What is the definition of the entropy of the Universe?

[AlexandreZani]

My understanding is that to define the entropy of a system what you have to do is as follows:

• Define the boundaries of your system.
• Define a set of "microstates" of the system.
• Define a partition of microstates of the system where each element of the partition is measurable and known as a "macrostate".
• The entropy of the system in some macrostate A is the logarithm of the measure of the microstates contained in mactrostate A.

Now, I often hear cosmologists say things like "the entropy of the early universe was low." What I understand that to mean is that for some specific partition of the microstates of the universe P, the state of the early universe was contained in some "small" element of that partition.

What I am trying to learn is: What is this partition P which is being implicitly referred to? And optionally, what is special about that partition that it is worth talking about as opposed to the very large number of alternative partitions you could define?

 

[Christopher Grayce]

They are comparing the number of microstates of the universe at its beginning with the number now, which is much, much greater. Perhaps you are confused because you are imagining that the correct number of microstates to count then was the number (if any) that the universe could also have been in, according to the Standard Model, at the time of the Big Bang. But that's not how we usually do it anyway. When you have a gas confined to a small cylinder, and then you open the cylinder, we say the entropy of the gas increases because now it can occupy all the states in the room, not just those in the cylinder. The fact that those states were inaccessible to the gas before we opened the cylinder is *why* the entropy was initially low. Similarly, the fact that the states the universe occupies now were inaccessible at the time of the Big Bang (because the universe was confined to almost a single point) is *why* the entropy of the universe has increased.

Remember there's no requirement in thermodynamics that you count the states at a given instant. Thermodynamics is not a dynamic theory, time plays no role in it. All that matters is how many microstates the system can access in such-and-such a macrostate, versus how many it can access in some other. It doesn't matter if something dramatic (like the passage of 14 billion years) has to occur to get from one macrostate to another.

 

[Grinkle]

the fact that the states the universe occupies now were inaccessible at the time of the Big Bang (because the universe was confined to almost a single point)

That is not known to be a fact. If the universe if of infinite extent now, then it was always of infinite extent. It was just a lot denser in the past - a dense infinite thing evolving to a less dense infinite thing. The OP is asking (to my reading) where to draw the boundary in an infinite universe so that one can begin discussing the entropy of that bounded region in a meaningful way.

No one knows if the universe is infinite or not - many argue that Occam's razor indicates it is infinite.

 

[PeterDonis]

the states the universe occupies now were inaccessible at the time of the Big Bang

We don't really know whether this is true or not, but if it is true, it's not for the reason you give. See below.

(because the universe was confined to almost a single point)

No. As @Grinkle has pointed out, if the universe is spatially infinite (and our best current model is that it is), it always has been. So it was at the time of the Big Bang too.

We actually don't really know how to count the microstates of the universe. However, we can make some general observations, the key one of which is that at the time of the Big Bang, the universe was much more uniform than it is now. And in the presence of gravity, with the possibility of gravitational clumping (which has indeed happened to a very great extent since the Big Bang), "uniform" means "low entropy", because there are many, many more ways for the universe to be lumpy, with very dense gravitationally clumped regions separated by large voids, than there are for it to be uniform, with the same density everywhere. This observation is the basis for cosmologists saying that the entropy of the early universe was low.

 

[AlexandreZani]

Why is clumpiness important? (Obviously, it's important in the sense that humans care about our existence which depends on lumpiness, but that's very subjective.) Specifically, I've often heard it said that the arrow of time is best understood as the product of the early universe having low entropy. But it seems weird to say that the arrow of time is a product of there being less clumpiness a long time ago than now. (Perhaps I'm thinking about this wrong)

 

[AlexandreZani]

Also, how is clumpiness defined? Something like variance of the distance between particles? Or maybe something like how warped spacetime is? I would imagine a uniform universe to have spacetime with something like a constant curvature while a lumpy universe would have a curvature which would take a very large range of values.

 

[PeterDonis]

Why is clumpiness important?

how is clumpiness defined?

To answer the second question first, "clumpiness" is basically the variation in density from place to place. In the early universe there was very little variation in density. Now there is a huge variation, somewhere between 40 and 50 orders of magnitude.

The reason it's important is that, as I said in post #4, there are many more ways for the universe to be clumped than there are for it to be uniform. So a clumpy universe has much higher entropy than a uniform one.

Another way to think about it is that we expect an increase of entropy to be associated with a spontaneous process, and in the presence of gravity, clumping is a spontaneous process; whereas "unclumping" is not. So in the presence of gravity, we expect the direction of increasing clumpiness to be the direction of increasing entropy.

I've often heard it said that the arrow of time is best understood as the product of the early universe having low entropy.

This is often said, yes, but it conceals a hidden assumption that is very likely not true. The hidden assumption is that there is some maximum entropy state of the universe (this is the sort of thing that is meant when the "heat death" of the universe is talked about). But if the universe is spatially infinite and will expand forever, there will not be any maximum entropy state; the entropy can continue increasing without bound, forever. And if that is the case, there would still be an arrow of time even in a universe that started off in a higher entropy state (for example, a hypothetical universe that started out the way our universe will be a trillion years from now).

But it seems weird to say that the arrow of time is a product of there being less clumpiness a long time ago than now.

That's just because you don't intuitively associate "less clumpiness" with "lower entropy". Once you have retrained your intuition to make that association seem natural, then the above quoted statement will seem natural too.

 

[Haelfix]

This is often said, yes, but it conceals a hidden assumption that is very likely not true. The hidden assumption is that there is some maximum entropy state of the universe (this is the sort of thing that is meant when the "heat death" of the universe is talked about). But if the universe is spatially infinite and will expand forever, there will not be any maximum entropy state; the entropy can continue increasing without bound, forever. And if that is the case, there would still be an arrow of time even in a universe that started off in a higher entropy state (for example, a hypothetical universe that started out the way our universe will be a trillion years from now).

You are correct in saying that this is an active area of research. Typically, the argument for the above goes something like the following. In a spatially flat universe with a positive cosmological constant, we will eventually asymptote to De Sitter space. The presence of the DeSitter horizon thus yields an entropy bound (the Bekenstein bound), which any comoving observer would be subject to. This of course would create a cosmological heat death in the sense that the free energy of the system would go to zero as the total entropy approaches a constant.