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.

Highlights

  • As I show in this paper,1 these accounts fail to give reasons why the Kac ring is used trustworthily to understand the treatment of irreversible phenomena in statistical mechanics while it is only made of idealisations

  • In a series of lectures published under the title “Probability and Related Topics in the Physical Sciences” ([27]), Kac aims to explicate the interrelation of dynamical law, probability, and initial conditions in physics, and notably to introduce statistical mechanics of irreversible phenomena

  • This paper argues that all of this is possible because the Kac model distorts the representation of a gas system underlying the H-theorem, and

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Summary

Introduction

Philosophical accounts of how models, albeit idealised, can teach us something about the world were first developed in the 1990s (e.g., [30, 31, 34]) and, since have flourished (e.g., [3, 4, 22, 28, 38, 42]) Most of these accounts argue that models can provide explanations of the phenomenon they are said to approximate, in virtue of the true components they contain, and in spite of the idealisations they are made of. As I show in this paper, these accounts fail to give reasons why the Kac ring is used trustworthily to understand the treatment of irreversible phenomena in statistical mechanics while it is only made of idealisations 2), and make it clear how the model reconciles both time reversibility and recurrence with “observable” irreversible behaviour, and enables us to evaluate interpretations of Boltzmann’s molecular chaos hypothesis Beforehand, I first present the historico-scientific context of the creation of the Kac ring (Sect. 2), and make it clear how the model reconciles both time reversibility and recurrence with “observable” irreversible behaviour, and enables us to evaluate interpretations of Boltzmann’s molecular chaos hypothesis (Sect. 3)

The H‐Theorem
The Reversibility Paradox and the Recurrence Paradox
The Kac Ring
Explaining Observable Irreversible Behaviour
Illustration of Time Reversibility and Recurrence
Trustworthiness of the Model
Targetless Toy Model
Approximation
Minimal Model
Scientific Caricature
Analog
Active Role of Idealisations
Descriptions of Macroscopic and Microscopic Levels
Multi‐realisability
The Molecular Chaos Hypothesis
Reversibility and Recurrence
Conclusion
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