Deterministic Fokker–Planck Transport

• Deterministic Fokker–Planck transport is a PDE framework that combines coherent propagation and effective diffusion to describe the evolution of physical fields.
• It augments traditional continuity equations with an effective diffusion term derived from coarse-graining unresolved fast dynamics, enabling analytical and numerical solutions.
• This framework is applied in X-ray phase-contrast imaging, speckle-tracking, and Talbot–Lau interferometry, improving phase retrieval and system inversion techniques.

A deterministic Fokker–Planck transport model describes the evolution of a physical field (such as wave intensity, particle density, or distribution functions) under both coherent (ballistic) and diffusive (scattering-induced) transport mechanisms, without explicit stochastic forcing. The framework arises where the fundamental physical processes can be recast as transport (continuity) equations with fluxes encoding both deterministic propagation and effective diffusion from unresolved dynamics—this is particularly instrumental in imaging, plasma transport, wave propagation in disordered media, and kinetic descriptions of particle ensembles.

1. General Formulation and Key Principles

The deterministic Fokker–Planck equation (FPE) augments a bare transport (continuity) equation with a “diffusive” flux, representing the net effect of small-angle scattering or unresolved microstructure. In a prototypical setting such as paraxial imaging, the field variable is the wave intensity $I(x, y, z)$, and the governing equation (specializing Eq. (4) in (Paganin et al., 2019)) is

$$\frac{\partial I}{\partial z} = -\frac{1}{k} \nabla_\perp\cdot\left[I\nabla_\perp \phi\right] + F(x, y)\,\nabla_\perp^2\left[D(x, y)\, I\right].$$

Here,

• $k = 2\pi/\lambda$ is the wavenumber,
• $$\nabla_\perp = (\partial/\partial x, \partial/\partial y)$$ is the transverse gradient,
• $\phi(x, y, z)$ is the local phase,
• $D(x, y)$ is an effective diffusion coefficient,
• $F(x, y)$ is the local fraction of energy undergoing diffusive (rather than coherent) transport.

The first term on the right describes deterministic coherent “lensing/prism” effects, while the second is a deterministic Laplacian acting on the field, representing the collective blurring or scattering by unresolved microstructure (Paganin et al., 2019).

2. Microscopic Derivation and Physical Interpretation

A key derivation invokes coarse-graining over unresolved features: for example, Fresnel propagation with sample exit-surface phase $\phi = \phi_{\text{slow}} + \phi_{\text{fast}}$, where $\phi_{\text{fast}}$ is an unresolvable Gaussian random field. After ensemble averaging, the emergent energy transport decomposes into (i) a coherent (specular) channel and (ii) a diffuse channel with fraction $F(x) \approx \sigma^2_\phi(x)$ for weak scattering.

An expansion under small defocus $\Delta$ yields $D(x, 0; \Delta) = L^2(x; \Delta)/\Delta$, with $L\sim \theta \Delta$ characterizing the microstructure correlation angle and $\theta \sim 1/\ell$ with $\ell$ the microstructure length scale. The net result is a deterministic PDE where randomness is encoded into $D(x, y)$, but the subsequent evolution is entirely deterministic (Paganin et al., 2019).

This structure—deterministic PDEs with drift and effective diffusion coefficients from unresolved fast/small-scale processes—is central in X-ray imaging, but also in classical and quantum kinetic theory (Paganin et al., 2019). The model admits higher-order generalizations (Kramers–Moyal expansion), capturing non-Gaussian, anisotropic, or kernel-structured scattering (Paganin et al., 2019).

3. Mathematical Structure: Kramers–Moyal and Beyond

If the underlying scattering or stochastic process is non-Gaussian or highly anisotropic, the flux term can be expanded via a Kramers–Moyal series: $$\overline{I}(z=\Delta) = I_s - \frac{\Delta}{k}(1-F)\nabla_\perp\cdot[I_s \nabla_\perp \phi_s] + F \sum_{M=2}^\infty \sum_{m=0}^M D_{m, M-m}^{(M)}(x, y; \Delta) \partial_x^m \partial_y^{M-m} I_s$$ where

$$D_{m, M-m}^{(M)} = \frac{1}{M!} \binom{M}{m} \iint (x'-x)^m (y'-y)^{M-m} K(x, y; x', y'; \Delta) dx' dy'$$

and $K$ is the scattering kernel (Paganin et al., 2019). In the “classical” regime, truncation at second order yields a standard Fokker–Planck equation; higher order terms are retained for structured or nonlocal kernels.

4. Boundary, Initial Conditions, and Solvers

Typical practice specifies

• Initial data at $z=0$: $I(x, y, 0)$, $\phi(x, y, 0)$, $D(x, y, 0)$ given by the physical object or field,
• Transverse boundary condition: $I \to 0$ as $|x|, |y| \to \infty$ (vanishing flux at infinity) (Paganin et al., 2019).

Both continuous and finite-difference forms are used. The finite-difference scheme for small step $\Delta$: $$I(z=\Delta) = I(0) - \frac{\Delta}{k} \nabla_\perp\cdot [I \nabla_\perp \phi]_{z=0} + F\Delta \nabla_\perp^2 [D I]_{z=0}$$ is suitable for imaging contexts or PDE-based solvers (Paganin et al., 2019).

5. Forward and Inverse Problems: Image Formation and Retrieval

The deterministic FPE framework handles both “forward” and “inverse” problems:

• Forward: Given $I_0(x,y)$, $\phi_0(x,y)$, and $D(x,y)$, solve the FPE to predict $I(x,y,z)$ at any downstream plane. For analytic models (e.g., single-material, constant $D$) closed-form solutions exist. For general $D(x,y)$, finite-difference or spectral methods are employed.
• Inverse: Phase retrieval (recovering $\phi$) is performed by measuring $I$ at several planes ($z=0$, $z=\Delta$) and solving the TIE-like part as a Poisson equation. With dual-propagation distances and under the projection approximation, one can simultaneously retrieve both phase shift and $D_{\rm eff}$: $$I(\Delta) = \left[1 - \Delta\left(\frac{\delta}{\mu} - D_{\rm eff}\right)\nabla_\perp^2\right] e^{-\mu T(x,y)}$$ from which $T(x,y)$ and $D_{\rm eff}$ can be reconstructed via analytical inversion (Paganin et al., 2019).

In structured illumination (mask-based) imaging, the formalism allows inversion for both the phase and position-dependent diffusion field using suitable mask data and Poisson solvers (Paganin et al., 21 Jan 2026).

6. Physical Examples and Numerical Implementation

Practical applications include:

• X-ray phase-contrast imaging: Unified treatment of coherent phase contrast and unresolved scattering (e.g., in biological tissues).
• Speckle-Tracking and SAXS: Generalization of geometric-flow and raster-scan methods with extra diffusion-field channels for blurring/sharpening signatures (Paganin et al., 2019).
• Talbot–Lau Interferometry: Finite-difference FPE used to analytically model phase-contrast with grating structures (Paganin et al., 2019).
• Shot-Noise and Negative Diffusion Effects: FPE naturally predicts local “sharpening” from negative diffusion near sharp sample edges (Paganin et al., 21 Jan 2026).

Numerical solution schemes involve finite-difference discretization, Poisson solvers for phase reconstruction, and moment-based fitting for diffusion coefficients (Paganin et al., 2019, Paganin et al., 21 Jan 2026). Dual- or multi-distance imaging allows one to separately identify phase and blur coefficients.

7. Scope of Applicability and Generalizations

The formalism applies to any system where

• Paraxial approximation holds,
• Unresolved speckle or partially coherent small-angle scattering occurs,
• Both phase-contrast (coherent) and blurring (diffusive) effects are significant.

Domains include visible light out-of-focus imaging, electron and neutron phase-contrast, Zernike phase contrast, grating-based and magnetic dichroism imaging—anywhere deterministic TIE-like and diffusion-like terms coexist (Paganin et al., 2019).

The diffusion operator may be scalar, tensorial, or nonlocal (Kramers–Moyal); negative diffusion coefficients are physically realized for edge-sharpening effects. No stochastic noise is present in the PDE: all randomness is subsumed into (deterministic) effective diffusion fields.

---

In summary, deterministic Fokker–Planck transport fuses coherent propagation and diffusive scattering into a unified, fully deterministic PDE framework, generalizing the transport-of-intensity equation. This yields a robust methodology for modeling, analyzing, and inverting imaging and transport systems where both phase and unresolved blurring phenomena are present, across a spectrum of physical modalities (Paganin et al., 2019, Paganin et al., 21 Jan 2026).