There are seven fast moving particles (in red) and six slow moving particles (in blue). you can see the demon that just closed the door between the two urns blocking a slow particle from moving from the right to the left urn. In general, we will assume n particles total with q in [0,1] such that qn particles are fast and (1 -q)n are slow. Eventually, all of the slow particles will be trapped in the right urn. Each step, pick a particle at random and move it to the other urn. Of course when the demon intervenes (when we try to move a slow particle from the right urn to the left one), the usual (Ehrenfest) dynamics are violated Let Xk be the total number of fast particles in the left urn at step k.

Exercise 1. Design the sample space Omega, the state space S and explain why each Xk is a random variable. Why is X={Xk}k in N a stochastic process?

Exercise 2. State the (possible) transitions of X for any state x in S and draw a transition diagram. For each transition describe what even occurs. Exercise 3. Calculate the transition probabilities for X={Xk}k in N and explain why X is a Markov chain on the state space S.

Exercise 4. Let Yk denote the number of slow particles present in left urn at time k. Write down the state space S for the process Y={Yk}k in N, state the (possible) transitions of Y for any state y in S, draw the transition diagram, calculate the transition probabilities and show that Y is a Markov chain.