The transition probability W(i to j)  between 3 states is:

 

W(1 to 1) = W(1 to 3) = 0, W(1 to 2) = 1

W(2 to 1) = 1/6, W(2 to 2) = 1/2,
W(2 to 3) = 1/3

W(3 to 1)=0, W(3 to 2)=2/3, W(3 to 3) = 1/3

 
 
(a)  Show that the solution for the probabilities is:

P₁ = 1/10, P₂ = 6/10, P₃=3/10
 

(b)  Use P0=(1,0,0) as the initial distribution, how fast the system converges to the equilibrium probability distribution?
 
 

• Consider a situation where we have only three possible states 1, 2, and 3 (e.g. discrete quantum states). The transition probability is given as W(1 to 2) = 1, W(2 to 1) = W(2 to 3) = W(3 to 2) = W(3 to 3) = 1/2, and W(1 to 3) = W(3 to 1) = W(1 to 1) = W(2 to 2) = 0.

(a) Is the Markov chain specified by the above transition probabilities ergodic?

(b) Calculate the equilibrium probability distribution Pi, where i = 1,2,,3.

(c) Does the equilibrium distribution satisfy detailed balance condition?