The Kac Ring or the Art of Making Idealisations
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The Kac Ring or the Art of Making Idealisations Julie Jebeile1 Received: 18 October 2019 / Accepted: 26 August 2020 / Published online: 7 September 2020 © The Author(s) 2020
Abstract In 1959, mathematician Mark Kac introduced a model, called the Kac ring, in order to elucidate the classical solution of Boltzmann to the problem of macroscopic irreversibility. However, the model is far from being a realistic representation of something. How can it be of any help here? In philosophy of science, it is often argued that models can provide explanations of the phenomenon they are said to approximate, in virtue of the truth they contain, and in spite of the idealisations they are made of. On this view, idealisations are not supposed to contribute to any explaining, and should not affect the global representational function of the model. But the Kac ring is a toy model that is only made of idealisations, and is still used trustworthily to understand the treatment of irreversible phenomena in statistical mechanics. In the paper, my aim is to argue that each idealisation ingeniously designed by the mathematician maintains the representational function of the Kac ring with the general properties of macroscopic irreversibility under scrutiny. Such an active role of idealisations in the representing has so far been overlooked and reflects the art of modelling. Keywords Scientific model · Idealisation · Model explanation · Macroscopic irreversibility · Kac ring · Bolzmann’s H-theorem · Molecular chaos hypothesis
1 Introduction Macroscopic phenomena are irreversible. For example, once poured into a same container, two gases do not split again. A coin dropped on the floor remains there, we see no coin jumping up by itself from the floor. And yet macroscopic phenomena result from a microscopic dynamics described by reversible laws, i.e., laws invariant under time transformation. * Julie Jebeile [email protected] https://www.juliejebeile.net/en 1
Institute of Philosophy & Oeschger Center for Climate Change Research, University of Bern, Bern, Switzerland
1Vol:.(1234567890) 3
Foundations of Physics (2020) 50:1152–1170
1153
Ludwig Boltzmann offered in 1872 an explanation of why macroscopic phenomena are irreversible despite having a reversible microscopic dynamics: this is the H-theorem. However, the H-theorem has been plagued by two important paradoxes that lie at the foundations of statistical mechanics: the “reversibility paradox” and the “recurrence paradox”. That is the reason why, later on, in 1959, mathematician Mark Kac introduced a model, called the Kac ring, as an “analog” of the classical solution of Boltzmann to explain macroscopic irreversibility ([27], p. 99). The Kac ring aims “to reconcile both time reversibility and recurrence with “observable” irreversible behavior” ([27], p. 73), and to offer and evaluate a proper interpretation of Boltzmann’s molecular chaos hypothesis. Thereafter it has undergone multiple developments (e.g. [10]), especially in quantum mechanics (e.g. [11–13, 41]). But the Kac rin
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