2026 Foundations of Thermodynamics Workshop: Finding Balance

In the third century BCE, during the time of Archimedes of Syracuse, one of the central concerns of the emerging discipline of mechanics was to understand how a simple yet widely used machine, the lever, works. Ancient thinkers sought to explain how a small weight, using a lever, can move a large one. They observed that a lever’s operation depends on both the weights involved and their distances from the fulcrum of the lever’s beam. The farther a small weight is from the fulcrum, the more effectively it can move a larger weight on the opposite side of the beam. As a geometer, Archimedes treats the law of the lever as a theorem that must be demonstrated. Thus, propositions 6 and 7 of the first book of Archimedes’ On the Equilibrium of Planes present a rigorous demonstration of the theorem that “two magnitudes, whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.” In my talk, I will draw on various ancient sources to explore how the concept of balance intertwines the disciplines of philosophy, geometry, and mechanics.

In the Principia, Newton provides mathematical treatments of various physical systems in static and dynamical equilibrium. I argue that Newton’s equilibrium reasoning is both (1) theoretically and (2) evidentially novel when compared to the treatments of equilibrium preceding the Principia (including his own). Newton’s equilibrium reasoning is (1) theoretically novel because equilibrium conditions are not derived from observation but taken to follow from a mathematical theory of forces. Such forces are taken to exist independently of the specific kinematic situations that instantiate and measure them (Janiak 2009). The equilibrium reasoning is (2) evidentially novel because it proceeds by Newtonian (de-) idealization (Smith 2014). Newtonian (equilibrium) idealizations are exact models of a theory, which are constructed not because they are true, but, if taken to be true, can license counterfactual inferences about unaccounted forces and their material causes. Such inferences may be drawn precisely when the idealizations disagree with measurements, leading to iterative revisions of the idealizing assumptions. I illustrate one powerful application of this method: Newton’s model of Earth’s shape derived from his theory of rotating gravitating spheroids in hydrostatic equilibrium. As the theory matured mathematically at the hands of Clairaut, Laplace, and Stokes, it allowed for increasingly sophisticated models of Earth’s shape and increasingly fine-grained inferences about its internal constitution and gravitational field. By the 1950s, we knew that both fundamental assumptions of the theory are false: Newton’s law breaks down outside of static and weak gravitational fields, and Earth’s surface is not in hydrostatic equilibrium. Yet, the previous revisions of equilibrium models provided us with the foundations of most modern geophysics.

Early modern and Enlightenment natural philosophy was the point of origin for modern thinking about principles of equilibrium. I explore notions of equilibrium integrated into a network of concepts of balance, conservation, optimality, and harmony vying for a place in the new natural philosophy by presenting both complementary and competing visions of the perfection of the natural world. My focal figure will be Leonhard Euler, discussed in connection to Descartes, Fermat, Bernoulli, Leibniz, Maupertuis, and others.

These physical principles were themselves standardly grounded epistemically in metaphysical and theological claims, but that relationship was reciprocal. Principles of equilibrium borrowed from antiquity, for instance, were used to justify metaphysical principles like the principle of sufficient reason. The most consequential connection was mediated by mathematical representation. Philosophers sought to show the utility and rigor of metaphysico-theological principles about the harmony and order of the natural world by representing them in the emerging mathematics of optimization. One consequence was a conceptual reformation: the conversion of metaphysical claims whose content was expressed in terms of the attributes of God into mathematical propositions whose content was expressed in terms of infinity and infinitesimals. Other, broader results were a dramatic reordering of the disciplines of metaphysics and natural science and, of course, a (miraculously?) fruitful mathematical framework for the latter.

What is known as the “thermodynamic arrow of time” is the ubiquitous tendency of systems, when left to their own devices, to move towards equilibrium, and, once there, to remain there. This is a temporally asymmetric phenomenon; the temporal inverses of such processes, which would involve a system moving away from equilibrium, are not observed in equal abundance. There are arguments that other temporal asymmetries, and in particular, the temporal asymmetry of causation, can be understood in terms of the thermodynamic arrow of time.

On the other hand, one should also attempt to explain the tendency of systems out of equilibrium to approach equilibrium. If one looks at detailed attempts to account for equilibration, involving not just a tendency to end up in equilibrium but also the sequence of states through which equilibrium is approached, and the rates of approach, they typically involve assumptions with a causal flavour, to the effect that states of systems that have not recently interacted are uncorrelated. Boltzmann’s Stosszahlansatz is an example; similar assumptions are at the core of other applications of the Boltzmann equation. Assumptions of this sort can be justified by reference to a common cause principle, to the effect that correlations are to be explained in terms of past interactions.

We seem to have two rival approaches to the relation between the thermodynamic arrow and the causal arrow: one that takes the thermodynamic arrow to be prior, and grounds the causal arrow on the thermodynamic arrow, and one that explains the thermodynamic arrow by reference to the causal arrow.

I will argue that these are only apparently rivals; the two arrows should not be thought of as standing in any sort of grounding, or priority relations. The thermodynamic arrow is a condition for the possibility of causal relations, and the causal arrow explains the tendency of systems to equilibrate. This is a vicious circularity only on a conception of metaphysics that, I argue, should be rejected.

I trace a historical lineage of "balance" or equilibrium via the development of fluctuation theorems, which, I show, grew by drawing on analogical reasoning across apparently disparate domains, whilst being grounded in some unifying principles of balance. The domains include thermoelectric circuits (Thomson 1854), thermal noise in resistors (Nyquist 1928), black-body radiation (Planck 1900), Brownian motion (Einstein 1905), and chemical equilibrium (Onsager 1931), leading to the development of Fluctuation-Dissipation Theorems (FDTs) and Onsager's Regression Hypothesis.

The fertile development across these domains was further built upon by Callen and Welton (1951), Kubo (1957), Martin and Schwinger (1959), and Haag, Hugenholtz and Winnink (1967) to yield correlation functions associated with FDTs, which were employed to provide a precise and general definition of equilibrium, namely, KMS states, for systems of any size. I explore further recent extensions of these ideas, such as by Jarzynski (1997) and Crooks (1999), leading to what are called generalised fluctuation theorems today, that supposedly extend the properties of "balance" to regimes far from equilibrium.

I demonstrate that all these successive developments, beginning right from the days of Thomson to modern fluctuation theorems, are best read as unfolding different shades of the idea of "balance" as discovered or applied in various settings, and not as independent discoveries. The main unifying principle, I show, is the Second Law of thermodynamics, which is supplemented by the related principles of detailed balance, local detailed balance and the structure of the (equilibrium) Gibbs state, all essentially reflecting the many shades of balance.

This bears on recent philosophical discussions about unifying the ideas of mechanical and thermal equilibrium, as in Roberts (2022), who argues that mechanical and thermal notions of balance come together in non-conservative systems, since both mechanical and thermal irreversibility can be explained by the existence of dissipative forces in open systems. I show that such a view of unifying balance is not quite correct, as it misses out on the finer structure of thermal balance and dissipation, as shown via FDTs. That is, thermal balance is built upon a rather distinctive foundation involving several layers of microscopic balance, including the Second Law, that is not found in purely mechanical forms of balance.

In this talk I will show that quantization originates in the invariant statistical structure of black-body radiation thermodynamics. By considering the neglected literature on 19th-century black-body radiation thermodynamics and building on the results of “Rewriting the Quantum ‘Revolution’” (Studies in History and Philosophy of Science, 109 (2025) 72–88), I will show that quantization follows from requiring logical compatibility between two physical principles: the second law of thermodynamics and Boltzmann’s principle. Albert Einstein’s 1905 “On a Heuristic Point of View Concerning the Production and Transformation of Light” is not an analogical argument about the question “What is light?”, as is traditionally maintained. It addresses a different question: “How must the laws of nature be constructed in order to rule out the possibility of bringing about perpetual motion?” The answer is quantization. This deepens our physical understanding of quantization, resolves longstanding tensions in the historiography, and provides a principled answer to Wheeler’s question “How come the quantum?”

We draw lessons from recent explanatory pluralism to argue that under certain circumstances events or occurrences can be explained by appeal to their consequences. We explore two modes of such consequence explanations—equilibrium explanations and teleological explanations. We argue that an effective explanation must capture the right modal relation between explanans and explanandum. This relation has three dimensions: dependence, sensitivity, and alternativity. We argue that in equilibrium systems and goal-directed systems, this modal relation holds between the consequences and the events that bring them about. Consequence explanations are not reducible to causal or ‘forward-looking’ explanations. They are autonomous and complementary.

In applied econometrics, some authors argue that reduced-form models are sometimes adequate (as opposed to requiring structural-form models) for policy analysis - for example, in this paper I take Raj Chetty’s “sufficient statistics” approach as a paradigmatic example. I (a) consider the relationship between the equilibrium concept, often deployed in model-based reasoning, and the strategic use of the envelope theorem to attain these reduced-form models, and (b) explore how these tools used together ground (or don’t ground) causal investigation.

There is a long history of the investigation and interrogation of equilibrium (stationary or static) solutions in General Relativity (GR). In this presentation, I review and characterize several episodes in this history. In the early decades of GR, only a small number of exact solutions were known (e.g.,  Einstein’s static universe, Schwarzschild’s solution) and their epistemic and methodological roles were limited. These solutions were largely treated as toy models: idealizations to be understood in a wholesale sense. Following what Eisenstaedt has called the “low water mark” of GR (1925-1955), and especially in the wake of the 1957 Chapel Hill conference, the role of exact solutions broadened. Solutions came to be studied more systematically (as evidenced by, e.g., Petrov and Pirani’s classification of solutions and the Kerr family of solutions). This shift allowed physicists to consider them as limiting or asymptotic cases of more realistic configurations and to use them as probes of the theory structure itself.  Finally, I offer several preliminary hypotheses to account for this shift in epistemic role. I consider whether it was driven by the proliferation of known solutions, the increasingly systematic exploration of the solution space, or broader historical and technological developments that reshaped how physicists understood the significance of idealizations in their theories.

Thermal equilibrium is a foundational concept of classical thermodynamics. However, integrable and near-integrable quantum statistical mechanics reveal its limits: not all its classical theoretical roles can be recovered. We argue that this failure is epistemically productive. Classical thermal equilibrium can guide theory even where it no longer straightforwardly applies, by serving as a conceptual anchor for identifying analogies and disanalogies in new domains. This anchoring clarifies the assumptions under which the old concept worked and supplies physical meaning to generalized equilibria, showing how disanalogies can guide, rather than defeat, concept extension.