OK, but then if I saw a video of Brownian motion which is for all intents and purposes time–reversible, should I conclude the time is in fact not moving forward?
Maybe another way to phrase the question: at the end of the universe when entropy has reached its highest state, does the time dimension of the universe disappear?
One could say "I spent, 1 hour looking at this still picture", but from the picture's perspective no time was spent, because the time in the world of the video is the delta between its frames, like the time in our world is expressed through increased entropy.
Inquiring time before or after the universe is like asking the question "what is the duration of this jpeg image?"
Even as the concordance model cosmos asymptotes to (vacuum) de Sitter, with fully evaporated black holes and less than one proton (assuming they're stable) per Hubble volume, there will be at least a sparse gas of cold photons from two sources: the relics (the cosmic microwave background) and the horizon radiation. A comoving observer, who has an isotropic view of these photons, would still in principle be able to detect the metric expansion in the spectrum of these extremely long wavelength photons. More vacuum being created means (Boltzmann) entropy is increasing: one can substitute microscopic sections of vacuum with each other and do so wholesale without getting anything but vacuum.
Importantly though, not all timelike observers in the ultra far future need to be comoving: they can be highly boosted with respect to the remaining sparse radiation, and can even be accelerated with respect to it. That means they'll see a dipole, bluer in one direction, more particles in the bluer direction. In turn that means have an easier time looking for the spectral signature of the adiabatically cooling relic photons and the horizon radiation, even if there is nothing else they can observe. (By Unruh and Hawking an ultra-accelerated observer could see and/or generate massive particles).
Therefore, "entropy can increase no more" depends choosing [a] a (large) subset of possible observers to the exclusion of others and [b] treating the cosmological frame as useful in picking out quantities like entropy when the sparse distribution of matter barely favours that frame's use. Neither is in the spirit of relativity. Switching to different set of coordinates can reveal an increasing volume of relatively empty space and a cooling of the remaining relic radiation and the relatively newer emission from the cosmological horizon (see Gibbons & Hawking (1977), Phys Rev D 15 or the sadly not very detailed <https://en.wikipedia.org/wiki/Gibbons%E2%80%93Hawking_effect>).
Therefore, in an FRW universe that asymptotes to de Sitter -- our standard cosmology -- there is no true vacuum state for a comoving observer, and there are many-particle states for various other observers. So at least Boltzmann entropy (S = k_B log W) can always increase and this increase is in principle measurable by whatever scientist is left in the almost void.
(Of course, our model cosmology gets updated from time to time with new discoveries, and it may turn out that we are not destined to asymptote to de Sitter, but at the moment as far as we know there is not much wiggle room without adding as-yet-unobserved features to our known universe.)
> video ... frame[s]
> no time was spent
The apparently paused frame (not much is going on in any direction) will grow from a very very dark grey (in the sense of "grey body radiation") to even darker grey while on "pause", as the (time-dependent) Hubble parameter decreases and as the CMB redshift (and the redshift of earlier-time cosmological horizon radiation) increases.
Ignored above: relic neutrinos, fluctuation theory and nonequlibrium statmech in general, observers with relativistic spin; these can all break the placidness of a far-future "tremendously low frequency" (sub-)radiotelescope's view. Also don't see much point in delving into speculative physics (like BTSM particle physics, quintessence and similar dark energy, or non-GR gravitation) looking for anything that undermines the above or which might support your analogy.
> Brownian motion which is for all intents and purposes time-reversible
How does the Brownian motion in a hot cup of tea compare to the Brownian motion in a room-temperature cup of tea? If you wait long enough, how does your hot tea not cool down?
(Or you can substitute an ideal gas in a container of volume V and the same ideal gas after adiabatic cooling after one increases V to say 4V while keeping everything else the same.)
> does the time dimension of the universe disappear
On theoretical grounds, no. All timelike and lightlike trajectories (that do not enter black holes) extend to the infinite future.
On practical grounds, maybe. In the standard model, our universe's volume keeps expanding into the infinite future, adiabatically cooling the matter (and radiation) within it, leading to among other things the redshifting of the cosmic microwave background. There is also cold (and cooling) radiation from the cosmological horizon. These photons may become very difficult, and in practice impossible, for some future scientist not accelerated with respect to the cosmic horizon to detect. However, strong acceleration, or a boost with respect to the cosmic frame, gives hotter and more particles in one direction, making them easier to detect.
So if even in the extreeeeeeeme future there exists the possibility of an accelerated or ultraboosted observer, can one say that the universe is truly frozen up to undetectaby small fluctuations in the energy-density of a (set of) chosen point(s)?
There are certainly ultraboosted observers of a sort now: cosmic rays ejected from extreme events like blazars and supernovae, including the famous "oh my god!" particle are examples. The GZK cutoff is a mechanism by which ultraboosted charged particles interact (given sufficiently long distances) with very low frequency photons, such as those from the cosmic microwave background. This mechanism should persist into the infinite future, so ultra high energy particles spit out of the last evaporating black holes might create interesting fireworks that way, for a very long time. They'd become rarer and rarer, but it's not clear they would ever totally vanish (it's something I'd have to actually calculate because I don't trust my intuition there!). Even if such naturally ultraboosted photons did become extinct, a future observer with an enormous particle accelerator could probably scatter charged particles off distant and very red photons. It might take a long time for the ultraboosted particle to interact with a very red photon and thus the roundtrip might be annoying for that future scientist, but at least there is a way to tell that the universe beyond its local laboratory is not completely empty (indeed with some care the scientist could continue to track the expansion history of the very red and very sparse universe of many trillions of years from now).
> highest state
The highest Boltzmann entropy would be complete vacuum, but that is unachievable because of horizon radiation, and therefore extreme observers (extremes of linear momentum against the horizons and the cosmic microwave background; extremes of spin angular momentum; extremes of acceleration) can still get a fairly exciting sky even as the universe empties out. They in turn might also produce exciting moments for bored observers of a very very dark sky.
Penultimately, "sky" and the equivalence principle: if there remains a planet with a high surface gravity after galaxies are gone and black holes have evaporated, any instrument placed on that will get a view with hotter/bluer and more particles than an low-mass instrument (like JWST mass) freely floating in space and seeing no (or only very very weak) dipole isotropy to the CMB.
And finally, that's not super fanciful - we already know some stars have been ejected from their galaxies (and very likely galaxy clusters), so a "dead" planet in its own Hubble volume is plausible. Maybe an observatory was built on a superearth before it escaped into inter-galaxy-cluster space, equipped with some reservoir of power that can last until the dark times.
So,
> entropy maximum
Given current knowledge, totally empty space is not a configuration our universe will relax into, although it'll be a very good approximation.
> time dimension
Here we return to the theory. Our universe is locally Lorentz-invariant, meaning there are three spacelike dimensions and one timelike one, and physics doesn't change for a system displaced very slightly in space or in time. There is no way to change that aspect of the universe without a fairly dramatic modification of extremely well tested parts of modern physics, and that necessarily includes a cut-off mechanism that keeps what we see here-and-now (and older distant things) locally Lorentz-invariant. (The other end of the universe has this issue too: if you want different dimensions or want break the translation invariance in our 3+1 dimensions near the hot big bang you have to make them something we don't notice here-and-now or in faraway things we see that are backlit by distant quasars).
Local Lorentz invariance everywhere leads to my paragraph near the top. It gives us three types of trajectory (spacelike, timelike, lightlike) and guarantees that the latter two types all go to the infinite future (if they do not enter black holes). <https://en.wikipedia.org/wiki/Causal_structure#Tangent_vecto...> for details.