Boltzmann's Dilemma: An Introduction to Statistical Mechanics via the Kac Ring
- 6 August 2009
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Review
- Vol. 51 (3), 613-635
- https://doi.org/10.1137/070705799
Abstract
The process of coarse-graining—here, in particular, of passing from a deterministic, simple, and time-reversible dynamics at the microscale to a typically irreversible description in terms of averaged quantities at the macroscale—is of fundamental importance in science and engineering. At the same time, it is often difficult to grasp and, if not interpreted correctly, implies seemingly paradoxical results. The kinetic theory of gases, historically the first and arguably most significant example, occupied physicists for the better part of the 19th century and continues to pose mathematical challenges to this day. In this paper, we describe the so-called Kac ring model, suggested by Mark Kac in 1956, which illustrates coarse-graining in a setting so simple that all aspects can be exposed both through elementary, explicit computation and through easy numerical simulation. In this setting, we explain a Boltzmannian “Stoßzahlansatz,” ensemble averages, the difference between ensemble averaged and “typical” system behavior, and the notion of entropy.Keywords
This publication has 10 references indexed in Scilit:
- Chaos and threshold for irreversibility in sheared suspensionsNature, 2005
- Drat such custard!Nature, 2005
- How many shuffles to randomize a deck of cards?Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2000
- Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropyPhysica A: Statistical Mechanics and its Applications, 1993
- New perspectives on Kac ring modelsJournal of Statistical Physics, 1987
- Shuffling Cards and Stopping TimesThe American Mathematical Monthly, 1986
- Loschmidt's and Zermelo's paradoxes do not existFoundations of Physics, 1974
- A Mathematical Theory of CommunicationBell System Technical Journal, 1948
- Ueber mechanische Erklärungen irreversibler Vorgänge. Eine Antwort auf Hrn. Boltzmann's „Entgegnung”︁Annalen der Physik, 1896
- Erratum to: Etude des surfaces asymptotiquesActa Mathematica, 1890