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This may be a fairly basic question as I don't have a strong background in physics. I intuitively thought that the universe must be able to be described by a Markov chain. That is, I thought you could feed the current state of the universe into a process and it would spit out the next state conditional on the laws of the universe. However, I have found no mention of the universe as a Markov chain outside of speculations on message boards.

Can the universe be described as a Markov chain or is there some reason to suggest that the next state in the universe is dependent on more than just the current state and constant universal laws?

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An important issue with this image is discreteness vs. continuity: Markov chains are inherently discrete, whereas quantum mechanics (and QFT, in a specific way) has continuous time evolution. –  Gerben Oct 5 '11 at 20:52
That's an excellent point. However, I was under the impression that a consequence of string theory (obviously it is just one among other hypotheses) was discrete space time--at least on a very small level. Even besides this, I was thinking that perhaps one could describe something like a continuous Markov chain as follows: x_t=f(x_{t-\delta}) and make that delta arbitrarily small to simulate continuity. Obviously, f(x_t) would have to be a continuous function itself for this to work. –  Vivek Viswanathan Oct 5 '11 at 20:57
I edited your question to talk about descriptions of the universe, rather than what the universe is - for one thing, physics doesn't necessarily concern itself with what the universe "is," but also, phrasing it this way invites responses that discuss the sort of discrete approximation you mentioned, even if the universe isn't actually discrete. –  David Z Oct 5 '11 at 21:21
You're absolutely right. My question was non-rigorously stated. –  Vivek Viswanathan Oct 5 '11 at 21:27
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4 Answers

The problem is that Markov chains are inherently lossy--- so in physics as it is commonly understood today, the answer is no. A Markov chain will always lose information about the initial state, as it relaxes to a stable distribution, while a quantum mechanical system does not do this. The modern understanding of a physical system is as a quantum markov chain, which is the same as a classical Markov chain with probability amplitudes taking the place of probabilities.

But I believe it is an open question if you can approximate quantum dynamics by a Markov dynamics, so that it resembles the real thing. This is related to this question and answer: Stochastic processes and wavefunction collapse

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Excellent answer. I just have a quick question. Do we need quantum Markov chains instead of regular Markov chains because of the necessary uncertainty of the state that comes with Heisenberg's uncertainty principle? –  Vivek Viswanathan Oct 6 '11 at 2:02
It's the opposite--- the uncertainty principle comes from the quantum markov-chain description of a point particle. There is no uncertainty in a quantum state--- it is a precisely defined thing. The uncertainties are in the values of position and momentum which you would get if you measure. Nobody calls it a "quantum markov chain", by the way, it's just called "quantum mechanics". But its the same thing as a Markov chain, except with amplitudes, and the transition matrix is now called the Hamiltonian. –  Ron Maimon Oct 6 '11 at 5:06
@Ron: inherently lossy? Can't a Markov chain be invertible? I understand that almost all eigenvectors vanish exponentially, but at large but finite $N$ you should still be able to find the initial condition. –  Gerben Oct 6 '11 at 16:29
@Gerben: the information is pushed into the small digits of the probability distribution, so that there is entropy production, the process is irreversible. But you are right in the abstract infinite accuracy limit--- if you know all the probability distribution to arbitrary accuracy, you never lose information. –  Ron Maimon Oct 6 '11 at 16:47
@Ron: thanks, I somehow never thought of this. –  Gerben Oct 6 '11 at 16:53
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I wanted to clear up a few misconceptions in the responses.

First, a markov chain system must be independent of its past. However, if the state space of the possible states is expanded to include the residual history, then a system which "remembers its past" becomes a markov chain if the state space itself is large enough to include all the states with the recorded past history.

Non linear phenomena are definitely markov chains, and the universe appears to be a markov chain provided you define the state space as the exact microstate.

Markov chains are not necessarily time continuous or time discrete.

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Systems where the past history is not important can be described by Markov chains. However when the system remembers the past we cannot use Markov chains for instance in non-linear phenomena.

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Why would non-linear phenomena be inherently non-Markov? Brownian motion isn't even differentiable at any point (and obviously isn't linear either), and it is Markov. –  Vivek Viswanathan Oct 7 '11 at 18:59
You can always encode the past history into a bigger phase space, a random walk on histories, rather than on positions. –  Ron Maimon Oct 7 '11 at 19:45
I didnt talk about differentiability at all. I used non-linear phenomena as an example where the system remembers the past history. However, it is clear that not all non-linear phenomena belongs to this category. –  armando Oct 9 '11 at 0:20
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For more thoughts on the universe being a Markov chain, please google "Markov Chain Universe". It is a truly fantastic web site dedicated to this topic. Who do you think made it? ;)

Whether or not you choose to accept the axioms of the theory, it is a mind bending experience to think about.

It is a lot like first understanding that the earth is a sphere, rather than a flat plane.

Except in this case, it is understanding that the Markov chain probability of the universe exists as an orthogonal metric to the measure understood by thermodynamic entropy.

Thus, there exists a completely new probability, which appears to correspond to our macroscopic concept of complexity.

Please stop by and take a gander and ask lots of questions.

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