There is growing evidence that for a range of dynamical systems featuring complex interactions between large ensembles of interacting elements, mutual information peaks at order/disorder phase transitions. We conjecture that, by contrast, information flow in such systems will generally peak strictly on the disordered side of a phase transition. This conjecture is verified for a ferromagnetic 2d lattice Ising model with Glauber dynamics and a transfer entropy-based measure of system-wide information flow. Implications of the conjecture are considered; in particular, that for a complex dynamical system in the process of transitioning from disordered to ordered dynamics (a mechanism implicated, for example, in financial market crashes and the onset of some types of epileptic seizures), information dynamics may be able to predict an imminent transition. In a system comprising a large number of interacting elements with an order/disorder phase transition, it is easy to argue that mutual information between elements must peak at an intermediate order: for a highly ordered system there is little indeterminacy about the state of individual elements and hence mutual information between elements will be small, while for a highly disordered system elements will behave near-independently and again mutual information between elements will be small. It also seems reasonable (particularly if long-range interactions are considered) to expect that the peak will occur at the phase transition-i.e. where susceptibility peaksand indeed this has been evidenced for a variety of complex systems, including the 2d lattice Ising model [1, 2], Vicsek's particle swarm model [3, 4] , random Boolean networks (RBNs) [5, 6] and financial markets [7] . For a complex dynamical system, it is again easy to argue that information flow between elements must peak in an intermediate order regime; here, however, it is less clear that the peak should coincide with the phase transition. We show that for a 2d lattice Ising model with Glauber dynamics [8] , information flow, as quantified by a global transfer entropy measure [9] , attains a maximum strictly in the disordered (paramagnetic) phase. We conjecture that this phenomenon is universal for a class of complex dynamical systems and discuss implications. We consider an isotropic ferromagnetic 2d lattice Ising model of size N = L × L with no external field . If the system is in state s = s 1 . . . s N , s i ∈ {+1, −1}, then the Hamiltonian is given by [22] H(s) = − <i,j> where < i, j > denotes a sum over the 2N unique pairs of lattice neighbours. We impose periodic boundary conditions, so that the system is homogeneous: for any sites i, j there is a symmetry τ -i.e. a permutation of sites leaving the Hamiltonian invariant-with τ (i) = j. At thermodynamic equilibrium, the Boltzmann-Gibbs probability to find the system in state s is where S denotes a random equilibrium state, β ≡ 1/T is the inverse temperature (units are considered normalized so that Boltzmann's constant = 1) and Z = For the kinetic model we consider discrete-time Glauber spin-flip dynamics [8] : at each time step a site i is selected uniformly at random and its spin flipped with probability. where ∆H i (s) ≡ H(s i ) − H(s) = 2s i j∈ν(i) s j is the energy difference between the spin-flipped and original state. Here a superscript i denotes flipping the ith spin and ν(i) denotes the indices of the lattice neighbours of site i. This (Markov Chain Monte Carlo) scheme satisfies detailed balance [12] and thus yields the Boltzmann equilibrium probabilities (2) at stationarity. Information-theoretic quantities may be framed in terms of statistical dependency. The basic measure of statistical dependency considered in this article is the mutual information I( X : Y | Z) between random
Barnett et al. REF show empirically that mutual information measured between pairs of neighboring spins peaks at the phase transition.
Information Flow in a Kinetic Ising Model Peaks in the Disordered Phase
Physical review letters
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Physics Medicine