Novel mapping in non-equilibrium stochastic processes

We investigate the time-evolution of a non-equilibrium system in view of the change in information and provide a novel mapping relation which quantifies the change in information far from equilibrium and the proximity of a non-equilibrium state to the attractor. Specifically, we utilize a nonlinear stochastic model where the stochastic noise plays the role of incoherent regulation of the dynamical variable x and analytically compute the rate of change in information (information velocity) from the time-dependent probability distribution function. From this, we quantify the total change in information in terms of information length L and the associated action J, where L represents the distance that the system travels in the fluctuation-based, statistical metric space parameterized by time. As the initial probability density function's mean position (μ) is decreased from the final equilibrium value (the carrying capacity), and increase monotonically with interesting power-law mapping relations. In comparison, as μ is increased from μ_∗ L and J increase slowly until they level off to a constant value. This manifests the proximity of the state to the attractor caused by a strong correlation for large μ through large fluctuations. Our proposed mapping relation provides a new way of understanding the progression of the complexity in non-equilibrium system in view of information change and the structure of underlying attractor.