How Are Markov Chains Applied to Cosmology?

[Vyresince]

Hello all,

This is my first post here and so I hope I am posting in the appropriate section. My question is actually related to both Physics and Discrete Mathematics. I'm writing a very general essay (no rigorous derivations or extremely involved mathematics) about Markov chains as they are applied to cosmology. I've spend the past 3 or 4 hours pouring over the resources offered by my college's library as well as those offered by the internet. Sadly, this search hasn't yielded too much. I found a few papers on Markov Chain Monte Carlo methods, but these papers tend to get very detailed, complicated, and long.

The main question I have for all of you helpful folks making up these forums is where I can go to find information on the basics of Markov Chains and how they are applied to cosmology. Do you have any papers or authors to suggest? Perhaps a relatively common book? Websites will also work. I just haven’t been able to find many good sources.

Also, my professor mentioned that there is a debate about whether or not Markov Chains/Discrete methods are applicable to cosmology due to the fact that the universe may or may not be a physical realization of infinity. If anyone knows where I could find resources on such a debate, it would be greatly appreciated. I have been unable to find anything other than vague hints at such a debate in my search for sources.

That about sums up my requests. Thanks in advance to all those that help/provide sources of information.

 

[Vyresince]

Perhaps I ask too much.

Well, it seems that my question may have been too broad and perhaps I didn't illustrate the efforts I have put forth (I'm basing this on the fairly high number of views and the lack of any responses, but it is possible I just haven't allowed sufficient time. If this is the case, I apologize for this extension to my post). At this point I have about three papers and one or two books from which I can derive the basics of Markov chains and how they are applied to cosmology. So I can now narrow my question down. What I have been completely unable to find almost ANY clear evidence of is an ongoing debate on the validity of applying Markov chains to cosmology in the Physics community. If anyone knows of such a debate or where I could find evidence of such a debate, I would be greatly indebted.

 

[BigIdiot]

Markov chains and cosmology

Hello Vyresince,

I pursued this very topic as my PhD dissertation around 16 years ago, and sadly it was soundly rejected by my advisory board thus ejecting me from the academic community. So you will hopefully take in stride any snide comments that may fall by the wayside while I relate my reply.

Markov chains normally include an observer, that is, the knowledge state of the observer (or lack of knowledge) is utilized to obtain probability matrices of expected path trajectory. However, if you look at a physical system observer free, you can see that a markov chain can be described which describes the trajectory of a physical system which has no observer. This chain describes the state of any physical system with a finite number of possible positions- the finiteness being defined by the quantum length of a subatomic particle may reside in.

In other words, you can consider the probability of the wavefunction of the universe as a whole- a cosmological question- and ask what this means. It apparently is the same thing as information and complexity- that is, the less probable markov state is the more informed state. (And my grand suggestion was that DNA is a form of low probability probability propagation, along with civilization itself)

For more on the history of this you might turn to Hartle and Hawking "Wave Function of the Universe" (1983) and Vilenkin "Quantum Creation of Universes" (1984)

There was an article in "Natural History" recently which discussed the countability of space time- it is necessary to have a finite space time discreteness to consider the countability (and applicability of finite markov chains). This was by Vilenkin. He certainly believes in the markov chain model of cosmology, although does not suggest it in that way.

You need not be constrained to finite Markov chains either, study some of the advanced papers by Kolmogrov and you may be able to sidestep the question of finiteness as being a necessary foundation for applying markov chains. You will have to address discreteness- for an infinite number of permutations of physical matter within a finite volume and your markov chain will become once again observer dependent (as you define some arbitrary criteria for considering configurations to be unique).

For considerations of breaking down space time into discrete and denumerable quantities, I would look to the philosophy of statistical mechanics. (The writers who expound on statistical mechanics).

If you have specific interest in any of the above ideas I can try to dig out some references for you, but it has been many years to be honest and so I only keep up with the state of knowledge on a tangential level. (Like noting the issue of Natural History and its specific denumeration of space time combinatorics)