Fundamental Statistical Mechanics

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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
Define a reduced distribution function that depends on x only: 1 ∫ W n (x) = dyρ n (x, y) 0 1 2 = ∫ dyρ n−1 (x 2,2y) + ∫ dyρ n −1 ((x +1) 2,2y −1) 0 1 1 2 Change to a variable y ′ = 2y in the first integral and to y ′ = 2y −1 in the second integral: Wn (x) = 1 2 1 ∫ 0 d ′ y ρn−1 ( x 2 , ′ y ) +ρ x +1 ( n−1 ( 2 , y ′ ) ) = 1 2 W ⎛ x n−1⎝ 2 ⎞ ⎠ + W ⎛ ⎛ x +1⎞ ⎞ n−1 ⎝ ⎝ 2 ⎠ ⎠ This is the model Boltzmann equation that is associated with the baker's transformation. We notice that the time is discrete rather than continuous, and that we have selected the x coordinate for some reason that is not yet clear. It is easy to check that if Wn does not depend on x then Wn remains constant in time. Thus there is an equilibrium distribution W 0 = constant , which corresponds to a uniform distribution on the unit x interval. The H -theorem is constructed in the same way as is done for the Boltzmann equation itself. We define 1 ∫ ( ) H n = dxW n (x)ln W n (x) 0 Then H develops in time as 1 Hn +1 = dx 1 2 Wn ( x ( 2 ) + Wn ( )ln ∫ 0 . x +1 2 ) 1 2 Wn ( x x +1 [ ( 2 ) + Wn ( 2 ) ) ] as the function F(y) = y ln y is convex, it follows that 1 a +b 2 ( F(a) + F(b) )≥F( 2 ) . Setting a = Wn ( x x +1 2) and b = Wn ( 2 ) we have: Hn +1 ≤ 1 2 dx W n ( x 2)ln Wn ( x 1 ∫ 2) 0 x +1 x +1 ( ( )+ Wn ( 2 )ln( W n ( 2 ) ) ) Change to ′ x = x 2 in the first term, and to ′ x = x +1 2 in the second term, we find: Statistical Mechanics Page 42
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Fundamental Statistical Mechanics

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