Fisher Regularised Information Hydrodynamics

• Fisher regularised information hydrodynamics is a framework that introduces convex Fisher information functionals into classical hydrodynamics to enforce smoothness, positivity, and conservation laws.
• It unifies hydrodynamic models, Hamiltonian field theories, and information geometry to create rigorous, reversible extensions applicable to quantum, kinetic, and wave dynamics.
• The approach facilitates applications ranging from quantum Madelung fluids to optimal transport and Fokker–Planck closures by ensuring regularisation through local, first-order, and convex extensions.

Fisher regularised information hydrodynamics is the synthesis of hydrodynamic frameworks, Hamiltonian field theories, and information geometry through the introduction of convex Fisher information functionals as local regularisers or flux densities. This theoretical structure rigorously unifies information-theoretic concepts (specifically, Fisher information) with continuum or fluid models, leading to rich interpretations of quantum mechanics, kinetic theory, nonlinear waves, and non-equilibrium thermodynamics. Across disparate domains—including quantum Madelung fluids, gradient flows in optimal transport, generalized Fokker–Planck closures, and the propagation of information in waves—Fisher regularisation enforces smoothness, positivity, reversible dynamics, and exact conservation laws. It also provides the only strictly local, first-order, rotationally invariant, and convex extension to classical hydrodynamics or Hamiltonian ensemble mechanics that admits both reversibility and projective linearity.

1. Fundamental Principles and Minimal Axioms

The core of Fisher regularised information hydrodynamics is characterized by the addition of a local Fisher information functional to classical hydrodynamical or Hamiltonian structures, typically through a strictly first-order, convex, rotationally invariant density:

$$I[\rho] = \frac{1}{2} \int_\Omega \frac{|\nabla\rho(x)|^2}{\rho(x)}\,dx = 2 \int_\Omega |\nabla \sqrt{\rho(x)}|^2\,dx$$

Axiomatically, any extension of classical (Hamiltonian) hydrodynamics on probability density $\rho(x,t)$ and conjugate phase $S(x,t)$ is subjected to:

• strictly local or first-order polynomial functionals (Dubrovin–Novikov class)
• phase (global $\mathrm U(1)$) invariance, enforcing that only $\nabla S$ can enter the Hamiltonian
• Euclidean invariance and probability conservation
• convex regularity: additional terms must be convex, local, and positive-definite in the gradient of $\rho$

The canonical Poisson bracket for admissible $F, G$ is

$$\{F, G\} = \int_\Omega \left(\frac{\delta F}{\delta\rho}\frac{\delta G}{\delta S} - \frac{\delta F}{\delta S}\frac{\delta G}{\delta\rho}\right) dx$$

The Fisher functional emerges uniquely: for a generic regularisation $F[\rho] = \int f(\rho) |\nabla\rho|^2 dx$, only $f(\rho) \propto 1/\rho$ (i.e., the Fisher information) generates a non-redundant, projectively linear, reversible term whose Euler–Lagrange derivative is $\propto -\Delta\sqrt{\rho}/\sqrt{\rho}$ (Dunkley, 5 Nov 2025).

2. Fisher Information in Quantum, Kinetic, and Wave Hydrodynamics

Quantum Madelung and Schrödinger Dynamics

In the Madelung hydrodynamic formulation, the density $\rho=|\psi|^2$ and phase $S$ of the quantum wavefunction $\psi$ yield continuity and momentum (Hamilton–Jacobi) equations with an additional "quantum potential"

$$Q(\rho) = -\frac{\hbar^2}{2m}\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}$$

whose expectation value is proportional to the Fisher information:

$\langle Q \rangle = \frac{\hbar^2}{8m}I[\rho]$

Here, Fisher information acts analogously to internal energy, regularising the hydrodynamical description and providing a thermodynamically meaningful, positive-definite penalty against disorder (spreading) (Heifetz et al., 2015, Dunkley, 5 Nov 2025). For spinful systems (Pauli fluids) and in the presence of electromagnetic interactions, the Fisher term is required to recover both the correct quantum stress tensor and the coupling between spin (internal degrees of freedom) and field gradients (Yahalom, 2018, Yahalom, 2023).

Information Flow in Linear Waves

In classical or quantum wave propagation, local and flux densities of Fisher information can be defined with respect to any material parameter $\theta$ (e.g., permittivity, permeability):

$$I(\mathbf{r}, t) = (\hbar \omega)^{-1} [\epsilon(\mathbf{r}) |\partial_\theta E_0(\mathbf{r}, t)|^2 + \mu(\mathbf{r}) |\partial_\theta H_0(\mathbf{r}, t)|^2]$$

$$\mathbf{j}(\mathbf{r}, t) = \frac{2}{\hbar\omega} \Re[ \partial_\theta E_0^*(\mathbf{r}, t) \times \partial_\theta H_0(\mathbf{r}, t) ]$$

These satisfy a local continuity equation up to source (due to spatial $\theta$-dependence) and absorption (dissipation) terms:

$$\partial_t I + \nabla \cdot \mathbf{j} = \sigma - \alpha$$

This propagation of Fisher information is precisely analogous to mass conservation, with source/sink terms corresponding to information generation/absorption (Hüpfl et al., 2023).

Fisher Regularisation in Kinetic and Fokker–Planck Models

In non-equilibrium kinetic theory, incorporating Fisher information as an entropic constraint into Fokker–Planck closure ensures:

• exact moment matching to low-order observables
• preservation of the H-theorem with dissipation rate $\propto$ Fisher information
• stabilization of higher-order gradients (hyperdiffusive regularisation)

The closure is constructed via variational maximization of entropy at fixed Fisher information and prescribed moments (Montanaro et al., 8 Sep 2025):

$$\mathcal{L}[f,...] = -\int f\ln f\, dv + \text{constraints} + \nu(I[f] - I_0)$$

yielding drift-diffusion operators with polynomial drift and leading higher-derivative (hyperdiffusive) corrections.

3. Gradient Flows, Optimal Transport, and Regularized Wasserstein Dynamics

The variational time-discretization of Wasserstein gradient flows (the JKO scheme) admits Fisher information as a natural regulariser:

$$\rho^{k+1} = \arg\min_\rho W_2^2(\rho,\rho^k) + 2\tau E(\rho) + \beta^{-2}\tau^2 I[\rho]$$

This Fisher regularisation strictly enforces positivity, strong convexity, and smoothness of density, eliminates the formation of vacuum, and permits efficient sequential quadratic programming (SQP) solvers owing to the uniform parabolicity injected by $I[\rho]$ (Li et al., 2019).

Under the Benamou–Brenier dynamic formalism, one recovers the Schrödinger bridge and entropic optimal transport structure,

$$\min_{\rho, m}\int_0^{1}\int [|m|^2/\rho + \beta^{-2}\tau^2|\nabla\log\rho|^2\rho]dxdt + 2\tau E(\rho)$$

with corresponding Euler–Lagrange equations and a precise connection to entropic interpolations.

4. Fluid-Mechanical and Thermodynamic Analogies

A recurring theme is the analogy between mass/hydrodynamic current and Fisher information density/flux:

Fluid quantity	Information-theoretic analogue
Mass density $\rho$	FI density $I$
Mass current $\rho v$	FI flux $\mathbf{j}$
Source/sink	Information generation/absorption ($\sigma$, $\alpha$)
Pressure/viscosity	Fisher penalty/internal energy

The quantum potential (Bohm term) becomes the force per unit mass driving dispersive effects, arising from the gradient of the Fisher information. The total energy density in Madelung fluids splits into macroscopic kinetic energy (from phase gradients) and internal energy (from amplitude gradients—the Fisher term) (Heifetz et al., 2015).

In thermodynamic terms, the Fisher information corresponds to a degree of "order," with its inverse acting as a measure of quantum disorder: as systems evolve or expand, Fisher information decreases, coding the increase of disorder and entropy.

5. Regularisation, Smoothing, and Numerical Stability

Fisher regularisation provides an explicit smoothing mechanism in all applications:

• In wave hydrodynamics, mollification via convolution or spatial averaging regularises singularities in the local field and ensures the existence of well-defined information flows (Hüpfl et al., 2023).
• In Fokker–Planck and hydrodynamic closures, the higher-order $\nabla^4$ (hyperdiffusion) terms stabilize under-resolved features, such as shock fronts, without smearing them excessively or violating primary transport coefficients (Montanaro et al., 8 Sep 2025).
• In variational Wasserstein gradient flows, the discrete Fisher functional is coercive, ensuring unique, positive, and smooth minimisers and fast quadratic convergence of scheme iterates (Li et al., 2019).

Practical implementations require the spatial and temporal discretization of Fisher terms via grid-based Laplacians or finite differences, and the choice of regularisation parameters must respect both physical scales (e.g., Planck constant in quantum mechanics) and numerics (e.g., mesh scale, time step).

6. Generalisations and Structural Necessity

The converse Madelung theorem establishes that, under minimal, physically and symmetrically motivated axioms, the only convex local regularisation that preserves strict reversibility, projective linearity, and all required conservation symmetries is Fisher information with precisely the Planckian coefficient (Dunkley, 5 Nov 2025). In particular,

• Any deviation away from the Fisher scale ($\alpha \neq \hbar^2/(2m)$) in the regularisation of the hydrodynamical equations breaks linear superposability, introduces dissipation, or loses Galilean invariance.
• Residual diagnostic measures (Hamilton–Jacobi and continuity errors, superposition stress tests) sharply minimize at the Fisher scale and nowhere else.

Extensions to multi-species and relativistic systems impose a unique identification of Planck’s constant $\hbar$ across all sectors, enforced through a global complex structure via the polar map $$(\rho, S) \mapsto \psi = \sqrt{\rho}\exp(i S/\hbar)$$ (Yahalom, 2023, Dunkley, 5 Nov 2025).

7. Extensions, Outlook, and Open Problems

Fisher regularised information hydrodynamics is a scalable paradigm with direct applications to quantum fluid dynamics, hydrodynamic limits of kinetic theory, information flow in complex media, and hybrid particle–continuum simulation strategies. Notable directions include:

• Generalisation to nonlinear or non-Hermitian systems by Fisher-regularised linearisation around background fields
• Embedding local Fisher continuity in information-geometric (e.g., Kubo–Mori) or Fisher–Rao structures for thermodynamics of open systems (Hüpfl et al., 2023)
• Hypergraph- or network-based analogues using discrete Laplacians to regularise transport on graphs (Li et al., 2019)
• Variational formulations of quantum–kinetic equations for particles with spin or internal degrees of freedom (Yahalom, 2023, Yahalom, 2018)
• Full relativistic and gauge-covariant fluid analogues for the Dirac equation, with Fisher regularisation applied to field amplitudes in multi-component bispinor fields (Yahalom, 2023)

Open questions persist regarding the representation and elimination of residual non-local couplings (e.g., Berry-phase-like connections in Dirac fluids), the quantification of disorder and entropy in nonstationary stochastic processes, and generalizations to higher geometric and algebraic settings.

Fisher regularisation thus serves as the uniquely privileged structure at the intersection of information theory, fluid mechanics, and quantum dynamics, enforcing both analytic regularity and deep physical constraints across domains.