nonequilibrium
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- 2
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I'll shortly explain what my reasoning is so far, but please ignore it if it comes across too jumbled:
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Let P denote the markov matrix associated with this problem, then I think I was able to argue that the probability that is asked for is equal to 1- \sum_{n=1}^{+\infty} P^n(0,0) where P^n(0,0) denotes the element in the first row of the first column of the n-th power of the Markov matrix.
And I then wanted to calculate P^n(0,0) for every n by trying to find a pattern in P^1(0,0), P^2(0,0), P^3(0,0), etc. I think I found one: define t_n = \left(t_{n-1} + \left( \frac{n+1}{n} \right)^2 \prod_{k=1}^{n+1} x_k \right) x_n (with t_0 = x_1) with x_n = p_{n,n-1}, then I think \sum_{n=1}^{+\infty} P^n(0,0) = \sum_{n=0}^{+\infty} t_n. But it seems near impossible to prove that 1 minus this expression equals \frac{6}{\pi^2} so I'm probably way off track...
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Any better suggestions? How would you approach this problem instead?