Generative Operators of Natural Informational Forces: A Unified Framework for MI-based Dynamics

抄録

Classical formulations typically describe natural forces in terms of potentials, vector fields, or variational principles that govern the dynamics of physical systems. However, recent information-theoretic perspectives suggest a more general characterization. Specifically, forces may be fundamentally understood through their role in reshaping probability distributions and altering informational dependencies among interacting systems.

In this work, we present a unified theoretical framework in which natural forces are characterized as specific components of generating operators acting on probability densities. For a Markovian evolution described by the differential equation
∂t p(t) = L p(t),
we identify the force-associated component, denoted L_att, as the canonical operator part that monotonically contracts the information radius
r(t) = H(X_t) + H(Y_t) − 2 MI(X_t; Y_t).
We further show that this contraction property corresponds precisely to the mutual-information gradient flow.

This operator-theoretic interpretation extends the notion of natural forces beyond classical geometric constructs to generators of informational coupling. It provides a rigorous, unified perspective that integrates interaction theories in physics, divergence contraction in information geometry, and the theory of Markov semigroups. The resulting framework offers a coherent, coordinate-free description of natural forces as operator components responsible for the creation, preservation, or enhancement of mutual information, independent of any privileged temporal or spatial parametrization.