G. Gottwald and M. Oliver,
Boltzmann's dilemma: an
introduction to statistical mechanics via the Kac ring,
SIAM Rev. 51 (2009), 613-635.
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 these notes, 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.
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The code for generating the figures in this paper is written
in Python using the Numpy and
Matplotlib extensions.
- figures.py for Figures
3-5
- scaling.py for Figure 6