probability distribution of patterns in Ising model. Examples 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 , then entropy coding would allow encoding a message using this chosen probability distribution. The stationary probability distribution of symbols for a given turns out to be . Hence, entropy production is , which is maximized for , known as the golden ratio. In contrast, standard random walk would choose suboptimal . While choosing larger reduces the amount of information produced after 0, it also reduces frequency of 1, after which we cannot write any information. A more complex example is the defected one-dimensional cyclic lattice: let say 1000 nodes connected in a ring, for which all nodes but the defects have a self-loop (edge to itself). In standard random walk (GRW) the stationary probability distribution would have defect probability being 2/3 of probability of the non-defect vertices – there is nearly no localization, also analogously for standard diffusion, which is infinitesimal limit of GRW. For MERW we have to first find the dominant eigenvector of the adjacency matrix – maximizing in: for all positions , where for defects, 0 otherwise. Substituting and multiplying the equation by −1 we get: where is minimized now, becoming the analog of energy. The formula inside the bracket is discrete Laplace operator, making this equation a discrete analogue of stationary Schrodinger equation. As in quantum mechanics, MERW predicts that the probability distribution should lead exactly to the one of quantum ground state: with its strongly localized density (in contrast to standard diffusion). Taking the infinitesimal limit, we can get standard continuous stationary (time-independent) Schrodinger equation ( for ) here. See also Principle of maximum entropy Eigenvector centrality Markov chain Anderson localization References External links Gábor Simonyi, Y. Lin, Z. Zhang, "Mean first-passage time for maximal-entropy random walks in complex networks". Scientific Reports, 2014. Electron Conductance Models Using Maximal