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m74z00219
2010-Apr-02, 10:39 PM
Hi all,

As I understand, the universe consisted of a neutral soup of hydrogen gas for a long while after recombination (millions of years as I understand). Why is the initial formation of stars not considered a violation of the second law of thermodynamics? I understand that gravitational collapse explains how the first stars formed - and ultimately the galaxies - but in light of the second law I don't understand how this process began in the first place.

Can anyone point out the flaw in my understanding? If the universe's energy was evenly distributed at this time, wasn't it already at its highest entropy?

Thanks,
M74

grant hutchison
2010-Apr-02, 11:50 PM
Edward Harrison says that most of the entropy in the Universe is contained in the cosmic microwave background. Roger Penrose says it resides in black holes. Either way, it is increasing only slowly, because of additional radiation added by stars or by the feeding processes of black holes.
Meanwhile, the Universe's capacity for entropy is increasing with the square of its expansion ratio. So, according to Victor Stenger, the Universe began in a state of maximal entropy for its volume, but its subsequent expansion has increased its capacity for entropy much faster than entropy has actually risen during the lifetime of the Universe. So there's a lot of room for more entropy. Gravitational collapse was therefore able to emit thermal photons into the cooling Universe, buying locally increasing order at the expense of globally increasing entropy.
Stenger estimates that the current entropy of the Universe is about 22 orders of magnitude less than its current carrying capacity, so there's still a lot of room to spare.

Grant Hutchison

Spaceman Spiff
2010-Apr-03, 02:31 AM
If the universe's energy was evenly distributed at this time,...


And so there is the answer to your question. Note the word "If" that I've emphasized. The early universe (and the matter/energy) therein was not evenly distributed. There are tiny, tiny fluctuations in the intensity cosmic microwave background radiation that show the level of matter density fluctuations near the epoch of matter-radiation decoupling. The small density fluctuations eventually began growing due to gravity.

As is always the case in our universe, it is the breaking of symmetry that results in structure and stuff happening. Perfect symmetry results in nothing (or maybe even less than nothing).

Spaceman Spiff
2010-Apr-03, 02:45 AM
The other issue is in the subtleties involved in defining entropy (and changes thereof) when gravity is important, especially gravity on the cosmic scales of space-time. This is also true of energy which is a conserved quantity in the limit of local space-time frames of reference). One of the subtleties that pop out is that a very smooth space-time universe has relatively very low entropy. Gravitational instabilities that grow into structure via collapse (like stars) result in an increase of entropy.

Precisely how the laws of thermodynamics "apply" on cosmic scales is not a settled matter, as far as I know.

Ken G
2010-Apr-03, 03:41 AM
Entropy is tricky, you have to keep track of all the changes. When a star forms, it compresses, which would tend to reduce entropy, but it also gets hot, which increases entropy, and it also transports light from its hot interior to the cool space around it, which increases entropy also. Basically, any process that happens spontaneously (i.e., happens with no encouragement from any external systems), and the reverse process does not happen spontaneously, is a process that increases entropy.

Incidentally, I'm not sure what is meant by the "capacity" for entropy-- there must be something held fixed to assess that concept (presumably total energy, including some kind of gravitational potential energy). Every photon that is part of a thermal distribution has the same entropy, it makes no difference what the temperature is. So when the universe expands, all its photons have the same entropy as they always had. There is no limit to how many thermal photons you can pack into a given volume, so there is no literal capacity for entropy, but I'm sure that Stenger is referring to the capacity given some assumptions that I'm not sure what they are. It sounds like gravity is playing a key role, where he is saying that as the universe expands, there is the potential for gravity to compress pockets, make them hot, and transport the radiation from the hot areas to the cool areas, ending up with a radiation field that is hotter (so has more photons, so more entropy) than you'd get with the adiabatically cooling CMB. If so, it means that the formation of stars is actually the process whereby the expanding universe is (unsuccessfully) "trying" to catch up with its growing entropy potential. I think the "capacity" must then be putting all the matter into black holes, and sharing the emitted light everywhere. If so, you just assume some light emission efficiency (maybe 0.3 or something), multiply that by mc2, turn all that energy into radiation, and then thermalize it over the volume available, and count how many photons you end up with-- relative to how many you have now in the CMB.

m74z00219
2010-Apr-03, 07:55 AM
Edward Harrison says that most of the entropy in the Universe is contained in the cosmic microwave background. Roger Penrose says it resides in black holes. Either way, it is increasing only slowly, because of additional radiation added by stars or by the feeding processes of black holes.
Meanwhile, the Universe's capacity for entropy is increasing with the square of its expansion ratio. So, according to Victor Stenger, the Universe began in a state of maximal entropy for its volume, but its subsequent expansion has increased its capacity for entropy much faster than entropy has actually risen during the lifetime of the Universe. So there's a lot of room for more entropy. Gravitational collapse was therefore able to emit thermal photons into the cooling Universe, buying locally increasing order at the expense of globally increasing entropy.
Stenger estimates that the current entropy of the Universe is about 22 orders of magnitude less than its current carrying capacity, so there's still a lot of room to spare.

Grant Hutchison

Thanks for your reply, Grant. Someone I forgot to consider expansion in my though experiment. More space equals more possible microstates. While I'm not familiar with Harrison, I will look into his work.


And so there is the answer to your question. Note the word "If" that I've emphasized. The early universe (and the matter/energy) therein was not evenly distributed. There are tiny, tiny fluctuations in the intensity cosmic microwave background radiation that show the level of matter density fluctuations near the epoch of matter-radiation decoupling. The small density fluctuations eventually began growing due to gravity.

As is always the case in our universe, it is the breaking of symmetry that results in structure and stuff happening. Perfect symmetry results in nothing (or maybe even less than nothing).

Yes, I see. Even a slight deviation from perfect distribution gives a particle "reason" to go one way rather than the other. Then, a cascade of order apparently follows from this. Entropy not increasing as the expanding universe increases the univere's entropic capacity. Very interesting :)



There is no limit to how many thermal photons you can pack into a given volume.

How could that be? If you increase the energy density of a given volume beyond a certain point, doesn't this lead to the formation of a black hole?



If so, it means that the formation of stars is actually the process whereby the expanding universe is (unsuccessfully) "trying" to catch up with its growing entropy potential.

This sounds very interesting Ken, but I'm not so sure I understand you. Would you mind breaking the idea down into more manageable chunks for me?

M74

WayneFrancis
2010-Apr-03, 12:10 PM
Edward Harrison says ... to spare.

Grant Hutchison

Great explanation.

grant hutchison
2010-Apr-03, 02:00 PM
... I'm sure that Stenger is referring to the capacity given some assumptions that I'm not sure what they are.Stenger works from black hole thermodynamics, with a nod to the Holographic Principle. Hence his radius-squared scaling for maximal entropy, which derives from the idea that black holes are maximally entropic, and that their entropy scales with the surface area of the event horizon.

Grant Hutchison

Ken G
2010-Apr-03, 02:08 PM
How could that be? If you increase the energy density of a given volume beyond a certain point, doesn't this lead to the formation of a black hole?
If you have a thermalized photon distribution, it is isotropic and homogeneous, so that's not a black hole. Instead, you are talking about the evolution of the scale factor of the universe as a whole, and you are wondering if too high of an energy density would cause the whole thing to collapse on itself. That depends on the value of the Hubble constant-- any energy density is possible if any Hubble constant is possible. But if we take the current Hubble value, then it's true that if we had too much energy density, the expansion would turn around into a crunch. So it's all about what is meant by the "capacity" for entropy-- there's only one entropy (which is essentially constant) that the universe actually has, so what are we allowing ourselves to imagine could be different to evaluate a concept of "capacity"? I think it must have to to with the matter distribution, and how many more photons could be made by collapsing all the matter into (inhomogeneous) black holes.

This sounds very interesting Ken, but I'm not so sure I understand you. Would you mind breaking the idea down into more manageable chunks for me?If I'm right about Stenger's concept of how much more entropy could be "contained" in the universe than we actually have, then the way to achieve that entropy is to take the rest-energy distribution of the matter, and turn maybe 30% of it into light by making black holes. The creation of stars is kind of a "first step" in that process, so represents a way that the universe is "trying" to catch up to its entropy potential. At the very least, it underscores the point that gravitational instabilities and the clumping it produces moves us toward a higher entropy, when the light that is generated is taken into account.

Ken G
2010-Apr-03, 02:15 PM
Stenger works from black hole thermodynamics, with a nod to the Holographic Principle. Hence his radius-squared scaling for maximal entropy, which derives from the idea that black holes are maximally entropic, and that their entropy scales with the surface area of the event horizon.

The entropy of the created black hole itself (not counting the light it generates) scales like its mass squared, so you get the most by taking all the mass in a Hubble sphere and making one very big black hole out of it. But that isn't how it would actually happen-- you'd have pockets of much smaller black holes due to gravitational instability. So another concept of "capacity" is the entropy you could actually get from the instability, and that should be dominated by the radiation generated, not by the black holes themselves-- it seems to me. But Stenger's concept seems to be the maximum you could get if you allow processes that wouldn't actually happen, just as a kind of benchmark. There is considerable freedom in defining a concept of "capacity."

grant hutchison
2010-Apr-03, 02:31 PM
But Stenger's concept seems to be the maximum you could get if you allow processes that wouldn't actually happen, just as a kind of benchmark.Yes, that's the spirit in which he offers it. A way of illustrating how the Universe could start in a state of maximal entropy, then increase in entropy over time (because the "permitted" maximum also increases over time).

Ken G
2010-Apr-03, 04:09 PM
It sounds like the difference has to do with fundamentally different ways that particles without rest mass, and those with rest mass, interact with entropy. Particles without rest mass all have the same entropy, if they are thermalized (let's pretend that the photons always find a way to thermalize), so there's nothing you can do to increase the capacity for entropy there unless you can make more photons. But particles with rest mass have a kind of "locked up" entropic potential, which comes both from the ability to convert much of their rest mass into photons (by making black holes), or by merging them all into one uber-black-hole (which has an entropy proportional to mass squared). Interestingly, that is also the entropy of the Hawking radiation after the uber-black-hole evaporates, since the number of photons created scales with the mass divided by the temperature, and the temperature scales inversely with mass, so together that's mass-squared scaling.

As for the radiation liberated during accretion of the black hole, the accretion temperature at the Eddington limit scales like M-1/4, so the light liberated while creating the black hole would have entropy scaling like M5/4. If you had N black holes of mass M/N, that entropy would scale like N-1/4M5/4, where M is the total mass taking part. So with either type of rest-mass/entropy connections, you get more entropy from a few big black holes than from a bunch of little ones. Ironically, that says that the cosmological principle is a low-entropy way to distribute rest mass, not a high-entropy way to do it (like it is with photons).

All this suggests that it is quite important, thermodynamically, that the universe went from a radiation-dominated phase to a matter-dominated phase. A universe that is always radiation dominated might tend to stay in a kind of perpetual "heat death", if its radiation stayed thermalized with its matter, because any local clumping would reduce entropy, and it might exhibit a slavishly unbending cosmological principle. But a universe that was always matter dominated would tend to have all its mass collect into an uber-black-hole, and be the opposite of the cosmological principle. A universe that goes from radiation dominated to matter dominated seems like the entropically motivated way to get a cosmological principle on large scales but interesting objects like stars and galaxies on smaller scales-- by the time the "capacity" for entropy was dominated by the matter, the timescale to achieve those maximally entropic conditions was much longer than the age of the universe.

Spaceman Spiff
2010-Apr-04, 01:39 AM
This paper appeared on last week's arXiv preprint server: Go with the Flow, Average Holographic Universe (http://arxiv.org/abs/1003.5952), by George F. Smoot. This is obviously a concept paper; it was written for the Gravity Research Foundation 2010 Awards for Essays on Gravitation. In it he discusses issues of entropy, horizons (black hole and cosmic), and a holographic universe, intersecting some of the comments posted here. I'm still trying to digest it.

George
2010-Apr-04, 03:15 AM
As for the radiation liberated during accretion of the black hole, the accretion temperature at the Eddington limit scales like M-1/4, so the light liberated while creating the black hole would have entropy scaling like M5/4. If you had N black holes of mass M/N, that entropy would scale like N-1/4M5/4, where M is the total mass taking part. So with either type of rest-mass/entropy connections, you get more entropy from a few big black holes than from a bunch of little ones. Ironically, that says that the cosmological principle is a low-entropy way to distribute rest mass, not a high-entropy way to do it (like it is with photons).

All this suggests that it is quite important, thermodynamically, that the universe went from a radiation-dominated phase to a matter-dominated phase. A universe that is always radiation dominated might tend to stay in a kind of perpetual "heat death", if its radiation stayed thermalized with its matter, because any local clumping would reduce entropy, and it might exhibit a slavishly unbending cosmological principle. But a universe that was always matter dominated would tend to have all its mass collect into an uber-black-hole, and be the opposite of the cosmological principle. A universe that goes from radiation dominated to matter dominated seems like the entropically motivated way to get a cosmological principle on large scales but interesting objects like stars and galaxies on smaller scales-- by the time the "capacity" for entropy was dominated by the matter, the timescale to achieve those maximally entropic conditions was much longer than the age of the universe. Nice post!

I was wondering if someone would toss in Dark Energy as a possible entropy generator countering the negative entropy gained by the expansion it seems to be driving?