Fundamental Statistical Mechanics

1
2
( )
Hn +1 ≤ ∫ d x ′ Wn ( x ′ )ln Wn ( x ′ ) + ∫ dxWn ( x ′ )lnW n ( x ′ )
0
That is, we obtain a H -theorem in the form
H n +1 ≤ H n
Note that H stays constant if W is a constant.
1
1
2
( )
Reverting for the moment to a physical system, a dilute gas, we know that the phase space
distribution function is the fundamental distribution which really determines the behaviour of an
ensemble of systems not in equilibrium and that the function which satisfies the Boltzmann
equation is the single particle distribution function, obtained by integrating over the variables of
all but one of the particles. Bogoliubov has argued that one can separate rapidly varying
functions from slowly varying functions, and the physically interesting functions change slowly
with time. The time scales that Bogoliubov thought were relevant in a gas are the duration of
binary collisions, the mean free time between collisions, and the time it takes a particle to travel a
macroscopic distance. Applying Bogoliubov's arguments to the baker's transformation we would
expect that the x variable is slowly varying while the y variable is rapidly varying.
What happens if we integrate the x coordinate rather than the y coordinate?
We need to look at the evolution of density differently
where
ρ n −1 (x, y) = ρ n (b(x), b(y))
⎧ ρn (2x, y 2) forx 1 2
1
∫
V n−1 (y) = dx ρ n−1 (x, y)
0
1
2
= ∫ dx ρn (2x, y 2) + ∫ dx ρn (2x −1,( y +1) 2)
0
1
1
2
= 1
2 d ′ x ρ 1
∫ n ( x ′ , y 2) +
0
1
1
∫ 2
0
= 1 2V ⎛ y
n⎝
2
⎞
⎠ + 1 2 V ⎛ y +1⎞
n⎝
2 ⎠
d ′ x ρn ( x ′ ,(y +1) 2)
Statistical Mechanics Page 43