Statistical Mechanics - Physics at Oregon State University

7.4. WHAT IS CHAOS? 147
surface. For the two-dimensional oscillator at a given energy U only three variables
are independent. Suppose that y is the dependent variable. In order to
get a two-dimensional representation of the orbit, we have to eliminate one additional
variable. We will choose x and consider all states of the system with
x = 0. Therefore we work in the plane (px, py). The energy U is equal to
1
2 (p2 x + p2 y) + 1
2y2 and hence the available part of phase space in this plane is
given by
p 2 x + p 2 y 2U (7.28)
A chaotic orbit would sample all values of the momentum within this circle.
The trajectories of the harmonic oscillator are very regular, however. If we
assume that x0 = 1 and px0 = 0, the condition x = 0 occurs at time t = 2πn
for n = 0, 1, 2, · · · At these times px = 0 and py = ±py0; hence only two points
are sampled!
Harmonic systems are too simple. There are three constants of motion.
Jx = x 2 + p 2 x and Jy = y 2 + p 2 y are conserved, and the phase angle between the
x and y part of the orbit is also constant.
Large systems.
This conclusion can be generalized to a system of N three-dimensional harmonic
oscillators. All orbits are non-chaotic, and such a system cannot be
described by statistical mechanical methods! If N is very large, however, and
the initial conditions of the oscillators are random, such a system does give the
correct statistical mechanical results if we measure variables as spatial averages!
Randomness has to be brought in, however. Either the initial states are random,
or each oscillator is perturbed randomly and continuously.
This trick only works when the oscillators are identical, and we essentially
map all coordinates into a six-dimensional phase space for a single harmonic
oscillator. Measurements which go beyond this simple projection of phase space
still have to be discussed carefully. Also, there are problems with correlations.
If there are no random perturbations the relative motion of any two oscillators is
completely determined by the initial conditions and hence cannot be described
by a statistical mechanical ensemble average.
Additional interactions.
The harmonic oscillator is easy since it only contains second order terms.
One has to include higher order terms to see interesting effects. A standard
example is the Hénon-Heiles potential
V (x, y) = 1
2 x2 + 1
2 y2 + x 2 y − 1
3 y3
(7.29)
Near the origin particles are still bound, because the third order terms are small.
In the y direction, however, the potential goes to minus infinity, and a quantum