spond to a matrix-vector equation. The matrix P is, thus, applied to transform the N-vector of coordinates = x E X, which we call . The result of such a transformation is an N-vector of coordinates p(x = x nn11), = x E X, which we call u'. The matrix-vector equation (Eq. (D7)) is, thus, summarized in the form: To take a practical example, consider the 2 x 2 transition matrix that corresponds to a two-state Markov chain (X being made of two events): (D10) Markov chains and the second law of thermodynamics where a, B are real constants. This means that this Markov process is time invariant. Replacing this definition in Eq. (D10), we obtain: (D11) Since the input coordinates satisfy M1 + M2=1 (being probabilities), we observe that the sum of the output coordinates is also unity, M' + M'2 = U1 + M2 = 1, which justifies our choice for the time-invariant transition matrix P (it is easily shown that this is actually the only one). This example will help us to illustrate yet another important concept. We have seen that a Markov process can be time invariant, meaning that the transition matrix has constant or unchanging coefficients. But this time invariance does not mean that the probability distribution does not change over time: we have just seen from our previous example that in the general case u # u, which means that p(x) at time tn+1 is generally different from p(x) at time tn. But nothing forbids the distribution from remaining unchanged over time. By definition, we shall say that the distribution u is stationary if the following property is satisfied: (D12) With the previous example, it is easily established that the stationary solution satisfies: (D13) with the condition a-B#1. Such a distribution is of the type p(x1) = B/M and p(x2) = 1 - B/M, where M = 1 - a + B, meaning that it is generally nonuniform. The specific case of a uniform stationary distribution is given by B = M/2, which gives p(x1) =p(x2)=1/2. = The lesson learnt from the above example is that time-invariant Markov processes have stationary solut