Yhe illustration of our setup is above. Yhere are seven fast moving par ticles (in red ) and six slow moving particles ( in blue). In the image you can see the demon that just closed the door between the two urns block~ ing a slow particle om movingom the right to the left urn. In general, we will assume n particles total with q 6 [0, 1] such that an particles are fast and (1 - q)n are slow. Eventually, all of the slow particles will be trapped in the right urn, a low entropy state (a high degree of order) and thus the system would violate the second law of thermodynamics. Other than the interventions by the demon the dynamics proceed pre~ cisely as before. Namely, at each time step, we pick a particle uniformly at random and move it to the other urn. Of course when the demon in tervenes (when we try to move a slow particle om the right urn to the left one), the usual (Ehrenfest) dynamics are violated. As in class, we let X k denote the number of fast particles in the left urn at time step k. Exercise 1. Design the sample space 5'2, the state space S and explain why each X k is a random variable. Why is X = {X k}keN a stochastic process? Exercise 2. State the (possible) transitions of X for any state x e S and draw a transition diagram. For each transition describe what even occurs. Exercise 3. Calculate the transition probabilities for X = {X k }keN and explain why X is a Markov chain on the state space 8. 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}keN, state the (possible) transitions of Y for any state y e S, draw the transition diagram, calculate the transition probabilities and show that Y is a Markov chain