   3.2 Double-counting II   3.1 Double-counting I   3.2 Double-counting II 4 Climate Science Examples: Comparative Confirmation in Practice   4.1 Confirmation due to better and worse best fits   4.2 Confirmation due to more and less plausible forcings values   4.1 Confirmation due to better and worse best fits   4.2 Confirmation due to more and less plausible forcings values 5 Old Evidence 6 Doubts about the Relevance of Past Data 7 Non-comparative Confirmation and Catch-Alls 8 Climate Science Example: Non-comparative Confirmation and Catch-Alls in Practice 9 Concluding Remarks

Explaining Thermodynamic-Like Behavior in Terms of Epsilon-Ergodicity

Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections in particular arguments based on the Kolmogorov-Arnold-Moser theorem and the Markus-Meyer theorem are beside the point. We then argue that something close to Boltzmann’s proposal is true: gases behave thermodynamic-like if they are epsilon-ergodic, that is, ergodic on the phase space except for a small region of measure epsilon. This answer is promising because there is evidence that relevant systems are epsilon-ergodic.

On choosing between deterministic and indeterministic models: underdetermination and indirect evidence

There are results which show that measure-theoretic deterministic models and stochastic models are observationally equivalent. Thus there is a choice between a deterministic and an indeterministic model and the question arises: Which model is preferable relative to evidence? If the evidence equally supports both models, there is underdetermination. This paper first distinguishes between different kinds of choice and clarifies the possible resulting types of underdetermination. Then a new answer is presented: the focus is on the choice between a Newtonian deterministic model supported by indirect evidence from other Newtonian models which invoke similar additional assumptions about the physical systems and a stochastic model that is not supported by indirect evidence. It is argued that the deterministic model is preferable. The argument against underdetermination is then generalised to a broader class of cases. Finally, the paper criticises the extant philosophical answers in relation to the preferable model. Winnie’s (1998) argument for the deterministic model is shown to deliver the correct conclusion relative to observations which are possible in principle and where there are no limits, in principle, on observational accuracy (the type of choice Winnie was concerned with). However, in practice the argument fails. A further point made is that Hoefer’s (2008) argument for the deterministic model is untenable.

On Defining Climate and Climate Change

The aim of the article is to provide a clear and thorough conceptual analysis of the main candidates for a definition of climate and climate change. Five desiderata on a definition of climate are presented: it should be empirically applicable; it should correctly classify different climates; it should not depend on our knowledge; it should be applicable to the past, present, and future; and it should be mathematically well-defined. Then five definitions are discussed: climate as distribution over time for constant external conditions; climate as distribution over time when the external conditions vary as in reality; climate as distribution over time relative to regimes of varying external conditions; climate as the ensemble distribution for constant external conditions; and climate as the ensemble distribution when the external conditions vary as in reality. The third definition is novel and is introduced as a response to problems with existing definitions. The conclusion is that most definitions encounter serious problems and that the third definition is most promising. 1   Introduction 2   Climate Variables and a Simple Climate Model 3   Desiderata on a Definition of Climate 4   Climate as Distribution over Time    4.1   Definition 1: Distribution over time for constant external conditions    4.2   Definition 2: Distribution over time when the external conditions vary as in reality    4.3   Definition 3: Distribution over time for regimes of varying external conditions    4.4   Infinite versions 5   Climate as Ensemble Distribution    5.1   Definition 4: Ensemble distribution for constant external conditions    5.2   Definition 5: Ensemble distribution when the external conditions vary as in reality    5.3   Infinite versions 6   Conclusion Appendix   1   Introduction 2   Climate Variables and a Simple Climate Model 3   Desiderata on a Definition of Climate 4   Climate as Distribution over Time    4.1   Definition 1: Distribution over time for constant external conditions    4.2   Definition 2: Distribution over time when the external conditions vary as in reality    4.3   Definition 3: Distribution over time for regimes of varying external conditions    4.4   Infinite versions    4.1   Definition 1: Distribution over time for constant external conditions    4.2   Definition 2: Distribution over time when the external conditions vary as in reality    4.3   Definition 3: Distribution over time for regimes of varying external conditions    4.4   Infinite versions 5   Climate as Ensemble Distribution    5.1   Definition 4: Ensemble distribution for constant external conditions    5.2   Definition 5: Ensemble distribution when the external conditions vary as in reality    5.3   Infinite versions    5.1   Definition 4: Ensemble distribution for constant external conditions    5.2   Definition 5: Ensemble distribution when the external conditions vary as in reality    5.3   Infinite versions 6   Conclusion Appendix  

Reconceptualising equilibrium in Boltzmannian statistical mechanics and characterising its existence

In Boltzmannian statistical mechanics macro-states supervene on micro-states. This leads to a partitioning of the state space of a system into regions of macroscopically indistinguishable micro-states. The largest of these regions is singled out as the equilibrium region of the system. What justifies this association? We review currently available answers to this question and find them wanting both for conceptual and for technical reasons. We propose a new conception of equilibrium and prove a mathematical theorem which establishes in full generality -- i.e. without making any assumptions about the systems dynamics or the nature of the interactions between its components -- that the equilibrium macro-region is the largest macro-region. We then turn to the question of the approach to equilibrium, of which there exists no satisfactory general answer so far. In our account, this question is replaced by the question when an equilibrium state exists. We prove another -- again fully general -- theorem providing necessary and sufficient conditions for the existence of an equilibrium state. This theorem changes the way in which the question of the approach to equilibrium should be discussed: rather than launching a search for a crucial factor (such as ergodicity or typicality), the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem.

Rethinking Boltzmannian Equilibrium

Boltzmannian statistical mechanics partitions the phase space of a system into macroregions, and the largest of these is identified with equilibrium. What justifies this identification? Common answers focus on Boltzmann’s combinatorial argument, the Maxwell-Boltzmann distribution, and maximum entropy considerations. We argue that they fail and present a new answer. We characterize equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilibrium thus defined corresponds to the largest macroregion. Our derivation is completely general and does not rely on assumptions about the dynamics or interparticle interactions.

Mind the Gap: Boltzmannian versus Gibbsian Equilibrium

There are two main theoretical frameworks in statistical mechanics, one associated with Boltzmann and the other with Gibbs. Despite their well-known differences, there is a prevailing view that equilibrium values calculated in both frameworks coincide. We show that this is wrong. There are important cases in which the Boltzmannian and Gibbsian equilibrium concepts yield different outcomes. Furthermore, the conditions under which equilibriums exists are different for Gibbsian and Boltzmannian statistical mechanics. There are, however, special circumstances under which it is true that the equilibrium values coincide. We prove a new theorem providing sufficient conditions for this to be the case.

A New Approach to the Approach to Equilibrium

Consider a gas confined to the left half of a container. Then remove the wall separating the two parts. The gas will start spreading and soon be evenly distributed over the entire available space. The gas has approached equilibrium. Why does the gas behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the system has to be ergodic for the approach to equilibrium to take place. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that these criticisms have dismissed Boltzmann’s answer too quickly and that something almost like Boltzmann’s answer is true: the approach to equilibrium takes place if the system is epsilon-ergodic, i.e. ergodic on the entire accessible phase space except for a small region of measure epsilon. We introduce epsilon-ergodicity and argue that relevant systems in statistical mechanics are indeed espsilon-ergodic.

Model tuning in engineering: uncovering the logic

In engineering, as in other scientific fields, researchers seek to confirm their models with real-world data. It is common practice to assess models in terms of the distance between the model outputs and the corresponding experimental observations. An important question that arises is whether the model should then be ‘tuned’, in the sense of estimating the values of free parameters to get a better fit with the data, and furthermore whether the tuned model can be confirmed with the same data used to tune it. This dual use of data is often disparagingly referred to as ‘double-counting’. Here, we analyse these issues, with reference to selected research articles in engineering (one mechanical and the other civil). Our example studies illustrate more and less controversial practices of model tuning and double-counting, both of which, we argue, can be shown to be legitimate within a Bayesian framework. The question nonetheless remains as to whether the implied scientific assumptions in each case are apt from the engineering point of view.