Entropy and the Big Bang

[Darwin123]

In this case we have a high temperature system that will evolve into a low temperature system over time (due to expansion). Without expansion that wouldn't happen: the average temperature would just remain the same.

So, my question is, does that evolution from high to low temperature imply an evolution from low to high entropy?

If so it seems odd, but entropy becomes dependent not just on the current state of the system but on possible future states.
...
Or to take another tack: say I have a model universe that's finite in size. I start it in a low entropy state and let it evolve for a while until it reaches thermal equilibrium. It's now in a high entropy state. Then we turn on expansion. It seems to me that I've just opened up the possibility for higher entropy and my system can go on evolving in interesting ways (rather than just having random thermal fluctuations).

But maybe I'm just rambling incoherently here.

It about both boundary, initial conditions and final conditions. Since we are talking about GR, where time and space are dimensions, this all comes down to a type of four-dimensional boundary conditions. There is more than one type of boundary condition associated with a finite universe. The initial and boundary conditions have to be set in order to get a well determined solution to any calculation.

Let me give an estimate using a heuristic model that involves both a simplified version of GR and a semiclassical version of quantum mechanics. Momentum is conserved in all collisions that occur away from the boundary of this finite universe.

Quantum mechanics says that matter is quantized. The Bohr atom has a certain diameter. So we can further simplify the semiclassical model of QR by using the hardball approximation for atoms. Every atom is an elastic sphere, with a nonlinear restoring force that is infinite when the atoms is squeezed into an infinitesimal volume. Every sphere has a zero pressure limit to its radius, so it settles to a certain radius in a vacuum. A universe filled with hardballs.

GR says that the speed of light is the largest speed information can go. However, lets simplify this by giving the universe an expanding boundary that somehow takes into account this condition. For instance, maybe the boundary is expanding at the speed of light in a vacuum.

Let us give this finite universe of yours periodic boundary conditions, so that one get to the other side of the universe by traveling more than the diameter of the universe. Conservation of quasimomentum is valid in this universe, analogous to the quasimomentum of electrons in a crystal. Alternatively, we can place some type of reflective boundary at the surface of the universe. Note that in the reflection model, the atoms are doing work on the boundary of the universe. The work transfers energy from the interior of the universe to the boundary.

The initial conditions are such that at the big band, there is no space between the hard balls. While there is some potential energy due to the elasticity of the hard ball, there is no room to move. The final conditions is a limiting case where on average each ball is an infinite distance apart.

The entropy can be defined as the negative logarithm of the number of states available for motion. The bigger the volume of space devoid of hard ball material, the greater the entropy.

Obviously, the entropy is largest at the big bang. There is no space where a hard ball can move into. The negative of the logarithm of zero is infinity. So at the Big Bang, the entropy of the universe is effectively infinite.

As the boundary expands, there is more and more space available for the balls to move into. Energy and quasimomentum are conserved as the boundary takes up more and more energy. However, the energy of the boundary is not available during collisions. So the work energy of the universe is decreasing.

The universe soon gets to a state where no ball is touching another. However, the entropy is still increasing because the unoccupied volume of the universe is increasing. As the universe expands, the frequency of collisions are decreasing. So less and less work is being done. The entropy increases as the number of collisions decrease.

You can use more sophisticated models. However, the universe may not be this way.

 

[Roboramma]

True. To get a sense of what "flavors" are available, take a look at interpretations of entropy for black holes: http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy#What_is_behind_the_black_hole_entropy

Beyond my pay grade.

Personally I prefer this option:

Black hole entropy counts the number of states or excitations of a fundamental string.
Strings in string theory have a variety of excitations, so there is a multitude of string states. Therefore, a string has entropy, which turns out to be proportional to its mass. This is quite in contrast with black hole entropy. However, an argument by Bowick, Smolin and Wijewardhana (1987) suggests that by adiabatically (i.e. sufficiently slowly) reducing the string coupling constant g , it is possible to shrink a black hole's size as well as to reduce its mass (while keeping its entropy constant) until eventually it gets to be the size of the string length scale ls when the black hole should not be distinguishable from a string. At the corresponding value of g , string and black hole entropy are quite similar (see e.g. Zwiebach 2004). This has been taken to mean that there is a one-to-one correspondence between black hole and string states, where both entities have the same entropy (Susskind 1993). This picture has been corroborated in the context of five-dimensional extreme black holes (Strominger and Vafa 1996). Hence black hole entropy can be understood in terms of string entropy.

But it's far beyond my pay grade as well.

 

[Roboramma]

Yes. This is a classic component to the ideal gas problem. If you have a container with a divider in the middle, and fill it with gas on one side and nothing on the other, what happens when you remove the divider? The gas won't lose any energy in the process, and it won't change temperature either, but it will expand into the entire chamber. Furthermore, that change is irreversible: the gas won't spontaneously return to one half of the container. Why? Because the expansion increased the entropy of the system.

HOWEVER...

If you don't expand the gas by removing the divider, but instead by allowing the gas to do work against the divider (ie, move it like a piston), then the change is reversible: the gas loses energy and lowers in temperature as it expands. The momentum contribution to entropy lowers, compensating the increase in the positional entropy. This process is reversible: get the gas back on one side by moving the divider back.

So, in regards to the universe as a whole: which expansion does it more resemble?

I'm honestly having a hard time with this: as the universe expands it cools, which seems to resemble your second case, but I don't think there's any action of the particles in the universe exchanging energy with something in order to drive the expansion. There's no divider for them to do work against, in which case it seems to me to more resemble the first case: space is expanding and the particles within the space are simply freely moving through that newly expanded space. Though perhaps I don't understand the mechanism for expansion?

So, which is it?

 

[Ziggurat]

I'm honestly having a hard time with this: as the universe expands it cools, which seems to resemble your second case, but I don't think there's any action of the particles in the universe exchanging energy with something in order to drive the expansion. There's no divider for them to do work against, in which case it seems to me to more resemble the first case: space is expanding and the particles within the space are simply freely moving through that newly expanded space. Though perhaps I don't understand the mechanism for expansion?

So, which is it?

I'm no expert on cosmology, but I think it's a mixture of both.

Light is spread fairly evenly throughout the universe. The expansion of the universe red-shifts light, lowering its energy in exactly the manner one would expect of the expanding piston scenario.

But matter is not evenly distributed: it clumps together because of gravity. The expansion of the universe doesn't cool matter very much because matter stays concentrated in these clumps during expansion. So I think that component of the expansion more closely resembles the removed divider.