Top 10 topics

• "Universal entropy, why did it start so low?"

It is often said that in a Universe with gravity, matter, and radiation a smooth initial state is very unlikely. (Penrose, 1984; Carroll, 2005; others) Thus the question is, is there some way of adding some physics to gravity, matter, and radiation, that makes the smooth initial state overwhelmingly likely?
The usual give-and-take on this subject seems to be: "add a bit of physics that makes things smooth" and then later realize that "if the entropy of this physics is calculated we still have a very unlikely IC set". (See Guths use of old inlfation in this style and the various rebuttals)

• "Is entropy a valid concept in a non-equilibrium situation?"

Entropy, a concept defined in equilibrium statistical mechanics, is partly defined by assuming that all states are equally accessible. If there are a significant number of states that are inaccessable, is it still valid?
In a non-equilibrium, the constraints are not just defined by limits in phase space, (position, energy, etc) but constraints in time as well. Consider two boxes connected by a stop-cock with gas leaking from one to the other. The state where the pressure is equalized is simply not available yet since the gas particles have not had time to get there yet.
Let us examine how evolution of the same gas in special relativity represents a similar constraint.

Another definition of entropy is the logarithm of the fraction of phase space occupied by a system.
S = ln(\Omega/\Omega_min)
where \Omega is the volume of phase space occupied and \Omega_min is the quantum granulation of that space. In the case of a momentum-position space, the smallest possible granulation size is given by planks constant.

Given these definitions let us examine the entropy of massive and massless particles in a special relativistic framework.
In relativity, particles are constrained to stay within their own light cones. Consider a dust cloud of massive non-interacting particles. Here the light cone does not provide any constraint as the horizon expands at a speed greater than any of the dust motes. Our non-relativistic dust cloud is free to expand unimpeded. Assuming an initial condition is given that describes the velocity and position of each particle to the precision allowed by the uncertainty principle the system will occupy a single voxel of phase space. Thus the fraction of phase space occupied is one and the entropy is zero.

Consider then a gas of photons. In this case the light rays are constrained to lie upon the surface of the light cone as it expands. Thus the available phase space increases. In this way the increase in surface area represents the increase in volume in the stop-cock example. The surface area of the light-cone goes as t^2, thus the fraction of available phase space goes as

where the third spatial dimension in phase space is on the order of the uncertainty in the location given by

Assuming most precise measurement possible I can reduce the light-cone phase space fraction to

where p is the momentum of the photon and delta p_Omega is the error in the most accurate photon solid angle measurement possible. The entropy is the logarithm of this function and we finally see an equals sign

Assuming short time periods, we can neglect cosmological redshift then the entropy increases logarithmically in time.

This increase in entropy is for the most part independant of any other dynamics present in the system. However in the early Universe the situation would be more complicated as photons were not free to travel but rather underwent many interactions with other particles. In the first case we had a non-interacting cloud of dust, where each particle continues unimpeded on its own geodesic. This represents a single voxel in phase space and thus the minimal entropy condition. More importantly the entropy is constant, representing a kind of equilibrium. The gas of non-interacting photons/relativistic particles shows an increase in entropy or measure of disorder simply through the passage of time, despite the fact that the particles undergo the same type of expansion as the non-relativistic particles.

• Can we answer this question with previous work into non-equilibrium thermodynamics?
• What constitutes equilibrium in this system?
  A proffessor of mine once offered his definition of equilibrium as: "Hard to define but you know it when you see it." Though when we used this on a test it recieved few points...
  
• "Is there a topological explanation for the low entropy?"
  How exactly does the initial 'singularity' work? Does it imply some unique form of space where only one wavefunction solution is possible?
• How does one describe the initial singularity? Are there special coordinate systems that simplify it?
• Can one define the set of all solutions given a certain topology of the singularity?
• Much investigation is being done on the global topological structure of space-time. Inflation is widely agreed to be a good solution to the flatness and horizon problems. Even a loose interpretation of the expansion necessary finds that 60 exponential e-foldings are necessary. This implies that a pre-inflation size scale on the order of the plank length 10^-33 cm inflates to approximately 10^10^12 cm in about a plank time. How then, in a universe so large can we ever expect to observe closed universe or wrap-around structure?
• How is it that energy is not conserved during inflation?
• How is entropy defined in curved space and where various quantum fields are present?
  

All I need now is to add all the references and I'll have a decent summary.