# Understanding ergodicity and what an ergodic system is

I am trying to understand the concept of ergodicity/ergodic system in physics, but because my understanding of phase space, its elements is a bit unclear,I have trouble understanding the former. Regarding ergodicity (in physics), in Wikipedia I read this:

A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system.

A point in phase space, represents a microstate as far as I understand. Also in the case of MCE or CE the microstates are eigenstates of the hamiltonian. That doesn't mean that it can't also be a superposition of the eigenstates of the Hamiltonian (please correct me if my understanding is faulty here). Now if a point (system) gets to visit the entire volume of the system, doesn't that imply that the microstate changes its energy, and isn't that in contradiction to the Liouville theorem where we say that the change of a system is governed by the Hamiltonian mechanics?

Edit: What does it mean for a system to spend time in a region of phase space?

The wikipedia article is talking about ergodicity in classical physics. This is where concepts like phase space are most relevant. Allow me to quote the first paragraph of the section you're reading:

The case of classical mechanics is discussed in the next section, on ergodicity in geometry. As to quantum mechanics, although there is a conception of quantum chaos, there is no clear definition of ergodocity; what this might be is hotly debated.

As alluded to, the emergence of ergodicity in quantum mechanics is an active topic of current research. If you are interested in how ergodicity relates to the energy eigenstates of an isolated system's Hamiltonian, you can start by reading about the eigenstate thermalization hypothesis.

Complementing Zack's answer for version 3 of question:

Now if a point (system) gets to visit the entire volume of the system, doesn't that imply that the microstate changes its energy, and isn't that in contradiction to the Liouville theorem

No, because, sometimes implicitly, by "entire volume" it's meant "entire accessible volume". So, for instance, the orbit is not required to (nor could it) visit states with different energies to be considered ergodic.

Edit: What does it mean for a system to spend time in a region of phase space?

If a trajectory $$\vec{x}(t)$$ in the phase space $$S$$ of a classical system traverses a region $$R \subset S$$, the length of the time interval $$t\in(t_\mathrm{in},t_\mathrm{out})$$ between going in and out of $$S$$ can be called the time spent in this region.

• Sorry for coming back, but I have one more question. The idea of "entire accessible volume" exists only once a system is in equilibrium, or is it wrong to say so? Because once equilibrium is reached the probability density function is non zero at a certain region and zero elsewhere. An example would be the MCE, where we have a constant value of the probability density function in the hyper-surface of the 6N-Dimensional sphere and zero elsewhere. So is it wrong to say that ergodicity is considered once a system is in equilibrium, where the accessible volume is well defined? Feb 10, 2022 at 10:48