Can the universe build itself back together after the Heat Death? We all know that due to the 2nd Law of Thermodynamics the universe after a certain long period of time will eventually have all its energy spread out homogeneously and reach a Heat Death where nothing interesting ever happens anymore.  However, given that the universe's lifespan is infinite, and the 2nd Law of Thermodynamics is a statistical law (meaning that decreases in entropy are low in probability but still possess nonzero chance), isn't it reasonable to assume that at some point, very far down the line in the Heat Death, the universe puts itself back together and forms stars, galaxies and everything interesting again?  Similar to how a monkey on a typewriter, given an infinite amount of time, will type out the complete works of Shakespeare, isn't the large- scale reduction in entropy just an extremely, extremely rare, but inevitable event which will eventually occur given an infinite timespan?
If so, the universe can never really die, can it?
 A: This was originally proposed by Boltzmann, who suggested that the entire universe could be a low-probability low-entropy fluctuation in a high-entropy world:

We assume that the whole universe is, and rests for ever, in thermal
  equilibrium. The probability that one (only one) part of the universe
  is in a certain state, is the smaller the further this state is from
  thermal equilibrium; but this probability is greater, the greater is
  the universe itself. If we assume the universe great enough, we can
  make the probability of one relatively small part being in any given
  state (however far from the state of thermal equilibrium), as great as
  we please. We can also make the probability great that, though the
  whole universe is in thermal equilibrium, our world is in its present
  state. It may be said that the worlds is so far from thermal
  equilibrium that we cannot imagine the improbability of such a state.
  But can we imagine, on the other side, how small a part of the whole
  universe this world is? Assuming the universe great enough, the
  probability that such a small part of it as our world should be in its
  present state, is no longer small.
If this assumption were correct, our world would return more and more
  to thermal equilibrium; but because the whole universe is so great, it
  might be probable that at some future time some other world might
  deviate as far from thermal equilibrium as our world does at present.

Are Boltzmann worlds plausible? Current cosmological models have unbounded futures and are at finite temperature due to horizon radiation, so it looks like they are a prediction. This doesn't sit well with some physicists, of course. As Susskind points ut:

This would mean not that the future is totally empty space but that
  the world will have all the features of an isolated finite thermal
  cavity with finite temperature and entropy. Thermal equilibrium for
  such a system is not completely featureless. On short time scales not
  much can be expected to happen but on very long time scales everything
  happens.

Smaller fluctuations are exponentially more likely than larger fluctuations. That means that occasionally field fluctuations will not just randomly produce particle-antiparticle pairs, but atoms, molecules, and entire organisms that briefly persist. This includes randomly generated observers that can have brains in arbitrary states ("Boltzmann brains", a headache to some philosophers and physicists) and domains that have low entropy (the Boltzmann worlds). But note the probability distribution: finding ourselves in a big universe with lots of remote galaxies is far, far less likely than being in a single solar system floating in nothingness. So our observations seem to rule out that we live in a Boltzmann world.
So if our observations rule out living in a Boltzmann world but theory predicts that most observers live in them, there is something awry. But there is no general agreement on whether anthropic arguments, the unboundedness of the future, the way we measure probability across "big worlds" or something else is the source of the contradiction. 
A somewhat related, but distinct, phenomenon is Poincare  recurrence. In systems that has a dynamic that maps their bounded phase spaces to themselves and conserve phase space volume the system state will after a sufficiently long but finite time return to a state very close to the initial state. It may hence seem plausible that the universe will eventually return to a state close to the initial state and history will repeat. 
Contrarily, the expansion of the universe means the phase space is expanding and the theorem does not apply. In a very real sense the expansion makes the evolution of the universe irreversible even if local regions may experience Boltzmann recurrences due to random chance and relative isolation. 
But see also Lubos Motl's answer to this question and Susskind's paper on de Sitter space. Motl claims the recurrence time is $\exp(10^{120})$ because by the holographic principle we have a dynamics on the horizon - which is unchanging in size. 
Whether Poincare recurrence will happen hence appears to hinge on somewhat unsettled theoretical issues.
