Expectation parameter representation of information length for non-equilibrium systems

To define a metric tensor which supplies the information length with geometric underpinning in the non-equilibrium systems, we introduce the expectation parameters instead of the externally controllable parameters as the coordinates of the statistical manifold of the system. In our formulation, the metric tensor is defined only from the stochastic variables without physical details of the system, and then, the physical properties are taken into account through the time evolution of the correlation functions. Thus, the expectation parameter expression makes it possible to analyze the information length and velocity even in extremely out-of-equilibrium systems. To illustrate usefulness of our methodology, we show the applications of our formulation to the simple three-states system and the spin-1 magnetic meanfield model whose time evolutions are described by the master equations.