Information flow within stochastic dynamical systems.

Information flow or information transfer is an important concept in general physics and dynamical systems which has applications in a wide variety of scientific disciplines. In this study, we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system. An explicit formula has been obtained for the resulting transfer measure, which possesses a property of transfer asymmetry and, if the stochastic perturbation to the receiving component does not rely on the giving component, has a form the same as that for the corresponding deterministic system. This formula is further illustrated and validated with a two-dimensional Langevin equation. A remarkable observation is that, for two highly correlated time series, there could be no information transfer from one certain series, say x_{2} , to the other (x_{1}) . That is to say, the evolution of x_{1} may have nothing to do with x_{2} , even though x_{1} and x_{2} are highly correlated. Information flow analysis thus extends the traditional notion of correlation analysis and/or mutual information analysis by providing a quantitative measure of causality between dynamical events.