% Consider the following markov model: % Time is discretized into units of length dt, say dt = 1 second. % The state-space will consist of a large 'ring' of 128 states, [0,...,128-1] % ordered in a periodic sequence, with the last state adjacent to the first. % In this periodic arrangement, any particular state k is the same as state k+128. % % For this system, imagine the following state-transition-matrix: % Given that the markov process begins a time-step in state k, % the probability P(j,k) of transitioning to another state j is 0, % unless j is within 4 steps of k (either to the right or left). % If |k-j| <= 4, then P(j,k) = 1/9. % % Find the equilibrium-distribution for this markov model. % How quickly does a typical ensemble converge to this equilibrium-distribution? % Plot the dominant 6 eigenvectors of the state-transition-matrix P. % % Now consider an ensemble initially concentrated at one point (e.g., state 64). % Plot the evolution of this ensemble over time.