[Bill S (OP)]

Please forgive the ponderous nature of this post; it has to do with my thought processes.

Consider entropy as a measure of disorder.  In general, we tend to think of disorder as increasing as things spread out.  The ubiquitous example of the cup (compact, ordered, low entropy) being smashed and thus, becoming fragments (spread out, disordered, higher entropy) comes immediately to mind.  However, is this always the case?

Consider next a cloud of dust in space.  This might be spread out and seemingly disordered, but because its particles have mass they will gravitationally attract one another.  Thus, as they clump together, they become less spread out and, intuitively, more ordered.  Does this mean that gravity can reverse the normal pattern of progress from low to high entropy? 

Entropy is defined as the logarithm of the number of microstates accessible to a system.  Could it be that under the influence of gravity, a system has more microstates when it has collapsed than it had before?

The answer would seem to be “yes”.  For example: A black hole, which is created out of a large amount of matter that has been condensed under gravity into a very small area, has, we are assured, a huge entropy; far greater than the entropy of the total matter in its non-compacted state.  This must mean that the available number of microstates increases as the matter becomes more compressed.

How can this be?

[chiralSPO]

entropy tends to increase in a closed system, but there are many examples of spontaneous processes in which entropy decreases locally. For instance, condensation of a gas to a liquid, or crystallization of a liquid, or the reaction of two molecules to form one. In these cases, the release of heat is able to overcome the entropic losses.

ΔG = ΔH – TΔS

if ΔG is less than 0, the reaction happens spontaneously so in cases with large negative values of ΔH are able to overcome positive values of ΔS (especially at lower temperatures).

The aggregation of matter to produce a planet results in a LOT of heat. The earth is STILL cooling 4 billion years later, and most of the mass is still molten! (radioactive decay also does contribute some heat, as well as very minor contributions from tidal heating).

[jeffreyH]

This page may also be a useful source despite the name.
http://physicsforidiots.com/physics/thermodynamics/

[Bill S (OP)]

Continuing my line of thought, I had wondered about the release of heat into space causing an increase in non-local entropy, but I was stuck on the configuration of microspaces in the before and after situations.  Intuitively, I was thinking there should be more freedom of movement, and therefore more potential microstates, in the open configuration than in the compressed.

Obviously it is not as simple as that; which is where the "why" came in.

"Physics for idiots"  Sounds like just what I need!

[evan_au]

Does this mean that gravity can reverse the normal pattern of progress from low to high entropy? 

Another qualitative way of looking at thermodynamics is "Over time, systems tend towards the lowest energy state".

In our little bit of the universe*, mass clumped together is in a lower energy state than mass spread out throughout space. So thermodynamics tends to favor this lower-energy state.
- A hot body (the center of the Earth) when compared to the temperature of space (near absolute zero) is in a high energy state:  you can produce geothermal energy. The state where the Sun and the Earth have cooled down to near absolute zero (and space is still near absolute zero) is in a lower energy state, so this is the preferred direction, even though it will take a long time to get there.

The hydrogen/helium clouds that formed our galaxy were able to radiate energy from collisions into space, allowing the galaxy to form (a similar process on a smaller scale formed the Sun and planets from dust clouds). However, it is thought that Dark Matter does not radiate energy via electromagnetism, so it forms a spherical halo around the galaxy, rather than the "fried egg" shape we see for the galaxy and solar system.

*However, on cosmic scales, it seems that a state with matter far apart has lower energy than a state with matter close together, driving the cosmic expansion of the universe. The source of this Dark Energy is still rather mysterious.

[Bill S (OP)]

In our little bit of the universe, mass clumped together is in a lower energy state than mass spread out throughout space. So thermodynamics tends to favor this lower-energy state…………However, on cosmic scales, it seems that a state with matter far apart has lower energy than a state with matter close together, driving the cosmic expansion of the universe. The source of this Dark Energy is still rather mysterious.

That seems to say that, locally, lower energy = hotter = higher entropy, but on cosmic scales, lower energy = cooler = higher entropy.

Am I misinterpreting it?

[jeffreyH]

Microstates and statistics are the key.
https://en.m.wikipedia.org/wiki/Microstate_(statistical_mechanics)

[Toffo]

Less room => Fewer microstates

More energy => More microstates

Compression of gas in a cylinder by a piston:
Less room and more energy => number of microstates stays constant

Compression of gas in a gas cloud  by gravity:
Less room and more energy => number of microstates stays constant

Compression of matter into a black hole by gravity:
Hmm .. well...  the matter is transformed to something else that has a very large number of microstates

[jeffreyH]

More reading on statistical entropy and the number of microstates. This discusses the distribution of energy. Since energy is quantised this makes sense.
https://ch301.cm.utexas.edu/section2.php?target=thermo/second-law/microstates-boltzmann.html

[Bill S (OP)]

Less room => Fewer microstates

More energy => More microstates

Compression of gas in a cylinder by a piston:
Less room and more energy => number of microstates stays constant

Compression of gas in a gas cloud  by gravity:
Less room and more energy => number of microstates stays constant

That seems to make good sense.

Compression of matter into a black hole by gravity:
Hmm .. well...  the matter is transformed to something else that has a very large number of microstates

Is there a wave of the hand to go with this? 

[Bill S (OP)]

https://ch301.cm.utexas.edu/section2.php?target=thermo/second-law/microstates-boltzmann.html

A thought provoking link, Jeffrey, thanks.  Here are some initial thoughts.

We can give each molecule one unit of energy (1 + 1 + 1 = 3 total). There is only one way to accomplish this. Each molecule needs to have one unit of energy.

Presumably, this is because no distinction can be made between molecules when they have no energy quanta (EQ) attached;

…. molecules a, b, and c are equivalent…..

thus, if each has an identical EQ, surely, they are still indistinguishable.

 

Alternatively, we can give all three units to just one molecule, and zero energy to the other two (3 + 0 + 0 = 3 total). There are three ways to accomplish this since we can give all the energy to either molecule a, or molecule b, or molecule c.

For molecules a, b, and c, the configurations must be: 3+0+0, 0+3+0 and 0=0=3.  However, if the molecules are indistinguishable, what is the difference between these three configurations? 

Finally, we can give 1 unit of energy to one of the molecules, and 2 units of energy to another one. It turns out there are six possible ways to accomplish this distribution of energy

Surely, the same reasoning must apply:  There can be no distinction between, for example,
a - 2, b -1, c -0 and a – 0, b – 2, c – 1.

I acknowledge that energy levels may come into the reasoning, but I'd like to get this out of the way first.

[jeffreyH]

You also have position to consider. Every particle has to be somewhere.

[Toffo]

Is there a wave of the hand to go with this? 

The rule is not that when matter is compressed, maybe by gravity, the number of microstates increases. There are counter examples.

The rule is that when matter is irreversibly, in a microstate-incresasing way,  compressed, the number of microstates increases.

We are told that compressing matter into a black hole is the latter kind of compression of matter.

Maybe it is so that when compression of matter is increased, inevitably at some point some irreversible thing occurs, to the matter, or to the space-time  surrounding the matter, or to both the matter and the space-time.

[Bill S (OP)]

You also have position to consider. Every particle has to be somewhere.

True, but if particles can be distinguished by position, there must be no such thing as indistinguishable fermions. 

[Bill S (OP)]

The rule is not that when matter is compressed, maybe by gravity, the number of microstates increases. There are counter examples.

Such as?

[jeffreyH]

For any two fermions we can arrange frames of reference that make them identical. So that they have the same spin, momentum and kinetic energy. This misses the point about entropy.

[Bill S (OP)]

For any two fermions we can arrange frames of reference that make them identical. So that they have the same spin, momentum and kinetic energy.

OK, but as you said, "Every particle has to be somewhere." Can two particles, that are otherwise identical, be distinguished on the basis of position alone?

[Toffo]

Such as?

Anything reversible is an example of number of microstates not increasing..

A gas cloud of ideal gas in space starts to collapse, ideal gas does not radiate radiation or particles at any temperature, so the gas cloud can not lose energy, so it must bounce back, so there was no increase of entropy at any point.

The aforementioned  gas cloud must not be so large that a black hole is formed.

[evan_au]

For any two fermions we can arrange frames of reference that make them identical. So that they have the same spin, momentum and kinetic energy.

Pauli's exclusion principle says that no two fermions (eg electrons) can be in exactly the same quantum state.
If you transform the frame of reference of one fermion, the other's state is also transformed (in the new frame of reference) so they are still distinct.

Of course, you need to define the scope of your quantum system...
See more at: https://en.wikipedia.org/wiki/Pauli_exclusion_principle

[jeffreyH]

For any two fermions we can arrange frames of reference that make them identical. So that they have the same spin, momentum and kinetic energy.

Pauli's exclusion principle says that no two fermions (eg electrons) can be in exactly the same quantum state.
If you transform the frame of reference of one fermion, the other's state is also transformed (in the new frame of reference) so they are still distinct.

Of course, you need to define the scope of your quantum system...
See more at: https://en.wikipedia.org/wiki/Pauli_exclusion_principle

Aha! You caught my sleight of hand.