Would you agree that ergodic is a type of random process? Essentially, it is a well mixed process, right?
Like I said, this might just be a language miss-match, but no, ergodicity can be produced by deterministic processes. Well-mixedness can also be produced by deterministic processes - as in the case of chaotic dynamics.
Are you sure about that?
For example a pseudorandom number is random for practical purposes. It is still deterministic, just we can’t predict it (without the algorithm and the seed), so we call it “random.” Technically, however, it is not producing any new information at all, because it is compressible down to the seed. I’m pretty sure that physicists still call it “random”, right?
Wikipedia gives these definitions of ergodic:
Notice, this defines ergodic as a type of random process, which is the way I learned it. It is not deterministic, but
it might be used to model a deterministic process. As an example of an ergodic process.
- A coin flip on an 50/50 unbiased coin. Over time, the time average is 50%, and the ensemble average is 50% too.
As an example of a non-ergodic process (from wikipedia).
- Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins first , and then perform a sequence of independent tosses of our selected coin. Let X [ n ] denote the outcome of the n th toss, with 1 for heads and 0 for tails. Then the ensemble average is ½ (½ + 1) = ¾; yet the long-term average is ½ for the fair coin and 1 for the two-headed coin. So the long term time-average is either 1/2 or 1. Hence, this random process is not ergodic in mean.
What seems clear is that ergodic is just a another type of random. It is, what I was saying before, a well mixed processes. Regardless I should modify this:
I should have written:
it seem some people are trying to claim that ergodic processes are not random processes. I can’t make any sense of this because ergodic processes are just one type of random, and in fact they are the usual type of randomness to which we refer. Every random variable obeys some distribution and said distribution is the pattern of the random variable. This is even true for MaxEnt distributions. What are they getting at that I am missing?
Does this make more sense @PdotdQ ?
I don’t know about the second link, it seems to be ergodic process as defined by econometrics and signal processing, which might not be how physicists and mathematicians define ergodicity.
I don’t know if you are familiar with measure theory, but from you first link, if you look under the “Formal Definition” section, you will see that none of the definitions require randomness. Note that “probability space” is just math-speak for “measure-space”, and does not actually require randomness to define.
Indeed, in physics one can obtain ergodic systems from non-ergodic systems in the following way: start with a non-ergodic system (we call them integrable) and modify the equation of motion (through modifying what physicists call the Hamiltonian) with a large enough (or nonlinear) perturbation. The system remains deterministic but becomes ergodic under the definitions in the first wikipedia link.
Interestingly, the perturbation needs to be large for this to happen (or nonlinear), otherwise, there is a theorem that shows that small perturbations will keep the system non-ergodic (KAM theorem).
As you found, random processes can be ergodic/non-ergodic, and these are the two examples you found with the coins. However, so can deterministic processes. In reality, there are 4 types of systems, (deterministic+ergodic), (deterministic+non-ergodic), (random+ergodic), and (random+non-ergodic).
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Well there you go. That clarifies things immensely. I’ve come at it from a probabilistic modeling point of view, which emphasizes using randomness a tool to model complex processes. Physics, obviously, cares more about deterministic systems. That explains why we saw it differently. Ergodic can be random or deterministic.
Regardless, in biology when we are saying mutations are “random” we are meaning they follow some statistical distribution. Usually distributions are ergodic in our models, even if they are non-ergodic in nature. Why someone would care either way in theology is beyond me.
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Well, my type of physicists care a lot about deterministic systems, but a lot of physicists spend their lives working on stochastic systems that are inherently random. I don’t want to discount their work by saying that physics don’t really care about them 
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What would you characterize Brownian Motion?
It is clearly theoretically deterministic, but certainly not so in practice. It can be modeled as a random walk (an ergodic process). It turns out this is very hard to even simulate directly in molecular dynamics as a deterministic process. So it may not even be deterministic in the context of simulation. Still, it apparently follows classical physics and should be in principle reversible and deterministic.
Is that a random or a deterministic process?
This is a philosophical question
I would say that the mathematical model of a Brownian motion model is stochastic and random. Real life Brownian motion is classical and not random. In the limit of large number of particles, the stochastic mathematical model of Brownian motion approaches the deterministic real life Brownian motion.
So I would say that it is both random and deterministic, depending on whether you are talking about the mathematical model or the IRL version.
I suppose that is my point exactly.
Or just a a causal pattern we cannot practically model, as in the case of Brownian Motion, even though we know in fact it is there.
What about when there is no causal pattern, like in quantum mechanical collapse?
We don’t know yet, do we? As I understand it, collapse can still be explained by a Bohmian mechanics. There have been some very interesting experimental results on this recently too, out of MIT I think. So it is possible there is a hidden deterministic process governing collapse, right?
Regardless, even if there is not, it might still be deterministic from God’s point of view.
Yes, but for Bohmian mechanics to be true we need to give up locality. This is not easy for a physicist to give up - perhaps even harder than determinism. Further, the relativistic generalization of Bohmian mechanics relies on an unobservable preferred frame - which in its own way runs contrary to the spirit of relativity. The field theory generalization (to bring QM->QFT) as I understand it is not fully understood and is still a topic of active research.
If one can even define a notion of determinism for a being outside of time like God, I suppose this could be true.
This is a lot of discussion to support a favorable view of “randomness” … whuch is conpletely beside the point here at PS.Org: we dont have to defend randomization because we dont need an unguided Universe.
We have already signed-up for a “god-guided” universe!
Some of us have. Some us haven’t.
I think most people here would agree that god-guided is not in conflicted with evolutionary science, even if that isn’t their personal opinion.
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It was mostly for a small number of people who have seemed confused about randomness.
So… you can definitely say that from a theoligical perspective, that there are things In The Universe that are RANDOM to God as you would understand God?
Just say yes, and ill re-adjust the wording of my complaints!
Yes - that amounts to the same thing: a lack of full knowledge. The molecules know exactly where they’re going, at least at the Newtonian level.
What’s our problem, then? Simply that we are not closely in touch with the molecules. We’re not even fully in touch with ourselves. If it was the case that we knew ourselves perfectly, and that in us the molecules lived, and moved, and had their being, it would be a different epistemological universe.
Stirring custard would then be reasonably analogous to lining up counters in a row.
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Still a big fan of ‘stochastic’ as a good enough term…
‘Ergodic’ seems unnecessarily ‘fussy’.
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Isnt stochastic just random too?
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I agree with this statement. I see no conflict between the randomness described by science and the theological belief that God is involved in the process in some way, even though I lack a belief in God. This non-conflict has existed within the sciences for a long time now, and I am quite proud that it appears to be continuing into the future.
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