1. Problem 15.35 Consider Problem 15.1 again. Let the Markov chain {Xn}

describe the number of type-1 particles in compartment A. Prove that the equi librium probabilities satisfy the recurrence relation pk,k−1πk = pk−1,kπk−1 Use this result to verify that πj = r j

r for k = 1,2,...,r.

r−j

/ 2r r

for j = 0,1,...,r. Remark:

the recurrence relation for the πk expresses that the system has the follow ing property when it has reached statistical equilibrium. Conditionally upon being in state k, the probability of coming from state k − 1 is the same as the probability of going to state k − 1(pk−1,kπk−1

πk

= pk,k−1), and the probability of coming from state k +1 is the same as the probability of going to state k +1(pk+1,kπk+1

πk

= pk,k+1). In other words, two outside observers using clocks in opposite directions will see probabilistically identical evolutions of the sys tem when the system is in statistical equilibrium. A Markov chain having this property is said to be a time-reversible Markov chain.