Suppose that [tex] M [/tex] molecules are distributed among two urns; and at each time point one of the molecules it chosen at random, removed from its urn, and placed in the other one. So this is a time-reversible Markov process right?(adsbygoogle = window.adsbygoogle || []).push({});

So [tex] P_{i,i+1} = \frac{M-i}{M} [/tex]. What do the limiting probabilities mean in words?

Like [tex] \pi_0 = \left[ 1 + \sum_{j=1}^{M} \frac{(M-j+1) \cdots (M-1)M}{j(j-1) \cdots 1} \right ]^{-1} [/tex]

[tex] = \left [\sum_{j=0}^{M} \binom{M}{j} \right]^{-1} = \left(\frac{1}{2} \right)^{M} [/tex]

and [tex] \pi_i = \binom{M}{i} \left(\frac{1}{2} \right)^{M}, \ i = 0,1, \ldots, M [/tex].

What do these really signify?

Source:Introduction to Probability Modelsby Sheldon Ross

Thanks

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# Bose-Einstein physics question

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