# I Classical chaos?

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1. Jan 27, 2017

Suppose we are talking about a purely classical phenomena (OK, nothing is purely classical, but suppose we consider quantum effects as insignificant, that is, we ignore them). In this context, I came across someone talking about "a particle in chaotic continuous motion as the particle is subjected to a force". For me, this sounds impossible to take literally, since chaotic in its strict sense (not in the sense of hidden variables) would mean the lack of all predictability of its motion beyond the light-speed barrier (that is, its motion would be random, but not necessarily all factors would be random, as in "random variable", except perhaps the ) , and the force would add the predictability to the motion. So, the questions:
is it possible to talk of a classical particle being both literally chaotically moving and subject to a force at the same time? If so, does this mean that "chaotically" and "randomly" are different concepts? If not, then what better term would there be for a particle whose motion would be chaotic/random were it not for the force, but then under the influence of the force, some aspects of its motion would be predictable and others not? (For example, in a fixed three-dimensional space, the path traced out in a certain plane was determinate, but the motion in the third dimension was literally chaotic?) "Determinate with caveats" doesn't work, and "restricted chaos" sounds like an oxymoron.

2. Jan 28, 2017

### 256bits

I am not sure what the part I highlighted in red is supposed to mean to the reader of your prose.
Can you explain, or perhaps re-phrase.

Chaotic does not mean lack of all predictability.
It means how far into the future can we predict how a system will perform, or what state it will be in, with a definite certainty.
The weather is an example of a chaotic system, wherein we can say with high certainty that in 5 minutes from now the conditions will be the same. As the time frame extends, the certainty drops.

Random means that we cannot predict the outcome of a system.

Why are you searching for new terms? They seem to have been accepted in the scientific community as having a known meaning, even if in everyday speech people will use the two as being one and the same.

3. Jan 28, 2017

Thanks for the clarification, 256 bits, and the link. That answers my question. (I go by the old "I'm told I should learn by my mistakes. Since I want to learn, I guess I'll have to go out and make some mistakes.")

4. Jan 28, 2017

### Staff: Mentor

Yes. Chaotic motion is a classical concept. It does not mean random, it means hard to predict. It means that small errors lead to large deviations in the future.

A good example of a chaotic system is a water mill with slowly emptying buckets. If the fill rate is slow, then this water mill will be non chaotic. It will turn at a steady rate, and if you come back an hour later you can accurately predict its state. If the full rate is fast then it will be chaotic, and if you come back an hour later your prediction of its state will be very inaccurate.

5. Jan 28, 2017

Thanks, Dale. The difference between random and chaotic was one which was not clear to me when I wrote the post. But if I now understand correctly, when your wrote
both conditions (unpredictability and high sensitivity to initial conditions) make up the definition of chaotic, whereas only unpredictability is sufficient for the definition of random. I would guess that one could formulate the following: the chaotic system is one in which small changes in initial conditions create a tendency towards randomness in the state change. Would this be a fair guess?

6. Jan 28, 2017

### Staff: Mentor

Yes. Are you familiar with the concept of phase space and phase space trajectories?

7. Jan 28, 2017

Only superficially. My rough definitions are: (a) a phase space for a system is a graph whose axes are all the variables (momentum, position, etc.) needed to describe the system; a phase space trajectory is the graph of time versus the rest of the variables. My guess of the connection with chaos is that the phase space trajectory would appear chaotic, and if stable, a fractal, for example a Lorentz attractor. However, my knowledge doesn't go deeper than this, but I would be grateful for any better indications.

8. Jan 28, 2017

### Staff: Mentor

Your rough definition is fine. So for a pendulum, one axis would be the angle, and the other would be the angular velocity. Any possible state of the pendulum is represented by a point in that phase space. And the laws of physics tell you how the system moves from one state to another, tracing out a line in the phase space.

So, instead of thinking about starting with a single point in phase space, think about starting with a small area of points, each one associated with its own trajectory line. One characteristic of phase space is that the trajectory lines never cross, split, or join. This leads to the fact that if you take the area of your initial patch, let them evolve for some time, then the area of the final patch will be the same as for the initial patch and it will still be all connected.

Although the area may be the same and it will be connected, in a chaotic system it can become very long and skinny and stretched out. This is why it is difficult to predict the behavior of a chaotic system far into the future. Trajectories that start out arbitrarily close can wind up arbitrarily far away in phase space.

9. Jan 28, 2017

Thanks very much, Dale. If I understand correctly, you are pointing me in the direction of an application of Liouville's Theorem. That is very instructive; I shall take that as my starting point for looking into this problem more deeply.

10. Jan 29, 2017

### Staff: Mentor

Yes, that is correct. Let us know if you have any other questions on the topic.