Page 24 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

10 1 The classical distribution function

evolution of the system is described by the motion of a point. Take a small re-
3N
gion of this space whose volume is denoted 3N p q centered at the point
(p, q). (Henceforth we denote (p, q) ≡ ({q i }, {p i }) and similarly ( p, q) ≡
({ q i }, { p i }).) Consider the interval of time t defined as

t(q 0 , p 0 , t 0 ; q, p, t; p, q) (1.11)
equivalent to the time which the point describing the system spends in the region
3N
3N p q around (q, p) between t 0 and t if it started at the point (q 0 , p 0 ) at time
t 0 .
3N
Now consider the fraction of time that the system point spends in 3N p q,
denoted w:

t
w(q 0 , p 0 ; q, p; p, q) = lim (1.12)
t→∞ t − t 0
which is the fraction of the total time between t 0 and t →∞ which the system
3N
spends in the region 3N p q around (q, p).
¯
Now we express the time average φ t of equation (1.9) in terms of w by dividing
the entire phase space into small regions labelled by an index k and each of volume
3N
3N p q:
¯
φ t = φ(q 0 , p 0 ; q k , p k ) w(q 0 , p 0 ; q k , p k ; p, q) (1.13)
k
We then suppose that w(q 0 , p 0 ; q, p; p, q) is a well behaved function of the
arguments ( p, q) and write
6N
∂ w
3N
w = q 3N p + ··· (1.14)
∂ 3N q∂ 3N p p= q=0
Defining
6N
∂ w
ρ(q 0 , p 0 ; q, p) = (1.15)
∂ 3N q∂ 3N p p= q=0
we then have in the limit p q → 0 that

¯ 3N 3N
φ t = ρ(q 0 , p 0 ; q, p)φ(q, p)d q d p (1.16)
which is of the form (1.10). Several of the smoothness assumptions made in this
discussion are open to question as we will discuss in more detail later.
¯
Equation (1.16) is most useful if φ t depends only on a few of the 6N initial
conditions q 0 , p 0 . Experimentally (and in simulations) it is found that the time
averages of many macroscopic quantities measured in equilibrium systems are very
insensitive to the way the system is prepared. We will demonstrate that under certain