# Entropy and Information

#### The_Brain

I read the new Scientific American which contains an article that talks largely about entropy and information. In it the author states that there is a minimum area that a certain amount of entropy can be contained in; this got me thinking. Could it be plausible that an increase in entropy over a certain limit could necessitate an increase in area? I was thinking along the terms that the universe could be enclosed by the minimum volume needed to harbor the amount of entropy it contains. Thus, since entropy is always increasing, the minimum volume needed to harbor that entropy must also increase, leading to the conclusion that the universe is forced to expand. Even farther out, since the level of entropy is growing in the universe at an increased rate (no idea if this is correct) then the universe if forced to expand more rapidly (accelerate). I have no idea if this is even plausible; it could be that when the level of entropy surpasses its enclosed system, the system fails. Thoughts, opinions, and criticisms are welcome. I'm just a high schooler who has no idea what he's talking about.

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#### jcsd

Gold Member
Your close, what would acrtual happen is a universe of a fixed size would eventually be in thermodynamic equilibrium, after this point it's entropy would not increase and remain comnstant (excepting the probailistic natur of entropy which would cause it's actual entropy to fluctuate and actually decrease fr short periods before going back to thermodynamic equilibrium).

#### marcus

Gold Member
Dearly Missed
Originally posted by jcsd
Your close, what would acrtual happen is a universe of a fixed size would eventually be in thermodynamic equilibrium, after this point it's entropy would not increase and remain comnstant (excepting the probailistic natur of entropy which would cause it's actual entropy to fluctuate and actually decrease fr short periods before going back to thermodynamic equilibrium).
jcsd here's a question that might interest you to think about:
in the example you offer here,
what would happen to the size of the universe and
to the amount of entropy in it
if one of the stars in it collapsed
to form a black hole?

I hope this extra wrinkle fits into the context of
what you were describing and is a meaningful question.
not sure how "size" is measured in this discussion,
I guess volume? The_Brain was talking about volume
and entropy.

#### jcsd

Gold Member
Well, blackholes are compliant with the laws of thermodynamics and have a surface temperature, so eventually the blackhole would evaporate, allowing the universe to go into a state of thermodynamic equilibrium.

Yes,I was just using size as synonym for volume.

#### marcus

Gold Member
Dearly Missed
Originally posted by jcsd
Well, blackholes are compliant with the laws of thermodynamics and have a surface temperature, so eventually the blackhole would evaporate, allowing the universe to go into a state of thermodynamic equilibrium.

Yes,I was just using size as synonym for volume.
so as long as there is one black hole in the universe, the entropy of the universe cannot be defined?
you have to wait 1060 years or something to give
every black hole time to evaporate and then finally you can
have a state of thermodynamic equilibrium and define the entropy?

It seemed to me that when a star collapses to form a black hole there is at that moment quite a substantial increase in entropy.
If you can, please elaborate. Several of us might benefit from
more discussion of this.

#### jcsd

Gold Member
No, the entropy of a black hole is well defined and is a function of the surface area of the event horizon, so the more massive the black hole the higher it's entropy. It was known for a while that blackholes must somehow radiate energy in order for them to comply with thermodynamics and it was Stephen Hawking's who by applying quantum physics showed how the mechanism of this radiation (Hawking radiation). Also of note is the application of superstring theory which gives an explantion on why black holes should have a definte entropy in terms of the number of possible arrangements of strings.

#### marcus

Gold Member
Dearly Missed
Originally posted by jcsd
No, the entropy of a black hole is well defined and is a function of the surface area of the event horizon, so the more massive the black hole the higher it's entropy. .....
That is what I thought, jcsd.
So now I am back to my original question.

You seem able to imagine defining the entropy of the universe.
What happens to the entropy of the universe
when a star collapses to form a black hole?

Does it change, by a lot, by a little? Does it stay the same?
Can you discuss this a bit for us?

#### jcsd

Gold Member
I don't know exactly what happens to the entropy of a closed system containing a region of gravitational collapse, except gravitational collapse is an apparently irreversible process, which is an indicator of a large increase in entropy.

Of course in these models we're just viewing the universe as a closed system of finite and fixed volume.

#### marcus

Gold Member
Dearly Missed
Originally posted by jcsd
I don't know exactly what happens to the entropy of a closed system containing a region of gravitational collapse, except gravitational collapse is an apparently irreversible process, which is an indicator of a large increase in entropy.

Of course in these models we're just viewing the universe as a closed system of finite and fixed volume.
I believe your intuition is absolutely right here! A black hole has an enormous amount of entropy per unit volume, within its event horizon. Far more than any star.

We might be able to make a rough order-of-magnitude estimate.

edit: Bravo! jcsd, I just came back to this thread and saw that you have in fact does this for 5 solar masses. Good show---nice to have a definite number in the picture

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#### jcsd

Gold Member
I've calculated the entropy of a Schwarzchild black hole weighing 5 solar masses and got the figure of S(bh)= 3.62 * 10^55 J K^-1 (SI units).

#### The_Brain

Originally posted by jcsd
I don't know exactly what happens to the entropy of a closed system containing a region of gravitational collapse, except gravitational collapse is an apparently irreversible process, which is an indicator of a large increase in entropy.

Of course in these models we're just viewing the universe as a closed system of finite and fixed volume.
Ok, so what if the model we used was one such as when a certain limit is reached, the system expands in volume? 2nd Law says that entropy can never decrease, so after the system expands it can only gain more entropy which would require more volume at a critical limit.

#### jcsd

Gold Member
Yep, but saying the volume will get larger is non-trivial, i.e. there is no reason to suppose it would.

You have to realize entropy, though it works with almost no exception (actually I should say scientists have observed decreases in entropy on a minute scale over a short period of time) are not absolute, they are statistical. This is why the law is phrased "The entropy of a closed system tends to increase over time" rather than "The entropy of a closed system will increase over time", infact after an arbitarily long period of time you would (statistically) expect to see a huge decrease at some stage in the total entropy of a closed system of fixed volume.

#### The_Brain

Originally posted by jcsd
infact after an arbitarily long period of time you would (statistically) expect to see a huge decrease at some stage in the total entropy of a closed system of fixed volume.
I thought the entropy of a closed system could never decrease?

#### jcsd

Gold Member
No, they're is always a very tiny chance that it will decrease, as I said before it is statistical.

#### vedder

Thinking about the original question, does (+)entropy imply an increase in volume? I mean, the equation does not seem to imply +volume as a necessity but...

Lets say we have a closed system defined as one liter of 100 degree H^2O in a perfect container(one that does not allow diffusion from it to it's surroundings). The information in that container will not become less organized will it? It seems to me that the information needs an area to spread into for it to become less organized. That would mean that The_Brain was pretty right on when he said "an increase in entropy over a certain limit could necessitate an increase in area".

#### jcsd

Gold Member
No, Vedder an increase in entropy MOST DEFINTELY does not mean an increase in area. The point is the 2nd law of thermodynamics only works because the universe is not in a state of thermal equilibrium. The conclusion should be that the 2nd law of thermodynamics will cease to hold at this point, not that it will contiune to hold and therefore the volume will increase.

#### vedder

I guess i was getting away from the intent of the original post a bit.

The point is the 2nd law of thermodynamics only works because the universe is not in a state of thermal equilibrium
Of course if we look at the equation for entropy it does not imply an increase in area. What i mean to say is that here, in this universe which is not in thermal equilibrium, entropy is the only thing i can think of, as a law, that would necessitate the expansion we observe.

I do think you are very correct to say this...

The conclusion should be that the 2nd law of thermodynamics will cease to hold at this point

#### The_Brain

vedder said:
What i mean to say is that here, in this universe which is not in thermal equilibrium, entropy is the only thing i can think of, as a law, that would necessitate the expansion we observe.
Yes, this is exactly what I am led to think. However there are people here who know much more about physics than I do so please, if anybody can, tell us why this might be false?

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