Can Isolated Systems Under an Energy Constraint Exhibit Chaotic Behavior?

[Gear300]

For systems under an energy constraint (a conserved Hamiltonian that is a function of the q's and p's in generalized coordinates), is it possible for it to behave chaotically (chaos by its dynamical definition), or will it always have some equilibrium it tends towards?

 

[Jolb]

Yes, even Hamiltonian systems can behave chaotically. The double pendulum is a quick and easy example. There are many others though.

 

[Gear300]

So for an isolated thermal system, such as a gas, we can effectively consider deviations from an equilibrium state; but these deviations should be small according to observation, meaning that the distribution of the parameters describing the system should be tightly bound to the equilibrium values (they should have a small standard of deviation). I'm not sure if it would be right to call this mildly chaotic, but for the most part, the equilibrium region is dominant. What is preventing the system from undergoing heavy chaotic behavior, such as aperiodic convections?

 

[Mute]

So for an isolated thermal system, such as a gas, we can effectively consider deviations from an equilibrium state; but these deviations should be small according to observation, meaning that the distribution of the parameters describing the system should be tightly bound to the equilibrium values (they should have a small standard of deviation). I'm not sure if it would be right to call this mildly chaotic, but for the most part, the equilibrium region is dominant. What is preventing the system from undergoing heavy chaotic behavior, such as aperiodic convections?

I think there are two parts to the answer to this question. The first is that you have an isolated system whose energy is conserved. If you had an energy flow you could get much more complicated behavior (turbulence, etc). Of course, this is the point of your question: if you can have chaos in a system with conserved energy, why don't we observe it here.

The answer to that, I think, at least in the particular scenario you put forward, is one of following the average dynamics of the system versus the dynamics of the individuals parts (e.g., gas molecules) of the system. While individual parts appear chaotic, I would guess the average properties of the system are much less so, and that is what we are looking at when we discuss gasses in equilibrium: the average properties of the system.

If you look at the animations on the wikipedia page for the double pendulum, I could be convinced that it looks like the long-time average of the pendulum motion is zero in the x-direction and the equilibrium height in the y-direction, with some fluctuations if we measure the averages only over a finite length of time. So, although the actual motion of the two coupled pendulums is chaotic, the average motion looks much more regular.

I'm not prepared to prove this rigorously at the moment, but this is what I imagine the resolution to your question is.

i.e., if we were to track the motion of every gas particle, we would observe chaotic behavior, but if we only track the average properties, we do not observe chaos due to the conservation of energy. We need to have an energy flow in the system to observe macroscopic chaotic dynamics.

 

[Gear300]

I agree with your answer. While the individual gas molecules follow chaotic behavior, the average behavior follows equilibrium behavior. What had me interested was the making it rigorous part. The Hamiltonian in consideration is generally a function of the q's and p's by the nature of the potentials. The implication is that the nature of the interactions is what makes such systems entropic. If the interaction potentials held higher order derivatives then we would probably see something different (though, at least in the Newtonian context, since the force is a second-order derivative in position, it would be odd seeing potentials of corresponding or higher order).

 

[AlephZero]

If you look at the animations on the wikipedia page for the double pendulum, I could be convinced that it looks like the long-time average of the pendulum motion is zero in the x-direction and the equilibrium height in the y-direction, with some fluctuations if we measure the averages only over a finite length of time. So, although the actual motion of the two coupled pendulums is chaotic, the average motion looks much more regular.

Isn't it obvious that the long term average will be 0 (at least, as measured in one inertial reference frame), from Newton's laws of motion?