Limits, asymptotic expansions, and perturbation theory in physics – ASYMPTOPHYS
Limits, asymptotic expansions, and perturbative methods in physics
This philosophy of science project studies the role of limits in physics, especially singular limits, which rely on idealized assumptions such as infinite numbers of particles. These limits raise questions about explanation, approximation, and the unity of physics. The project addresses them through perturbation theory and asymptotic expansions.
Understanding the Role of Infinite Idealizations and Asymptotic Methods in Physics
The ASYMPTOPHYS project addresses a central question in the philosophy of science: how should we understand the role of idealizations in physics? Physical models often use assumptions that are known to be false, but that nevertheless seem necessary to explain certain phenomena. For example, to describe the transition from an ice cube to liquid water, physicists assume an infinite number of molecules, even though a real ice cube always contains only a finite number. Why is such an assumption useful, or even indispensable? This question has generated an important debate in the philosophy of physics. Some authors argue that infinite idealizations play an essential role in scientific theories; others, by contrast, regard them as merely practical tools that could in principle be replaced by more realistic descriptions. The project engages with these issues by starting from a distinction that is central to these debates: the distinction between two types of mathematical limits. In regular limits, the behavior of a real system remains close to that of the idealized system. In singular limits, by contrast, taking the limit gives rise to new behaviors that are qualitatively different from those of finite systems. Such limits therefore raise a specific problem: they seem to show that some phenomena cannot be understood as the simple approximation of a finite system. This difficulty also concerns the relations between physical theories. Can one theory, for example, be reduced to another, or can we explain how a more general theory gives rise to a more specific one? Limit procedures often play a central role in such relations. But when the limit is singular, this reduction becomes problematic, because the equations or behaviors obtained at the limit no longer simply resemble those that precede them. The hypothesis of the project is that these problems can be better understood by studying asymptotic methods and perturbative theories. These methods, widely used in physics and applied mathematics, make it possible to tackle complex problems through series expansions, which may be either convergent or divergent. The overall aim of ASYMPTOPHYS is therefore to analyze the role of these methods in order to better understand infinite idealizations and the relations between physical theories. The project pursues two main objectives: first, to examine cases already discussed in the literature, such as the rainbow problem and quantum field theory; and second, to extend the analysis to other domains, especially fluid mechanics, in particular through Chapman-Enskog expansions, which connect the kinetic theory of gases to the equations of hydrodynamics.
The project is based on the recruitment of a postdoctoral researcher for two years, making it possible to deepen the study of asymptotic methods, particularly in the field of quantum field theory. Other cases are also examined, especially in fluid mechanics, in order to better understand how certain fundamental equations can be derived from more general models. The collective work is structured around regular meetings, the reading of articles, the preparation of publications and conference papers, as well as the organization of seminars, workshops, and international conferences. The project thus fosters numerous exchanges between philosophers, physicists, and mathematicians, and contributes to the writing and publication of several scientific works.
The results of the project are listed on the HAL page dedicated to the ANR project.
The project has led to several publications in scientific journals and edited volumes. In particular, it resulted in the editing of a special issue of Synthese, entitled “Models, Computation, and Representation,” co-edited with Cyrille Imbert and Sorin Bangu. Several articles directly related to the project’s themes have been published or accepted, notably on approximation methods in physics, perturbative expansions in quantum field theory, perturbative causality, rigour in theoretical physics, the uses of the Ising model, and finite-size scaling theory. These publications appeared in journals such as Synthese, European Physical Journal H, Cahiers philosophiques, and Studies in History and Philosophy of Science.
The project also gave rise to numerous presentations at international conferences, seminars, and workshops. These talks addressed asymptotic expansions, divergent series, relations between physical theories, fluid mechanics, quantum field theory, turbulence, mathematical rigour in physics, and the metaphysics of scales. They were presented in major scientific venues, including the EPSA, PSA, BSPS, and SPS conferences, as well as at universities and research institutes in Paris, London, Munich, Bonn, Bergen, Bristol, Groningen, Waterloo, and New Orleans.
Among the most significant results are the presentations devoted to singular limits and intertheoretic relations in fluid mechanics, delivered in particular at PSA 2024 and BSPS 2024, as well as the work on the foundations of quantum field theory and on the role of divergent perturbative series. The project thus helps to strengthen the international visibility of the research carried out, while producing original results at the intersection of philosophy of science and philosophy of physics.
The perspectives opened up by the project first concern the analysis of the notion of approximation in asymptotic methods. This question will be further explored within the framework of the ANR GRASP project, “Explanatory Progress and the Gradable Extension of Understanding in the Mathematized Science,” led by Cyrille Imbert. One of its objectives will be to analyze how asymptotic methods contribute, in a gradual way, to the prediction, explanation, and understanding of physical phenomena. The work carried out will also provide the basis for future publications and for the consolidation of a broader research program on the relations between models, approximations, and physical theories, notably through the Twin Research Scholars project between the CNRS and the University of Toronto. Finally, thematic connections with other recent projects should foster new collaborations around questions of scales, limits, and reduction between scientific theories.
This research project in philosophy of science aims to investigate the role of limits in physics theorizing. For two decades, there are intense discussions due to the use of ‘singular limits’ in physics. For instance, although the number of molecules of water in a glass is finite, our contemporary physical theories require the false assumption that this number tends to infinity to explain some phenomena. The singular limits also challenge the unity of physics, by preventing to reduce some physical theories to more fundamental ones. This research project aims to tackle these epistemological problems by investigating the conceptual framework of asymptotic expansion and perturbation theory. With a team mainly composed of philosophers of science, of physics and mathematics, I shall offer novel insights on this hot topic in philosophy of science with three research tasks dedicated to physical modelling, the reduction of hydrodynamics, and approximation in physics.
Project coordination
Vincent Ardourel (Centre national de la recherche scientifique)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
Partnership
IHPST Centre national de la recherche scientifique
Help of the ANR 223,787 euros
Beginning and duration of the scientific project:
February 2023
- 36 Months
