Entropy of the whole universe In "Thermal Physics", Charles Kittel proves that entropy always increases in systems when the degree of freedom are increased (adding particles, adding energy, expanding volume, etc ).
I started to think about the entropy of the whole universe. Since by definition the universe contain all the particles, and energy is constant, i expect the entropy to be constant. (Forgetting the expansion of the universe for the moment)
here: Can entropy of Universe be constant?
it is said that the entropy of the universe can in fact be constant if the universe is closed. 
Assuming that it is, then we already are at the maximum entropy, but why are we not at the heat death of the universe ?
Did the universe didn't have enough time to evolve ? (it is not at the most probable state of system yet)
Or is the expansion of the universe crucial ?
I'm also curious about which other factors could increase the entropy of the whole universe, besides expansion.
 A: This is a good question. In fact, as I'll explain, it leads right to some topics of current research.
First, an important fact about entropy: if you look carefully at the proof Kittel gives, it should only be applicable to a situation in which you add degrees of freedom to a small system that is exchanging energy with some heat bath. However, if you consider an isolated system in which you know as much information as possible about every part, the total entropy of this system is not necessarily larger than the entropy of a small part of the system. In fact, if you know everything possible about a closed system its entropy is zero, since you know the microstate, while a subsystem may have non-zero entropy.
Anyway, as you say, let's imagine a hypothetical universe that is roughly static, and has some region that is interacting. In addition, there is a far-off region that is nearly isolated, and only connected by some probe that allows it to watch the rest of the "universe" without significantly disturbing it. From the perspective of an observer in this far-off region, the rest of the universe is essentially closed, but still appears to be evolving toward equilibrium according to principles of thermodynamics. This is a bit of a puzzle. Thermodynamics is usually defined in an open system, especially for a system which can exchange energy with some exterior heat reservoir, and without this it is not clear, for example, what role entropy plays. This problem actually gets worse, if anything, when you try to describe your universe with quantum mechanics. Roughly speaking, in quantum mechanics you can never lose information about an isolated system, and losing information is necessary for entropy to increase.
You may be surprised to hear that the resolution to this question is actually not completely settled! However, a popular proposal goes by the name of the "Eigenstate Thermalization Hypothesis." This says, in essence, that some (but not all) interacting closed systems follow an evolution such that any small piece of the system sees the rest of the system as a heat bath, and as a result each part approaches a state that looks like an equilibrium. If this were true of the universe, it would mean that the entropy of the whole thing does not change, but if you look at any subset of the universe then the entropy of that subset always increases in the same way an open system would. As I mentioned at the top, this is possible because the definition of entropy is such that a subsystem can have a larger entropy than a total closed system.
As suggested by the word 'hypothesis,' this is more of a proposal than an ironclad explanation. However, we can actually do experiments that show that this happens in some cases. Ultracold atomic gas experiments can create an interacting system that is very, very close to completely isolated- it is levitated by electric and magnetic fields, in a nearly perfect vacuum, and too cold to emit any significant radiation. Nonetheless, if you disturb these systems out of equilibrium, and wait for awhile before you check them, in some cases they will return to what looks like a new thermodynamic equilibrium, in a sense appearing as if their entropy increased even though the system is closed. This shows that at least in some cases this kind of behavior is possible, but research is ongoing to try to understand under what circumstances it is true. For the universe as a whole, I think it is fair to say that all bets are off.
A: Penrose has something to say on the matter. Namely, that our universe started off in an absolutely amazingly low entropy state.

However, we are considering a closed universe so eventually it should
  recollapse; and it is not unreasonable to estimate the entropy of the
  final crunch by using the Bekenstein-Hawking formula as though the
  whole universe had formed a black hole. This gives an entropy per
  baryon of 10^43, and the absolutely stupendous total, for the entire
  big crunch would be   10^123.   This figure will give us an estimate
  of the total phase-space volume V available to the Creator, since this
  entropy should represent the logarithm of the volume of the (easily)
  largest compartment. Since 10^123 is the logarithm of the volume, the
  volume must be the exponential of 10^123, i.e.   V = 10^10^123.   in
  natural units! (Some perceptive readers may feel that I should have
  used the figure e^10^123, but for numbers of this size, the a and the
  10 are essentially interchangeable!) How big was the original
  phase-space volume W that the Creator had to aim for in order to
  provide a universe compatible with the second law of thermodynamics
  and with what we now observe? It does not much matter whether we take
  the value W = 10^10^101 or W = 10^10^88 given by the galactic black
  holes or by the background radiation, respectively, or a much smaller
  (and, in fact, more appropriate) figure which would have been the
  actual figure at the big bang. Either way, the ratio of V to W will
  be, closely   V/W = 10^10^123.   This now tells us how precise the
  Creator's aim must have been: namely to an accuracy of one part in
  10^10^123.   This is an extraordinary figure.

A: Increase in Entropy signifies that the system is going into higher disorder state. Even if you assume that universe has constant energy and particles, it doesn't mean the change in Entropy is zero.
Second, even if change in Entropy is zero, it doesn't mean the Entropy of the system is at maximum. If you take an ideal reversible process, the change in Entropy would be zero but the Entropy of the system wouldn't be maximum. Maximum Entropy means no macroscopic object: Only photons and particles flying around freely. There can be no stars, planets or intelligent life in maximum Entropy state. And, this is where the universe is heading.
A: Prior to posting this answer, I reviewed the definition of thermodynamic entropy.
Entropy is a measure of the number of “specific ways” in which a thermodynamic system may be arranged. If this is wrong, let me know.
Since there is no way to calculate or measure this, the answer must be theoretical. 
If the universe is infinite, then the entropy would also be infinite.
If we are talking about the only universe we can really discuss, the known universe. This is the universe we can see, detect, measure, etc.; anything beyond that is in fact unknown and we can’t say for certain it even exists until technology enables us to detect more of it.
If we define the universe as the “known” universe and put a bound on it, there would be a finite number of “specific ways” (based on the definition above) in this finite universe. Intuitively this would indicate there would be a “maximum entropy”.
Outside the “finite” universe would be nothing more than a “thought-provoking exercise”.
