Friday, May 5, 2017

Relation between Entropy and information

This is sort of a personal note that is probably not as useful or interesting as my past posts on this.

General relativity shows mass to be some sort of convolution of space-time. This is trying to see if the convolution can be viewed strictly as the presence of information, without needing to have a concept of mass or space-time.

S/(kb x ln(2)) = number of Yes/no question needed to determine what state a physical system is in.

Temperature x ln(2) x kb = Joules of heat per particle per yes/no question that determine what state it is in.

The above two statements satisfy Q = S x T.

If Joules are given to a single particle, and if the kb is simply converting from temperature to joules (so that it is not needed when working with joules), then the minimum number of yes/no questions needed (the most wisely chosen, splitting the remaining volume in half at each step) to determine where it is in a square box of volume d^3 is

ln [(p x d)^3) / (1.46 x h)^3 ] / ln(2)

where p = SQRT(2 x mass x Joules).

1.46 x h = hbar/(2 x sigma^2) from uncertainty principle. (the 1.46 may have some error)

Let's say all the joules came from a mass so that a photon exists in the box instead of a particle. And let's say we know it is somewhere in the box in time interval t.

Using meters= i x c x seconds and p=h/wavelength for a photon, the number of yes/no questions needed to position a photon in the space-time box is:

ln [ ( (h/w x d)^4 x ic / (1.46 x h)^4 ] / ln(2)

ln( ic / (1.46w/d)^4 ) /ln(2)

where w = hc/joules and d includes an extra meters to account for a box in space-time.

ln(i) = 1.5708


y/n questions = {1.57 - 3ln(c) + 4ln(joules x d/(1.46hc)]} / ln(2)

Being less precise now:
ln(1.46^4) =~ ln(i). I believe there is some error in my 1.46, so it could be exact. So approximately

yes/no questions = log2 (c (d/w)^4)
= log2(c) + 4log2(d/w)

where the log2(c) is just a units conversion that should be log2(1) = 0. hc might be viewed as a conversion from 1/joules to meters. But anyway, I have
y/n questions = log2(d/w) per dimension which in hindsight is obvious. It's sort of the definition of the w.

To give a feel for what I might be trying to say: this is the amount of information needed to describe the state of energy (and therefore mass) in a space-time box. The lack of h is interesting because it's usually the starting point for quantifying the number of states. I let joules and hc cancel each other. Reversing the logic: if I have the ability to perceive a certain number of bits, then this is how much mass and energy I can perceive in a certain amount of space-time.

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