Fundamental Statistical Mechanics

EXAMPLE: Consider the map given by
φ (x) = x + α mod(1)
If α is a rational number, nm where n and m are integers, then the mapping repeats the initial
point after m iterations. A trajectory starting from an initial point will be periodic and will not
spend equal time in regions of equal (Lebesgue) measure, and hence the trajectory is not dense
on the unit interval. In this case the system is not ergodic. But if α is irrational no trajectory is
periodic and the system is ergodic! To see this consider the time average of a phase function on
a trajectory starting at x
1
B (x) = lim
N →∞ N
N −1
∑
n= 0
B(φ n (x))
1
= lim
N→∞ N
N−1
∑
n=0
B(x + nα)
Consider the Fourier series for B(x): B(x) = a j e 2πijx
∞
∑
B(x + nα) = a j e
j =−∞
2πijx+ 2πijnα
Substituting into the time-average gives
1
B (x) = lim
N →∞ N
1
= lim
N →∞ N
N −1
∞
∑ ∑
n= 0 j =−∞
∞
∑
j=−∞
2π ijx+ 2πijnα
a je a j e 2πijx
e 2πijnα
N −1
∑
n= 0
Statistical Mechanics Page 38
∞
∑
j =−∞
1
= a0 + lim
N →∞ N
∞
∑
j =−∞
j ≠0
a j e 2πijx
1− e 2πijNα
2 πijα
1 − e
The terms with n ≥ 1 don't survive the limit N →∞, thus B (x) = a 0 for α irrational. In this
case the time average is a constant - independent of the starting point x and the system is
ergodic, where the ensemble average is the average over the circle with the usual Lebesgue
measure,
1
∫
B (x) = dxB(x) = B(x) mc
0