Problem 1: (Ehrenfest model) Gas molecules move about randomly in a box which is divided into two halves symmetrically by a partition. A hole is made in the partition Suppose there are 20 molecules in the box. Think of the partitions as two urns (urn 0 and 1) containing balls labeled 1 through 20. Molecular motion can be modeled by choosing number between 1 and 20 at random and moving the corresponding ball from the urn it is presently into the other. This is a historically important physical model introduced by Ehrenfest in the early days of statistical mechanics to study thermodynamic equilibrium. The Ehrenfest model has also been applied to study the dynamics of social network formations. Let X denote the number of molecules at the left partition of the box (urn 0) after n transitions. In HW4, you modeled {Xn: n > 0) as a DTMC with S = {0,1,2,..., 20) and 19/20 1/20 19 20 1/20 2/20 0 1 2 0 1 2 3 18 19 20 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 ... 0 0 20 0 0 0 0 0 0 0 0 0 0 : : : : 0 0 0 0 0 0 ... 0 0 O 0 0 0 0... 0 1 0 1 0 P= : 19 20 You also calculated the steady state distribution: 1 1 Steady State Distribution of X 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 5 10 15 20 Simulate the first 10000 transitions of the chain. The initial state of the system X, is unknown. Assume that X, could be any one of the values 0,1,2,...,20 with equal proba- bilities. Copy your MATLAB code. Use the default random generator in your simulation (rng('default'):). Attach the histogram of the simulated path. Attach the histogram of the frequency distribution. It should match with the theoretical steady-state distribution shown above. (40 Points)