Not with a Bang

Another question this week from my grandfather-in-law-to-be, George: How does entropy work on the scale of the universe? Doesn't gravity reduce entropy by pulling things together?

Before I get into this, I should first explain what entropy is.  I've mentioned it before as a type of randomness, but it's a bit more complicated.  First, there's the idea of microstates and macrostates.  Imagine we have a box with 10 particles bouncing around inside, and we take a photo at two different times:

 These look basically the same, even though the particles are in different places.  These are two microstates belonging to the same macrostate.  Now we take another photo, and find this:

This is qualitatively different, and is therefore a different macrostate.  However, there are far fewer ways to put all the particles in one corner like that, so the number of microstates is much smaller.  Thermodynamics assumes that all microstates are equally likely, which is why we usually only see the macrostates that have many of them.

Entropy is a way of measuring how many microstates a configuration has.  It's defined as

where k_B is Boltzmann's constant, and Ω is the number of microstates.  The second law of thermodynamics says that entropy always increases in any process – In a way, it defines the progress of time.  A common example is a box with two gasses separated by a divider.  If you remove the divider, the gasses will mix, and there are astronomically more ways for them to be mixed than separated, so they will not return to the original state.

So now, back to the original question.  It seems as if increasing entropy would imply the universe would smooth out to an equal amount of matter everywhere, but gravity acts against that.  The key is that when gravity brings lots of matter together, you can end up with a black hole.  It is believed that black holes carry entropy proportional to the area of their event horizons, which in turn is proportional to the mass that has fallen into them.  Therefore by collecting matter, a black hole is not violating the second law.

That leaves us with a rather sad conclusion called the heat-death of the universe.  If entropy continues to increase, it will reach some maximum value, at which point the universe will be a uniform temperature at all points.  Differences in temperature are required to extract energy from a system, so heat-death implies that no processes can occur after that point.

This is the way the world ends
This is the way the world ends
This is the way the world ends
Not with a bang but a whimper.
        –T. S. Eliot