Tuesday, December 27, 2011

The Heat Death of the Universe

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Time's Arrow. The previous post is essential to understanding this post.

In most of the discussion of nonsense in this series, the nonsense stems from a poor understanding of physics.  This post introduces some nonsense that must be taken seriously.  Perhaps, this nonsense, also stems ultimately from a poor understanding of physics.  The people with the poor understanding this time, however, are some of the most brilliant minds in physics.

The School-Book Story

This discussion starts with the school-book story of the heat death of the universe.  By calling it the "school-book" story I do not mean to pooh-pooh it too much.  In fact, it is most likely the correct story.  Much of this post, however, will focus on caveats and complications to the story as it is usually told.

In thermodynamics, the universe is defined as the system and its surroundings.  We have seen that the second law requires that for any change the total entropy of the system and the surroundings must increase or stay the same.  As time goes by, therefore, the entropy of the universe increases.

The consequence is that the universe will eventually head toward its maximum entropy, and there is no going back. The universe will become a very boring place.  All the stored energy that could be used to do useful work will be dispersed as heat, and the universe will become a very cold, well-mixed, dispersed, undifferentiated place. The universe will be at thermodynamic equilibrium.  There will not be hot places and cold places.

Criticism of the School-Book Story

As stated above, the school-book story is a good story.  In fact, it is probably right, but is the school book story definitely true?  To make such an assertion, it is necessary to examine some of the holes in the story.  It is also necessary to enter the world of speculative science.  First I turn to the obvious hole.

By defining the universe as the system plus the surroundings, I played a neat trick.  I essentially defined the universe as being a closed system in the thermodynamic sense (we have to be careful here because cosmologists are talking about something else when they refer to a closed universe).

What if the universe in not a closed system?

When examining a small system it is not hard to see the logic of the idea that the change in energy of the system is equal and opposite to the change in energy of the surroundings. If energy is conserved, such a conclusion must be true, but what about the edges of space and time?  What about extreme conditions that might exist somewhere in the universe.  Do we really know that the universe is closed?

If the universe were not closed, one of two things must be the case. 1) There must be someplace outside the universe from which energy and/or matter can be transferred to or from the universe, or 2) energy is not really conserved.  Let us examine each of these possibilities.

Is the Universe All There Is?

In a semantic sense, it is easy to say yes.  The word "uni" means one, meaning there is only one universe composed of everything that is.  It is important to understand, however, that this answer is based on a definition.

If there are places outside what we commonly call the universe, we need a new word for what we call the universe.  Unfortunately,  semantics aside, the common way to discuss this issue is to brush aside the etymology of the word "universe," and allow for the possibility that our universe is not all that there is.

If there are truly places outside our universe that can transfer energy to our universe, then the universe is not a closed thermodynamic system, and we cannot assert that the heat death will take place. Although, see the section below on multiple universes.

On the other hand, consider whether everything put together, the universe combined with everything outside the universe is a closed system?  If so, then there would be an inevitable heat death of this super-universe that truly includes all the stuff.  What happens if the collection of all the stuff is actually infinite?  What if there are regions that actually obey different physical laws?

Conservation of Energy and Noether's Theorem

One of the most profound theorems in physics is Noether's Theorem.  Noether's Theorem is a way of deriving conservation laws from fundamental symmetries.

For example, the conservation of momentum can be related to the symmetry of space.  It does not matter where we perform an experiment: we can translate the axes, and the laws of physics still apply.  This symmetry can be shown to lead to the conservation of momentum.

In three dimensional space the laws of physics apply no matter which way the axes are oriented; in other words we can rotate the experiment in space without altering the results.  This fact implies that angular momentum is a conserved quantity. (I'm proud to be a student of the man who literally wrote the book on angular momentum!).

Conservation of energy is a result of the symmetry of time (note that we are not talking about backward-forward symmetry as discussed in the previous post entitled Time's Arrow.).  To quote Feynman
In quantum mechanics it turns out that the conservation of energy is very closely related to another important property in the world, things do not depend on the absolute time.  We can set up an experiment at a given moment and try it out, and then do the same experiment at a later moment, and it will behave in exactly the same way.  Whether this is strictly true or not, we do not know.  If we assume that it is true, and add the principles of quantum mechanics, then we can deduce the principle of the conservation of energy. It is a rather subtle and interesting thing, and it is not easy to explain. (Feynman, I-4-7).
So the assumption that energy is conserved depends on the assumption that absolute time does not matter. If energy is not conserved, then the cosmos is not strictly a closed system, and the second law does not require  the increase of entropy.

We live in a time in which time seems to maintain such symmetry; thus we experience conservation of energy. Is it possible that at the very beginning of the universe, or at the very end that time may behave differently?  Such a notion is probably complete nonsense.

There is no reason to posit that time behaves any differently.  There are no observable events that are explained by such a notion.  Perhaps, one day such an observation may be made, however.

If energy is conserved, and we consider the whole cosmos of however many "universes,"  or other entities there may be, then the school-book story of the heat death of the universe seems to hold up.  Still, there are some added subtleties that need exploring.

Some of the ideas that I will examine in this post are a little different from normal science because they involve potential models of the cosmos that make predictions that are almost by definition not observable by us.  If a model makes a prediction that we cannot observe, it is more speculation than science. Still, it is necessary to consider such ideas, if we want to assert that we know the fate of the universe.

Many Worlds

The many-worlds interpretation of quantum mechanics (QM) was proposed to explain the finding of QM with regard to the collapse of a wavefunction.  In Everett's words:
Alternative 5: To assume the universal validity of the quantum description... The general validity of pure wave mechanics, without any statistical assertions, is assumed for all physical systems, including observers and measuring apparata. Observation processes are to be described completely by the state function of the composite system which includes the observer and his object-system, and which at all times obeys the wave equation...
The idea is that instead of probability determining to which alternative a quantum system must collapse, all the alternatives are in fact real.

To properly understand this idea requires an understanding of quantum statistics, in which a system can be in a superposition of two (or more) states before a measurement determines the state. In classical mechanics, we can flip a coin and not know the result, but there is a result. In QM, the result is fundamentally undetermined before the measurement.  The wavefunction is 50% heads and 50% tails.

This idea is so bothersome that the idea that both results actually occur but in different worlds has been proposed.

I should state up front that I am not happy with this theory because it does not make any predictions that would have a different result if the theory is true or if it is not. It does not appear to be a falsifiable theory.

Nevertheless, there are people a lot smarter than I am, who take this notion seriously. So, this is a piece of nonsense that we need to admit as possibly true.

To examine the consequences of the many world hypothesis on the heat death of the universe, let us take a shortcut and pretend that we can model QM by a coin toss. There are some subtle differences between coins and particles governed by quantum mechanics. Such particles come in two flavors: 1) fermions, particles that obey Fermi-Dirac statistics, and 2)bosons, particles that obey Bose-Enistein statistics.

If we have two pennies, we can distinguish them from one another.  The same is not the case for fermions or bosons.  In fact, the wavefunction that describes them must not be able to distinguish between them.

If this idea seems like an insignificant difference, it is worth pointing out that the very existence of covalent bonding in molecules owes to this fact. The universe would look very different if electrons did not obey Fermi-Dirac statistics.

Having said all that, I am now going to ignore it, and treat the collapse of a wavefunction just like a coin toss; except that in the many-worlds interpretations both results of the coin toss take place in different actually existing worlds.

If we flip a coin a large number of times, most of the worlds will look very similar. Consider my post on fluctuations to better understand the statistics involved.

Consider the fact that in one of the worlds, heads comes up every time.  In this world, entropy would not increase.

It is important to realize, two facts, however. The probability that the world in which only heads comes up is the world in which we live is vanishingly  small. Consider also, that if we consider the cosmos to consist of all of those worlds put together, that the tendency toward maximum entropy will be observed.

The many-worlds scenario does not contradict the idea of the heat-death of the universe, but it certainly complicates the issue.

Multiple Universes

Serious minds have also proposed the idea of multiple universes, perhaps infinitely many universes that exist throughout all time and constantly pop into existence. Perhaps, not all of these universes follow the same physical laws that our universe follows. Consider, however that a subset of them do, or perhaps they all do.

We cannot assert much about a universe that obeys different physical laws; so let us constrain our imagination to consider universes that have similar laws.

What if there are infinitely many universes very similar to our own? What if the number of universes is so large that it dwarfs the number of ways to arrange all of the particles in the universe?

Is it possible that in some of those universes entropy decreases with time?

Even if that is the case we must again consider two facts: 1) the probability that we live in such a universe is so improbable that it is essentially impossible. 2) The collection of all of the universes taken together would still increase their entropy.

Again, this scenario does not exactly contradict the story of the heat death of the universe, but it certainly complicates it.

Gravity and Entropy

To sum up this post, the school-book story of the heat death of the universe is most likely a good story, but there are certain seemingly nonsensical, but possible arrangements of the cosmos that make the story a bit simplistic. There is much that I have not discussed.  We live in an expanding universe, but what if we did not? What about black holes?

The next post entitled Gravity and Entropy addresses some of these questions.

Sources