Let us first look at a simple nontrivial situation: Fibonacci coding, where we want to transmit a message as a sequence of 0s and 1s, but not using two successive 1s: after a 1 there has to be a 0. To maximize the amount of information transmitted in such sequence, we should assume uniform probability distribution in the space of all possible sequences fulfilling this constraint. To practically use such long sequences, after 1 we have to use 0, but there remains a freedom of choosing the probability of 0 after 0. Let us denote this probability by q , then entropy coding would allow encoding a message using this chosen probability distribution. The stationary probability distribution of symbols for a given q turns out to be ρ ρ = ( 1 / ( 2 − − q ) , 1 − − 1 / ( 2 − − q ) ) . Hence, entropy production is H ( S ) = ρ ρ 0 ( q log ⁡ ⁡ ( 1 / q ) + ( 1 − − q ) log ⁡ ⁡ ( 1 / ( 1 − − q ) ) ) , which is maximized for q = ( 5 − − 1 ) / 2 ≈ ≈ 0.618 , known as the golden ratio. In contrast, standard random walk would choose suboptimal q = 0.5 . While choosing larger q reduces the amount of information produced after 0, it also reduces frequency of 1, after which we cannot write any information.