Recall the Ehrenfest gas molecules model from class. There are two compartments
ID: 3178668 • Letter: R
Question
Recall the Ehrenfest gas molecules model from class. There are two compartments that together contain N molecules. Each unit of time, one of the N molecules is chosen at random and moved to the other compartment. We form a Markov chain X_n, which is defined as the number of molecules in the first compartment at time n. Give the transition matrix and show that the MC is irreducible and positive recurrent. Find and solve the detailed balance equation for the distribution pi = (pi_i: 0 lessthanorequalto i lessthanorequalto N). Assume that N = 2M is even. What is the long run proportion of time when both compartments have the same number of molecules? Use the Stirling formula to make your answer more meaningful. Suppose that at time 0 the first compartment has no molecules. What is the average waiting time for this situation to happen again?Explanation / Answer
i)Suppose the two urns contain a total of 6 molecules i.e N=6, and the possible states for the Markov chain are 0, 1, 2, 3, 4, 5, and 6. Notice first that if there are 0 molecules in urn A at time n, then there must be 1 molecule in urn A at time n + 1, and if there are 6 molecules in urn A at time n, then there must be 5 molecules in urn A at time n + 1. In terms of the transition matrix P, this means that the columns in P corresponding to states 0 and 6 are:
P0=[0 1 0 0 0 0 0], P6=[0 0 0 0 0 1 0]
If there are i molecules in urn A at time n, with 0 < i < 6, then there must be either i 1 or i + 1 molecules in urn A at time n + 1. In order for a transition from i to i 1 molecules to occur, one of the i molecules in urn A must be selected to move; this event happens with probability i/6. Likewise a transition from i to i + 1 molecules occurs when one of the 6 i molecules in urn B is selected, and this occurs with probability (6 i)/6. Allowing i to range from 1 to 5 creates the columns of P corresponding to these states, and the transition matrix for the Ehrenfest model with N = 6 is thus:
P=
STATES
0
1
2
3
4
5
6
0
0
1/6
0
0
0
0
0
1
1
0
1/3
0
0
0
0
2
0
5/6
0
1/2
0
0
0
3
0
0
2/3
0
2/3
0
0
4
0
0
0
½
0
5/6
0
5
0
0
0
0
1/3
0
1
6
0
0
0
0
0
1/6
0
To prove irreducibilty,we can see from the transition matrix that we can reach from any state to each state with some probabilty and that will be more clear through transition diagram.Hence it is irreducible.
ii) Balanced equation:
=P or (P-I)=0
STATES
0
1
2
3
4
5
6
0
0
1/6
0
0
0
0
0
1
1
0
1/3
0
0
0
0
2
0
5/6
0
1/2
0
0
0
3
0
0
2/3
0
2/3
0
0
4
0
0
0
½
0
5/6
0
5
0
0
0
0
1/3
0
1
6
0
0
0
0
0
1/6
0
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.