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

Abstract

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 strict non-positivity of the diffusive dissipation term is proved. A vorticity-independence theorem is proved: purely rotational velocity fields do not alter the order measure. Explicit geometric sign criteria for each deformation term are derived, enabling prediction of order generation or destruction. Through exact solutions of the Burgers shock layer, a pointwise identity between the compression and divergence-gradient contributions is uncovered, underlying the three-term balance. The shear-dissipation balance is verified by two-dimensional turbulence simulations. It is further shown that variational steady states of the associated Wasserstein gradient flow imply decomposition balance. This framework provides a diagnostic and predictive tool for analyzing ordering dynamics in non-equilibrium systems.