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