arXiv subjects

Claude Zeller

Publications and source records attributed to Claude Zeller.

6 recordsLinked to original sources

arXiv2026

First-return statistics and the point-spread function of a Henyey-Greenstein random walk - a far-field stable law

A particle launched normally into a half space, scattering by the Henyey-Greenstein (HG) phase function, exits after a first-return time whose count of flights n approaches the distribution-free Sparre Andersen exponent, f(n) ~ n^{-3/2}, with an amplitude fixed by the launch rather than by the theorem. Conditioned on n, and past a correlation-set transient, the lateral exit position is Gaussian with variance proportional to n, a scaling we test directly. Subordinating that Gaussian displacement to the one-sided 1/2-stable first-return time gives, in the diffusion limit, an isotropic 1-stable scaling limit: the far tail of the point-spread function (PSF) for a normally incident pencil beam is the two-dimensional Cauchy tail p(r) ~ r^{-3}, the large-radius limit of the half-space Poisson kernel. The index alpha = 1 is the product of the central limit theorem and the distribution-free first-return exponent, so no phase function admitting the same diffusion limit - finite step variance, short-range angular correlation - can move it. Absorption weights each path by exp(-mu_a L) in its total length L, truncating the subordinator and producing the screened radial structure (mu_eff + 1/r) exp(-mu_eff r)/r^2 carried by each source term of the Farrell-Patterson-Wilson (1992) diffusion dipole; the finite-depth two-source dipole is not derived here. Keeping the launch-depth factor that the tail calculation discards continues the same kernel into a finite core of radius 1.7 l*, with no additional fitted parameter. A scale-dependent index alpha_eff from the empirical characteristic function tracks the similarity parameter gamma, separating phase functions matched in g_1 by 0.15-0.22 near one transport mean free path. Results are Monte Carlo at g = 0 to 0.9 with up to 10^6 walkers.

physics.optics↗

arXiv2026

Finite slab first passage statistics of Henyey Greenstein scattering

A photon entering a plane parallel scattering slab performs a random walk and eventually escapes through one of the two faces or is absorbed. The standard model employs a Henyey Greenstein phase function (HG) and an exponential step length distribution (Exp). Slab reflectance, transmittance, absorptance, and emergent angular distributions can be calculated in terms of random walk statistics. A central result is that the slab calculations factor into the order resolved first passage statistics of a half space combined with the a factor for the slab thickness. Absorptance is derived from order resolved walk statistics using the absorption rate. Two approaches are used. In the Monte Carlo (MC) approach, an extremely long random walk with many steps is efficiently generated without regard to any boundaries. The intersection of this walk with a large collection of target objects creates an ensemble of excursions of the objects. The MC approach relies explicitly on the memoryless property of Exp so that the portion of the first and last steps inside the object follow the same length distribution as the walk steps. The details of each excursion are recorded and any statistics can be extracted from the database of excursions. In particular, first passage statistics are extracted from this ensemble. In this work the objects are slabs with different positions and thicknesses. In the radiative transfer (RT) approach the slab is divided into thin layers with scattering treated to first order in each layer. The RT equations are then directly integrated over the slab to give the desired first passage statistics, reflectance, transmittance, and absorptance. The two methods agree to the Monte Carlo precision over the tested range of random walk parameters.

physics.optics↗

arXiv2026

Geometric Realism Without Angular Resolution Structural Classification of Multilayer Kubelka-Munk Theory within Radiative Transport

Kubelka-Munk (KM) theory provides a two-flux description of radiative transport in layered scattering and absorbing media. Despite its wide use in the coatings, paper, paint, and textile industries, the theory has often been regarded as a phenomenological model whose connection to the full radiative transfer equation (RTE) remains unclear. Under the standard steady-state, plane-parallel, azimuthally symmetric assumptions, we show that multilayer KM theory is exactly a rank-2 Galerkin projection of the RTE onto hemispherical basis functions. The projection is idempotent with an infinite-dimensional kernel, and its rank is preserved under multilayer composition -- so no amount of layer stacking can recover angular information discarded by the projection. We derive the KM coefficients as hemispherical moments of the transport operator and compute the projection error for representative scattering media (g from 0 to 0.85), finding that the reduced optical thickness tau* = tau(1-g) governs KM accuracy. The projection-error framework explains the well-documented accuracy of compositional multilayer models in printed media and shows where higher-order methods become necessary. The result places KM theory on rigorous footing as a legitimate -- if low-resolution -- transport approximation rather than an ad hoc phenomenology.

physics.optics↗

arXiv2026

Bridge Scaling in Conditioned Henyey-Greenstein Random Walks

We study fixed-length bridge paths -- half-space excursions that start and end at a planar boundary -- for three-dimensional random walks with Henyey-Greenstein scattering angles and exponentially distributed step lengths, using Monte Carlo simulation over asymmetry parameter g from 0 to 0.95 and path lengths from 4 to 200 steps. The key structural feature is that the walk evolves on a two-dimensional Markovian state space (depth, direction cosine) rather than the scalar depth coordinate alone. Four anomalies with respect to classical Brownian-excursion theory are reported. The mean amplitude scales super-diffusively, as path length to a power of 0.57--0.58 for isotropic scattering, nine standard deviations above the Brownian prediction of 0.5, with no sign of convergence out to 200 steps. The diffusion coefficient scales as the transport mean free path to the power 0.415 rather than the predicted 1.0. The midpoint depth distribution is Rayleigh rather than half-normal, consistent with a two-dimensional Bessel process. The bridge-conditioned mean direction cosine converges to minus two-thirds at the final step, independently of the asymmetry parameter and initial direction -- the classical Milne result anchored by the H-function moment identity. All anomalies are attributed to the two-dimensional state-space structure. The two anomalous exponents sum to approximately unity, suggesting a common geometric origin. Whether this constitutes a permanent universality-class shift or an anomalously slow crossover to Brownian-excursion behaviour remains the primary open question.

cond-mat.stat-mech↗

arXiv2025

First-return statistics in bounded radiative transport: A Motzkin polynomial framework

A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media.

physics.optics↗

arXiv2019

Light scattering as a Poisson process and first-passage probability

A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately achieves first-passage, leaving the medium or it is absorbed. The Kubelka-Munk model describes a flux of particles moving perpendicular to the surface of a plane-parallel medium. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length \(\lambda\). The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks in the alternating path is a step-length distribution-free combinatoric expression. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in an expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path combinatorics.

cond-mat.stat-mech↗