The Kac Ring Model

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The Kac ring is a simple, explicitly solvable model that illustrates the process of passing from microscopic, time-reversible behavior to macroscopic, thermodynamic behavior [1]. In this model, sites are arranged around a circle, forming a one-dimensional periodic lattice. Each site is occupied by either a black ball or a white ball. A fraction of the bonds joining neighboring sites have a "tunnel." From time to time each ball moves to the clockwise neighboring site, changing its color if the ball crosses a tunnel.

Contributed by: Daniel Díaz Simón and Andrés Santos (October 2016)
Open content licensed under CC BY-NC-SA



In this model, a "microstate" at time is characterized by quantities: the color (black or white) of balls sitting on each of the sites. A "macrostate" at time is characterized by a single coarse-grained quantity: the difference between the numbers of white and black balls, relative to the total number . The microscopic dynamics are reversible [2] and recurrent [3] (the microstate repeats itself after time steps). However, under a molecular chaos assumption (the fraction of white balls just about to cross a tunnel is assumed to be at any time) [4], the irreversible evolution equation is expected to be correct in the limit .

In this Demonstration, all the balls are white initially. You can control the total number of sites , the fraction of bonds with a tunnel and the time range to display. You can choose to follow the temporal evolution of the macrostate or the microstate.


[1] G. A. Gottwald and M. Oliver, "Boltzmann's Dilemma: An Introduction to Statistical Mechanics via the Kac Ring," SIAM Review, 51(3), 2009 pp. 613–635. doi:10.1137/070705799.

[2] Wikipedia. "Loschmidt's Paradox." (Oct 3, 2016)'s_paradox.

[3] Wikipedia. "Poincaré Recurrence Theorem." (Oct 3, 2016)é_recurrence_theorem.

[4] Wikipedia. "Molecular Chaos." (Oct 3, 2016)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.