Fisher Information Dynamics: A Kinematic Framework for Phase Space Ordering with Applications to Shock Layers and Turbulence

Abstract

Order here denotes the sharpness of probability-density gradients as quantified by Fisher information, distinct from thermodynamic order parameters. A kinematic framework for phase space ordering is established based on the Fisher information production rate, which possesses a rigorous thermodynamic foundation through the de Bruijn identity, Stam inequality, and dissipation theorem. A four-term decomposition into isotropic contraction, traceless shear, divergence gradient, and boundary flux is derived from the continuity equation. Extending to the Fokker-Planck equation, the non-positivity of the diffusive dissipation term is proved, with strict negativity for non-uniform densities. A vorticity-independence theorem shows that purely rotational velocity fields do not alter the order measure, and explicit geometric sign criteria enable prediction of where order is generated or destroyed. Through exact solutions of the Burgers shock layer, a pointwise identity between compression and divergence-gradient contributions is uncovered, underlying the three-term balance. In two-dimensional incompressible turbulence, a statistical two-term equilibrium (vanishing time-averaged production) is established as the null hypothesis: the conservative reference state from which three-dimensional turbulence departs through vortex stretching. Variational steady states of the associated Wasserstein gradient flow imply vanishing of the total production rate. This framework provides a diagnostic and predictive tool for analyzing ordering dynamics in non-equilibrium systems.