Charlotte Werndl

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. 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

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.

Entropy is ubiquitous in physics, and it plays important roles in numerous other disciplines ranging from logic and statistics to biology and economics. However, a closer look reveals a complicated picture: entropy is defined differently in different contexts, and even within the same domain different notions of entropy are at work. Some of these are defined in terms of probabilities, others are not. The aim of this chapter is to arrive at an understanding of some of the most important notions of entropy and to clarify the relations between them, After setting the stage by introducing the thermodynamic entropy, we discuss notions of entropy in information theory, statistical mechanics, dynamical systems theory and fractal geometry

The received wisdom in statistical mechanics is that isolated systems, when left to themselves, approach equilibrium. But under what circumstances does an equilibrium state exist and an approach to equilibrium take place? In this paper we address these questions from the vantage point of the long-run fraction of time definition of Boltzmannian equilibrium that we developed in two recent papers. After a short summary of Boltzmannian statistical mechanics and our definition of equilibrium, we state an existence theorem which provides general criteria for the existence of an equilibrium state. We first illustrate how the theorem works with a toy example, which allows us to illustrate the various elements of the theorem in a simple setting. After a look at the ergodic programme, we discuss equilibria in a number of different gas systems: the ideal gas, the dilute gas, the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas. In the conclusion we briefly summarise the main points and highlight open questions.

Equilibrium is a central concept of statistical mechanics. In previous work we introduced the notions of a Boltzmannian alpha-epsilon-equilibrium and a Boltzmannian gamma-epsilon-equilibrium. This was done in a deterministic context. We now consider systems with a stochastic micro-dynamics and transfer these notions from the deterministic to the stochastic context. We then prove stochastic equivalents of the Dominance Theorem and the Prevalence Theorem. This establishes that also in stochastic systems equilibrium macro-regions are large in requisite sense.

Boltzmannian statistical mechanics (BSM) partitions a system’s space of micro-states into cells and refers to these cells as ‘macro-states’. One of these cells is singled out as the equilibrium macro-state while the others are non-equilibrium macro-states. It remains unclear, however, how these states are characterised at the macro-level as long as only real-valued macro-variables are available. We argue that physical quantities like pressure and temperature should be treated as field-variables and show how field variables fit into the framework of our own version of BSM, the long-run residence time account of BSM. The introduction of field variables into the theory makes it possible to give a full macroscopic characterisation of the approach to equilibrium.

We present a definition of equilibrium for Boltzmannian statistical mechanics based on the long-run fraction of time a system spends in a state. We then formulate and prove an existence theorem which provides general criteria for the existence of an equilibrium state. We illustrate how the theorem works with toy example. After a look at the ergodic programme, we discuss equilibria in a number of different gas systems: the ideal gas, the dilute gas, the Kac gas, the stadium gas, the mushroom gas and the multi-mushroom gas.

This article argues that common intuitions regarding (a) the specialness of ‘use-novel’ data for confirmation and (b) that this specialness implies the ‘no-double-counting rule’, which says that data used in ‘constructing’ (calibrating) a model cannot also play a role in confirming the model’s predictions, are too crude. The intuitions in question are pertinent in all the sciences, but we appeal to a climate science case study to illustrate what is at stake. Our strategy is to analyse the intuitive claims in light of prominent accounts of confirmation of model predictions. We show that on the Bayesian account of confirmation, and also on the standard classical hypothesis-testing account, claims (a) and (b) are not generally true; but for some select cases, it is possible to distinguish data used for calibration from use-novel data, where only the latter confirm. The more specialized classical model-selection methods, on the other hand, uphold a nuanced version of claim (a), but this comes apart from (b), which must be rejected in favour of a more refined account of the relationship between calibration and confirmation. Thus, depending on the framework of confirmation, either the scope or the simplicity of the intuitive position must be revised. 1 Introduction2 A Climate Case Study3 The Bayesian Method vis-à-vis Intuitions4 Classical Tests vis-à-vis Intuitions5 Classical Model-Selection Methods vis-à-vis Intuitions 5.1 Introducing classical model-selection methods 5.2 Two cases6 Re-examining Our Case Study7 Conclusion.

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory. In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this paper clarifies von Neumann's contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the paper is that von Neumann's argument described in his letter to Ulam is the very first proof that spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic.

There are two theoretical approaches in statistical mechanics, one associated with Boltzmann and the other with Gibbs. The theoretical apparatus of the two approaches offer distinct descriptions of the same physical system with no obvious way to translate the concepts of one formalism into those of the other. This raises the question of the status of one approach vis-à-vis the other. We answer this question by arguing that the Boltzmannian approach is a fundamental theory while Gibbsian statistical mechanics is an effective theory, and we describe circumstances under which Gibbsian calculations coincide with the Boltzmannian results. We then point out that regarding GSM as an effective theory has important repercussions for a number of projects, in particular attempts to turn GSM into a nonequilibrium theory.

Gases reach equilibrium when left to themselves. Why do they behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that some of the main arguments against Boltzmann's answer, in particular, arguments based on the KAM-theorem and the Markus-Meyer theorem, are beside the point. We then argue that something close to Boltzmann's original proposal is true for gases: gases behave thermodynamic-like if they are epsilon-ergodic, i.e., ergodic on the entire accessible phase space except for a small region of measure epsilon. This answer is promising because there are good reasons to believe that relevant systems in statistical mechanics are epsilon-ergodic

A gas prepared in a non-equilibrium state will approach equilibrium and stay there. An influential contemporary approach to Statistical Mechanics explains this behaviour in terms of typicality. However, this explanation has been criticised as mysterious as long as no connection with the dynamics of the system is established. We take this criticism as our point of departure. Our central claim is that Hamiltonians of gases which are epsilon-ergodic are typical with respect to the Whitney topology. Because equilibrium states are typical, we argue that there follows the desired conclusion that typical initial conditions approach equilibrium and stay there

Many examples of calibration in climate science raise no alarms regarding model reliability. We examine one example and show that, in employing Classical Hypothesis-testing, it involves calibrating a base model against data that is also used to confirm the model. This is counter to the "intuitive position". We argue, however, that aspects of the intuitive position are upheld by some methods, in particular, the general Cross-validation method. How Cross-validation relates to other prominent Classical methods such as the Akaike Information Criterion and Bayesian Information Criterion is also discussed.

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 system's 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, the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem

Boltzmannian statistical mechanics partitions the phase space of a sys- tem into macro-regions, 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 maxi- mum entropy considerations. We argue that they fail and present a new answer. We characterise equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilib- rium thus defined corresponds to the largest macro-region. Our derivation is completely general in that it does not rely on assumptions about a system’s dynamics or internal interactions.

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 Introduction2 Climate Variables and a Simple Climate Model3 Desiderata on a Definition of Climate4 Climate as Distribution over Time4.1 Definition 1: Distribution over time for constant external conditions4.2 Definition 2: Distribution over time when the external conditions vary as in reality4.3 Definition 3: Distribution over time for regimes of varying external conditions4.4 Infinite versions5 Climate as Ensemble Distribution5.1 Definition 4: Ensemble distribution for constant external conditions5.2 Definition 5: Ensemble distribution when the external conditions vary as in reality5.3 Infinite versions6 ConclusionAppendix.

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.

This article argues that common intuitions regarding (a) the specialness of ‘use-novel’ data for confirmation and (b) that this specialness implies the ‘no-double-counting rule’, which says that data used in ‘constructing’ (calibrating) a model cannot also play a role in confirming the model’s predictions, are too crude. The intuitions in question are pertinent in all the sciences, but we appeal to a climate science case study to illustrate what is at stake. Our strategy is to analyse the intuitive claims in light of prominent accounts of confirmation of model predictions. We show that on the Bayesian account of confirmation, and also on the standard classical hypothesis-testing account, claims (a) and (b) are not generally true; but for some select cases, it is possible to distinguish data used for calibration from use-novel data, where only the latter confirm. The more specialized classical model-selection methods, on the other hand, uphold a nuanced version of claim (a), but this comes apart from (b), which must be rejected in favour of a more refined account of the relationship between calibration and confirmation. Thus, depending on the framework of confirmation, either the scope or the simplicity of the intuitive position must be revised.

Gibbsian statistical mechanics (GSM) is the most widely used version of statistical mechanics among working physicists. Yet a closer look at GSM reveals that it is unclear what the theory actually says and how it bears on experimental practice. The root cause of the difficulties is the status of the averaging principle, the proposition that what we observe in an experiment is the ensemble average of a phase function. We review different stances toward this principle, and eventually present a coherent interpretation of GSM that provides an account of the status and scope of the principle.