# Entropy in chaos dynamics

I'm curious about how entropy is defined within chaos theory. Are there analogous laws similar to the second law of thermodynamics? How do we define steady-state or equilibrium within the state space of a system governed by chaos theory? Is the dynamics considered reversible in principle? These questions highlight my lack of understanding, and I would greatly appreciate it if someone could kindly explain these concepts or provide any related information that would clarify them.

• This answer provides good references physics.stackexchange.com/a/461281/226902. In general, if you know everything about a classical dynamical system (chaotic or not) then there is no entropy (you can think of entropy as "ignorance"). Moreover, isolated chaotic systems are reversible (three masses orbiting each other is an example). Mar 16 at 22:41

I think those questions you are asking are pretty much exactly what "foundation of statistical mechanics" deals with. This is because either quantum or classical, you can view the time evolution of a physical system as a continuous time dynamical system (obeying Newton's equation of motion or the Schrodinger equation). I can't give a full lecture here, but I can try to point out to some key words or concepts.

Are there analogous laws similar to the second law of thermodynamics?

The second law of thermodynamics should rather be something that is derived from chaos theory setups. For example, a box of air molecules is a dynamical system with approximately $$6\times 10^{23}$$ degrees of freedom, that happens to be chaotic. Can we derive the fact that (for the vast majority of initial configurations) after sufficient time evolution, the system's macroscopic observables converge to a static value (that is uniquely determined by some few observables of the initial state)? is the question of deriving thermodynamics and statistical mechanics from dynamical systems (Newtonian or quantum mechanics, to be precise).

But if you don't want to go that deep, I can also point out that there's a concept called Kormogorov-Sinai entropy that characterizes how much chaotic a trajectory is in a dynamical system, and assigns "entropy" to it. How this KS entropy actually connects to thermodynamic entropy though, could be fairly subtle.

One of the most common ways to introduce statistical mechanics is to consider Liouville's theorem. This considers the setup where you start from a probability distribution of initial configurations, and then have a deterministic time evolution. In this case, if the dynamical system is chaotic, the Shannon entropy of your state increases as a function of time.

How do we define steady-state or equilibrium within the state space of a system governed by chaos theory?

That's the very core of the "foundations of statistical physics", and the eigenstate thermalization (ETH) is one of the approaches people have been looking into in the past decade or so. The basic idea of ETH (and some other approaches such as small effective dimension or typicality) is that the vast majority of states actually look macroscopically identical. This is in some sense the biggest addition to the simple "dynamics system" framework. We are always dealing with extremely large number of degree of freedom. So, words like macroscopic (observables) pop up.

Is the dynamics considered reversible in principle?

In many setups (unitary time evolution or Newtonian dynamics without dissipation), yes, you are right, and the truly interesting thing is that we still observe macroscopically irreversible processes associated with the increase of entropy, for large systems.

I think many of the references provided in Quillo's link would also be very helpful.

• Thanks! I found your explanation really enjoyable. Are you suggesting that chaotic dynamics underpins thermalization (in the first part of your answer)? It seems to me that reaching equilibrium might require dynamics governed by chaos theory as a necessary condition.
– Omid
Mar 17 at 16:21
• Yup! Maybe I wasn't clear enough.. >chaotic dynamics underpins thermalization This is exactly what I meant. As a matter of fact, it is fairly well known that integrable (thus explicitly non-chaotic) systems do not converge to the Gibbs state! They are known/believed to converge to a state called generalized Gibbs state (a quick google search with those key words gave me arxiv.org/pdf/1304.5374.pdf but I'm pretty sure the ref by pseudo-goldstone also explains that in depth). So yes, from these facts, it's pretty clear that chaos is very deeply related to thermalization. Mar 23 at 5:57

In the context of quantum chaos and eigenstate thermalization, I believe this review has many of the answers you are looking for: https://arxiv.org/abs/1509.06411