Re: Philosophical questions concerning physical Markov chains

On Nov 29, 8:24 am, Haines Brown <bro...@xxxxxxxxxxxxxxxxxxxxxxx>
wrote:
I have four elementary (definitional) questions about Markov chains.

I understand that Markov Chains are probabilistic by definition because
each successive state is not unequivocally determined by the prior state
but only constrained by it, and so the result is that the chain as a
whole is probabilistic.
I find it useful to write the deterministic equations in terms of
the usual variables (i.e., space and time, etc.), and add a variable
called "realization." The realization is a variable that contains all
the information necessary on the initial conditions of the system. The
system is completely deterministic if one includes the "realization."
However, calculating probabilities and their moments includes an
averaging over "realizations." This is called an ensemble average.
For a little description of "realization," read:
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics
of Turbulence. Volumes I and II (Dover, 1978). Although the books are
heavy mathematical reading, just the concept of realization helped me
sort through some complicated statistics.
Remember, in most topics other than quantim physics the
development of a system is always deterministic. The problem is that
the "realization" is a hidden variable. The development of the system
is so critically dependent on the realization, that each realization
is effectively uncorrelated with any other realization. However, the
system is determined precisely for any one realization.
I think that will help you with Markov chains.
.