# Questions about chaos.

I will talk about the quantum chaos.
Quantum chaos comes from the classical chaos. unfortunately, the concept of trajectory is forbidden in quantum theory. So it is very difficult to define the quantum chaos.
Now many works use lots of methods to define it, for example, Random-Matrix theory, Level repulsion, and so on.
Now I have a question.
1. For a classical chaotic system, what phenomenon can we detect in the experiment, and then we can say it is chaos?
2. There are lots of definitions of quantum chaos. I don't know which one is most popular? Similarly, what phenomenon can we detect in the experiment, and then we can say it is chaos?

Thank you!

## Answers and Replies

I will talk about the quantum chaos.
Quantum chaos comes from the classical chaos. unfortunately, the concept of trajectory is forbidden in quantum theory. So it is very difficult to define the quantum chaos.
Now many works use lots of methods to define it, for example, Random-Matrix theory, Level repulsion, and so on.
Now I have a question.
1. For a classical chaotic system, what phenomenon can we detect in the experiment, and then we can say it is chaos?
If I remember well, one can usually say that a system is a chaotic one if there exist a non zero positive Lyapunov exponent. This leads to the existence of a characteristic time beyond which the predictability is lost and where the dynamics becomes similar to a stochastic one.
2. There are lots of definitions of quantum chaos. I don't know which one is most popular? Similarly, what phenomenon can we detect in the experiment, and then we can say it is chaos?
I would say that the definition using Random-Matrix theory conjectures about the probability distribution of the gap between two consecutive energy levels is the most popular one. Experimentaly one "just" have to measure these levels and make statistics.

Thank you for your answer.
Then I have another question. For a chaotic system the poincare section is chaos. Then how can we know the system is chaos in the experiment? That is, which property of the system can we use to measure the chaos in the experiment?
Thank you!