Fundamental Statistical Mechanics

9. The Baker's transformation
9.1 Transformation and Properties
We consider an example which well illustrates the application of ideas from chaotic dynamical
systems to statistical mechanics, the baker's map. We take the phase space to be the unit square
0 ≤ x, y ≤1. The transformation consists of two steps: first the unit square is contracted by a
factor 2 in the y direction and expanded by a factor 2 in the x direction; then the rectangle is cut
in the middle and the right half placed on top of the left half. This doesn't change the volume,
and the transformation is reversible.
To write an express the transformation mathematically we need to distinguish between x < 1 2 and
x ≥ 1 2 , so if (x,y) → ( x ′ , y ′ ) = b(x,y).
For x < 1 2
⎧ x ′ = 2x
⎨
⎩ y ′ = y 2
and for x ≥ 1 2
⎧ x ′ = 2x −1
⎨
⎩ y ′ = (y +1) 2
The inverse of the baker's transformation (x,y) → ( x ′ , y ′ ) = b −1 (x, y)
For y < 1 2
⎧ x ′ = x 2
⎨
⎩ y ′ = 2y
and for y ≥ 1 2
9.2 A Model Boltzmann Equation
⎧ x ′ = (x +1) 2
⎨
⎩ y ′ = 2y −1
It is possible to derive a "Boltzmann equation for the time-reversible baker's transformation and
to show that a H -theorem holds for this equation. Here then is one example where the program
of Boltzmann can be carried out in detail. The price we pay for the simplicity of the model is a
lack of physical motivation. We will have to supply some of that as we go along.
Consider a density-function ρ(x,y) on the unit square, that satisfies a Liouville equation for
discrete time:
where
ρ n (x, y) = ρ n −1 (b −1 (x), b −1 (y))
⎧ ρn−1 (x 2,2y) for y 1 2
Statistical Mechanics Page 41