The arrow of time is usually defined by the direction in which entropy increases. In a closed system, if there's a max entropy that the system can reach, does that mean time stops or at least become undefined at the max entropy state?
See also: For an isolated system, can the entropy decrease or increase?
One may still describe the behavior of such a system at maximal entropy using a theory that does use the concept of time, or the time coordinate $t$. But it is true that operationally speaking, the passage of time ceases to exist because the equilibrium associated with the maximum entropy is incompatible with the existence of thinking observers.
Note that whenever someone measures time, remembers, thinks, plans, or does things, the total entropy has to strictly increase. All the macroscopic observables are stationary in the maximum-entropy state which means that nothing macroscopic is happening.
It is important to distinguish between the time and the flow of time.
The time, $t$, is just a coordinate like $x$, $y$ and $z$ that we use to specify points in spacetime. The time coordinate doesn't have an arrow any more than $x$, $y$ or $z$ have arrows. The time axis has a negative and positive direction, just like the spatial coordinates, but at normal energies all the fundamental equations we use in physics are time symmetric i.e. they don't distinguish between the future and the past.
But of course we can move in any direction in space, while our experience tells us that we have to move through time at one second per second. This is the flow of time. There are endless arguments about whether the flow of time is a quirk of human perception or something fundamental, but we should perhaps pass over those here.
You posit a system that is at maximum entropy. Entropy is a statistical quantity, and if you look at the microstructure of your system you'll see it is still changing with time. Gas molecules are still whizzing around and colliding with other gas molecules, and their interactions are still described by functions of time. So in this sense time still carries on just as it did before.
However when you consider large scale properties, like pressure or density, if the system is at maximum entropy these cannot change significantly - or more precisely it is exceedingly improbable that they will change significantly. So for example no macroscopic clock can tick, because the operation of a (real i.e. non-reversible) macroscopic system is always accompanied by an increase in entropy.
But does this mean the flow of time has stopped? I think this is largely semantics rather than physics. At the microscale the system is still evolving with time even though macroscale systems like you and I could not experience any flow of time. My view would be that t is not useful to say the flow of time has stopped in such a system. To make that statement just means you have not looked at the system closely enough.
This is my first answer here so please don't expect a lot. I watched the BBC show yesterday just about this question. It is called Wonders of the Universe, episode 1 - Destiny.
According to the Professor Brian Cox, yes, arrow of time will stop eventually when maximum entropy is reached. It's going to happen in unimaginable amount of time. Literally unimaginable. If you count every atom in our Universe as one year — it would be not enough atoms to describe this number. Eventually every star is going to die. Then every black hole. Entropy is always increasing. In the end it would be nothing but cold photons floating in the empty space. When this happens maximum entropy is reached and time as we know will no longer exist.
