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 (mu) is decreased from the final equilibrium value mu(*) (the carrying capacity), L and J increase monotonically with interesting power-law mapping relations. In comparison, as mu is increased from, mu(*), 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 mu 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.