FUGE and Entropy

So, I had toyed with the idea of trying to work out the entropy of a FUGE universe by considering states.  First, I went with a very rough approximation similar to this:

The first instance has only one possible state, noting that it’s representing an instanton (which, yes, should be a sphere of radius r=l_Pl, but this is just a rough approximation).  Then there’s the second instance and it gets tricky.

The simplistic (also known as “wrong”) way to look at it is to imagine placing two instantons in two of the available slots.  There are, therefore, Ω=8×7=56 different configurations or, more generally, Ω=n³×(n³-1)×…×(n³-n).  This is approximately equivalent to Ω=(n³-(n+1)/2)ⁿ – with the approximation error decreasing as the value of n gets larger (by the time that n=10, the error in the approximation is only 0.004%).  Note also that as n increases such that n³>>(n+1)/2, n³-(n+1)/2≈n³.

If we consider these as microstates (which we should not, because it is wrong to doso), then we could say that the entropy is:

S=k_BlogΩ=k_Blog((n³)ⁿ)=3n.k_Blog(n)

Then if n=æ≈8.07×10⁶⁰, because we are assuming a FUGE universe (see also below), S=1.5×10⁶³.k_B=2.04×10⁴⁰J/K.

Casting around the internet, I find that the entropy of the universe is of the order of 10¹⁰⁰-10¹⁰⁴J/K.  Therefore, there is something wrong with my approximation.  I did note that the magnitude of the error is close to æ, 2.04×10⁴⁰×8.07×10⁶⁰=1.64×10¹⁰¹, hinting that S∝n².

Remember that I used a simplistic approach.  I used notional cubes, rather than a sphere that expands.  Also, you clearly can’t just place instantons into slots, or rather, once you get a volume that is greater than one instanton (or one Planck sphere), you also have the option to spread the energy out.  And this increases the number of possible states.  The question then is by how much is the number of possible states increased?

Using the same sort of approach as in The Conservatory - Notes on the Universe, it would appear that the number of arrangements of mass-energy distribution could be quantised, which means that we cannot just shave infinitesimal amounts off the mass-energy in an instanton and redistribute it.

Consider then that, for each value of n, it would be possible to split each instanton worth of mass energy into n components and then the number of states would be in terms of those components.  In the second instance above, assuming for ease that the components cannot be collocated, this would be Ω=8×7×6×5=1860.  For the third, it would be Ω=27×26×25×24×23×22×21×20×29=1.7×10¹².  More generally, this is approximately equivalent to Ω=(n³-(n²-1)/2)^n.n.  Again, when n³>>(n²-1)/2, we can remove the second term, so:

S=k_BlogΩ=k_Blog((n³)^n.n)=3n²k_Blog(n)

And so, after substituting n=æ≈8.07×10⁶⁰, S=1.2×10¹²⁴.k_B=1.66×10¹⁰¹J/K.  So we end up in the right ballpark, but are we close enough?

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Then I thought about it different way.  An instanton is, effectively, a black hole and the entropy of a black hole is given by:

S_BH=k_BA/4l_Pl²

The area on the surface of a sphere is A=4πr², so for an instanton A=4πl_Pl ² and so:

S_instanton=k_B4πl_Pl ²/4l_Pl²=k_Bπ=4.34×10^-23J/K

A FUGE universe is equivalent to a blackhole at any time and when the age of the universe is t=æ.t_Pl the radius is r=æ.ct_Pl=æ.l_Pl, so:

S_universe=k_B.4π(æ.l_Pl)²/4l_Pl²=k_Bπ.æ²

Given that for t=13.787 billion years, æ≈8.07×10⁶⁰, that would make the entropy S_universe≈2.04k_B×10¹²²=2.83×10⁹⁹J/K.  This seems to be a bit higher than normally calculated, where the value tends to be in the order of k_B×10¹⁰³ but, by strange coincidence, Charley Lineweaver (of The Mass of Everything fame) co-wrote a paper in 2010 with Chas A Egan – A Larger Estimate of the Entropy of the Universe – which has, in the abstract, the following (where k is k_B): “We calculate the entropy of the current cosmic event horizon to be S_CEH=2.6±0.3×10¹²²k, dwarfing the entropy of its interior, S_{CEH int}=1.2^+1.1_−0.7×10¹⁰³k.”

The difference appears to be due to a different method of calculation which effectively uses a different radius, they used 15.7 Glyr, as compared to my 13.787 Glyr.  Note that (15.7²/13.787²=1.297)*2.04=2.64.

The question then is, why was the cosmic event horizon set at 15.7 Glyr?  Looking at equation (46) in the paper, it seems to be based on a variable and/or non-unity scale factor, noting that Figure 1 includes text that indicates an age of the universe of 13.7Gyr.

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The bottom line is that, if this is a FUGE universe, then the current entropy is ~2.83×10⁹⁹J/K.  It is also worth noting that, when expressed in terms of Planck units, entropy of the universe at any time t=n.t_Pl has a magnitude of π.n².

We could introduce a “Planck entropy”, being the entropy of a Planck black hole (also known as an instanton), S_Pl=k_Bπ(J/K).  In terms of such a derived unit, the current entropy is effectively the square of the age:

S_universe =æ².S_Pl