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this is my first question here, and I'm trying to self-learn physics from Kip Thorne's 2017 textbook "Classical Physics".

IF I understand the ergodic hypothesis correctly, it is simply the statement that given a "long enough" time, a system would visit all of the phase space. Interpreting the time that the system spends in each state as proportional to a probability, we can calculate the averages of physical quantities as time averages instead of ensemble averages.

Now, given a stationary random variable, how would I use the ergodic hypothesis to argue that

$$\lim_{t -> \infty} P_2(y_2, t| y_1,0)=p_1(y_2)?$$

(Question 6.1 of Kip Thorne's textbook)

In essence, the equation says: given that the random variable $Y$ takes value $y_1$ at $t=0$, the probability that $Y=y_2$ at $t>0$ (LHS) is equal to the probability that $Y=y_2$ at any given time.

I guess that means for this random variable $Y$, knowing that $Y$ takes a certain value in time does not give us new information.

But what does that have to do with the ergodic hypothesis?

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  • $\begingroup$ Can you define $P_{2}$ and $p_{ 1}$ please? $\endgroup$
    – user194811
    May 6, 2018 at 14:27
  • $\begingroup$ He seems to be referring to ergodicity of a Markov process, which is not the same thing as the ergodicity of an Hamiltonian flow that you seem to have in mind. For a Markov chain, for example, being ergodic means being irreducible, aperiodic and positive recurrent. Under these assumptions, one can then prove that there is a unique stationary probability measure and that the distribution of the chain converges to it as time tends to infinity. The situation is completely similar for more general Markov processes. $\endgroup$
    – Yvan Velenik
    May 6, 2018 at 16:32
  • $\begingroup$ I manage to get a copy of the book. He's indeed talking of ergodicity of a Markov process. Unfortunately, he is discussing it in a very informal way, which makes things look more complicated than they really are. I would recommend that you take a (rigorous) book on, say, finite-state Markov chains to understand these things in a simple context first. Then, you'll easily understand his informal discussion. A good, simple book is that of Haggstrom. $\endgroup$
    – Yvan Velenik
    May 6, 2018 at 16:39
  • $\begingroup$ @YvanVelenik I see! And you're correct --- I was thinking about a Hamiltonian flow, but perhaps the author had other ideas. I will move on from this question, and read about Markov chains / processes elsewhere instead. Thanks for pointing out the direction! :) $\endgroup$
    – Eve L
    May 7, 2018 at 18:04

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