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 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 (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.     

Historical development of Teichmüller theory

Originally, the expression “Teichmüller theory” referred to the theory that Oswald Teichmüller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmüller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmüller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmüller theory and the development of its early major ideas.     

Three principles of unity in Newton

I discuss three principles of unity available in Newton’s physics, appealing to space and time, causal interaction, and law-constitution respectively. I compare these three approaches with respect to aggregation (how a collection of entities can compose a whole) and multiplicity (how the world as a whole can contain a multiplicity of genuine unities), outlining the problems faced by the first two approaches and arguing that the third looks a promising candidate for further philosophical investigation.     

How physics flew the philosophers' nest

We all know that, nowadays, physics and philosophy are housed in separate departments on university campuses. They are distinct disciplines with their own journals and conferences, and in general they are practiced by different people, using different tools and methods. We also know that this was not always the case: up until the early 17th century (at least), physics was a part of philosophy. So what happened? And what philosophical lessons should we take away? We argue that the split took place long after Newton's Principia (rather than before, as many standard accounts would have it), and offer a new account of the philosophical reasons that drove the separation. We argue that one particular problem, dating back to Descartes and persisting long into the 18th century, played a pivotal role. The failure to solve it, despite repeated efforts, precipitates a profound change in the relationship between physics and philosophy. The culprit is the problem of collisions. Innocuous though it may seem, this problem becomes the bellwether of deeper issues concerning the nature and properties of bodies in general. The failure to successfully address the problem led to a reconceptualization of the goals and subject-matter of physics, a change in the relationship between physics and mechanics, and a shift in who had authority over the most fundamental issues in physics.     

On geometric objects,the non-existence of a gravitational stress-energy tensor,and the uniqueness of the Einstein field equation

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a “geometric object”, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy. The main theorem proven underlying all the arguments is considerably stronger than the standard result in the literature used for the same purposes (Lovelock's theorem of 1972): it holds in all dimensions (not only in four); it does not require an assumption about the differential order of the desired concomitant of the metric; and it has a more natural physical interpretation.     

On information-theoretic characterizations of physical theories

Clifton et al. (Found. Phys. 33 (2003) 1561) have recently argued that quantum theory is characterized by its satisfaction of three information-theoretic axioms. However, it is not difficult to construct apparent counterexamples to the CBH characterization theorem. In this paper, we discuss the limits of the characterization theorem, and we provide some technical tools for checking whether a theory (specified in terms of the convex structure of its state space) falls within these limits.     

Leibnizian relationalism for general relativistic physics

An ontology of Leibnizian relationalism, consisting in distance relations among sparse matter points and their change only, is well recognized as a serious option in the context of classical mechanics. In this paper, we investigate how this ontology fares when it comes to general relativistic physics. Using a Humean strategy, we regard the gravitational field as a means to represent the overall change in the distance relations among point particles in a way that achieves the best combination of being simple and being informative.     

Subjective probability and quantum certainty

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system.     

Space–time philosophy reconstructed via massive Nordström scalar gravities? Laws vs. geometry,conventionality, and underdetermination

What if gravity satisfied the Klein–Gordon equation? Both particle physics from the 1920–30s and the 1890s Neumann–Seeliger modification of Newtonian gravity with exponential decay suggest considering a “graviton mass term” for gravity, which is algebraic in the potential. Unlike Nordström׳s “massless” theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman–Cunningham conformal group. It therefore exhibits the whole of Minkowski space–time structure, albeit only indirectly concerning volumes. Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein׳s principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity(s): matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric (barely or on long distances) in the mass term. What is the ‘true’ geometry, one might wonder, in line with Poincaré׳s modal conventionality argument? Infinitely many theories exhibit this bimetric ‘geometry,’ all with the total stress–energy׳s trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers–Pirani–Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities—indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein׳s equations along the lines of Einstein׳s newly re-appreciated “physical strategy” and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity (massless spin 2, Pauli and Fierz found in 1939). The Putnam–Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th–21st century space–time philosophy in the light of particle physics. An appendix reconsiders the Malament–Weatherall–Manchak conformal restriction of conventionality and constructs the ‘universal force’ influencing the causal structure.Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space–time theory that would have blocked Moritz Schlick׳s supposed refutation of synthetic a priori knowledge, and how Einstein׳s false analogy between the Neumann–Seeliger–Einstein modification of Newtonian gravity and the cosmological constant Λ generated lasting confusion that obscured massive gravity as a conceptual possibility.     

Linear structures,causal sets and topology

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin׳s more general framework and I characterise what Maudlin׳s topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin׳s theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin׳s theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin׳s framework. This fact poses a challenge for the proponents of Maudlin׳s theory to prove it fruitful.     

Cell density-induced changes in lipid composition and intracellular trafficking

Cell density is one of the extrinsic factors to which cells adapt their physiology when grown in culture. However, little is known about the molecular changes which occur during cell growth and how cellular responses are then modulated. In many cases, inhibitors, drugs or growth factors used for in vitro studies change the rate of cell proliferation, resulting in different cell densities in control and treated samples. Therefore, for a comprehensive data analysis, it is essential to understand the implications of cell density on the molecular level. In this study, we have investigated how lipid composition changes during cell growth, and the consequences it has for transport of Shiga toxin. By quantifying 308 individual lipid species from 17 different lipid classes, we have found that the levels and species distribution of several lipids change during cell growth, with the major changes observed for diacylglycerols, phosphatidic acids, cholesterol esters, and lysophosphatidylethanolamines. In addition, there is a reduced binding and retrograde transport of Shiga toxin in high density cells which lead to reduced intoxication by the toxin. In conclusion, our data provide novel information on how lipid composition changes during cell growth in culture, and how these changes can modulate intracellular trafficking.     

Interpreting the Wigner–Eckart Theorem

The Wigner–Eckart theorem is central to the application of symmetry principles throughout atomic, molecular, and nuclear physics. Nevertheless, the theorem has a puzzling feature: it is dispensable for solving problems within these domains, since elementary methods suffice. To account for the significance of the theorem, I first contrast it with an elementary approach to calculating matrix elements. Next, I consider three broad strategies for interpreting the theorem: conventionalism, fundamentalism, and conceptualism. I argue that the conventionalist framework is unnecessarily pragmatic, while the fundamentalist framework requires more ontological commitments than necessary. Conceptualism avoids both defects, accounting for the theorem’s significance in terms of how it epistemically restructures the calculation of matrix elements. Specifically, the Wigner–Eckart theorem modularizes and unifies matrix element problems, thereby changing what we need to know to solve them.     

Have we lost spacetime on the way? Narrowing the gap between general relativity and quantum gravity

Important features of space and time are taken to be missing in quantum gravity, allegedly requiring an explanation of the emergence of spacetime from non-spatio-temporal theories. In this paper, we argue that the explanatory gap between general relativity and non-spatio-temporal quantum gravity theories might significantly be reduced with two moves. First, we point out that spacetime is already partially missing in the context of general relativity when understood from a dynamical perspective. Second, we argue that most approaches to quantum gravity already start with an in-built distinction between structures to which the asymmetry between space and time can be traced back.     

Advanced optical imaging in living embryos

Developmental biology investigations have evolved from static studies of embryo anatomy and into dynamic studies of the genetic and cellular mechanisms responsible for shaping the embryo anatomy. With the advancement of fluorescent protein fusions, the ability to visualize and comprehend how thousands to millions of cells interact with one another to form tissues and organs in three dimensions (xyz) over time (t) is just beginning to be realized and exploited. In this review, we explore recent advances utilizing confocal and multi-photon time-lapse microscopy to capture gene expression, cell behavior, and embryo development. From choosing the appropriate fluorophore, to labeling strategy, to experimental set-up, and data pipeline handling, this review covers the various aspects related to acquiring and analyzing multi-dimensional data sets. These innovative techniques in multi-dimensional imaging and analysis can be applied across a number of fields in time and space including protein dynamics to cell biology to morphogenesis.     

Time isotropy,Lorentz transformation and inertial frames

Homogeneity of Euclidean space and time, spatial isotropy, principle of relativity and the existence of a finite speed limit (or its variants) are commonly believed to be the only axioms required for developing the special theory of relativity (Lorentz transformations). In this paper, however, it is pointed out that the Lorentz transformation for a boost cannot actually be derived without the explicit assumption of time isotropy (viz. time-reversal symmetry) which is logically independent of the other postulates of relativity. Postulating time isotropy also restores the symmetry between space and time in the postulates of relativity (i.e. time and space share the same symmetries then). Time isotropy also helps explain naturally one key general feature of the fundamental physical laws, viz. their time-reversal symmetry. But inertial frames are defined in influential texts as frames having space-time homogeneity and spatial isotropy only. Inclusion of time isotropy in that definition is thus suggested.     

The Foundations of Projective Geometry in Italy from De Paolis to Pieri

 In this paper we examine the contributions of the Italian geometrical school to the Foundations of Projective Geometry. Starting from De Paolis' work we discuss some papers by Segre, Peano, Veronese, Fano and Pieri. In particular we try to show how a totally abstract and general point of view was clearly adopted by the Italian scholars many years before the publication of Hilbert's Grundlagen. We are particularly interested in the interrelations between the Italian and the German schools (mainly the influence of Staudt's and Klein's works). We try also to understand the reason of the steady decline of the Italian school during the twentieth century. (Received Febuary 25, 2000)     

Essentially narrative explanations

This essay argues that narrative explanations prove uniquely suited to answering certain explanatory questions, and offers reasons why recognizing a type of statement that requires narrative explanations crucially informs on their assessment. My explication of narrative explanation begins by identifying two interrelated sources of philosophical unhappiness with them. The first I term the problem of logical formlessness and the second the problem of evaluative intractability. With regard to the first, narratives simply do not appear to instantiate any logical form recognized as inference licensing. But absent a means of identifying inferential links, what justifies connecting explanans and explanandum? Evaluative intractability, the second problem, thus seems a direct consequence. This essay shows exactly why these complaints prove unfounded by explicating narrative explanations in the process of answering three interrelated questions. First, what determines that an explanation has in some critical or essential respect a narrative form? Second, how does a narrative in such cases come to constitute a plausible explanation? Third, how do the first two considerations yield a basis for evaluating an explanation offered as a narrative? Answers to each of these questions include illustrations of actual narrative explanations and also function to underline attendant dimensions of evaluation.