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.
Related
- On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening
- Kinematic Control and Dephasing Dynamics of Quantum Resources in e+e−→τ+τ−
- Generative AI in Turbulence: Foundations, Applications, Validations and Benchmarks
- Pair-exchange dynamics of magnetohydrodynamic turbulence
- A Multi-Model Non-Intrusive Reduced-Order Framework for Parametric Erosion Prediction via Kinematic Cross-Moment Compression
BibTeX
@misc{wu2026fisher,
title = {{Fisher Information Dynamics: A Kinematic Framework for Phase Space Ordering with Applications to Shock Layers and Turbulence}},
author = {Yingchuan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05021},
eprint = {2609.05021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05021},
}
RIS
TY - UNPB
TI - Fisher Information Dynamics: A Kinematic Framework for Phase Space Ordering with Applications to Shock Layers and Turbulence
AU - Yingchuan Wu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.05021
UR - https://arxiv.org/abs/2609.05021
AB - 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.
ER -