The Dynamics of Information Fields

We introduce a formal framework for the dynamics of information fields, defining information content as I(x,t)=−log(⁡P(x,t)) and deriving a general partial differential equation that captures wave-like propagation, diffusion, and nonlinear gradient interactions. The framework recovers classical diffusion in the overdamped limit and extends naturally to quantum systems, yielding a nonlinear Schrödinger equation with a density-dependent potential. This approach provides insight into how local variations in a scalar field which represents information, probability, or density, modulate the propagation of perturbations, with potential applications in Bose–Einstein condensates, nonlinear optics, and collective phenomena in complex systems.