A Conservative Log-Size Master Equation for Fragmentation PBEs: Jump Transport, Drift--Diffusion Asymptotics, and PSD Inference

TL;DR

This paper develops a conservative log-size master equation for fragmentation PBEs, linking jump transport, drift-diffusion asymptotics, and PSD inference, with applications in inverse modeling and numerical validation.

Contribution

It introduces an exact conservative transport equation in log-size for fragmentation PBEs, with a structure-preserving operator and inverse modeling methods.

Findings

Derived an exact log-size master equation for fragmentation PBEs.

Reduced jump transport to drift-diffusion in small-jump regimes.

Validated inverse PSD reconstruction with numerical benchmarks.

Abstract

Fragmentation population-balance equations (PBEs) describe how particle size distributions (PSDs) evolve under breakage and daughter fragment redistribution. From a standard self-similar fragmentation class we derive an \emph{exact conservative transport equation in log-size} for the \emph{normalized mass fraction}: a state-dependent \emph{pure-jump} master equation (nonlocal internal-coordinate mass transfer). We also give an explicit Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) factorization whose diagonal sector reproduces this master equation, used here as an \emph{optional} structure-preserving operator representation and constrained parameterization for inverse modeling (rather than a computational necessity). In a controlled small-jump regime, the nonlocal jump transport reduces to a drift--diffusion (Fokker--Planck) operator in log-size space. Under detailed-balance conditions this operator admits the standard symmetrization to a self-adjoint Schr\"odinger-type spectral problem, enabling compact parametric hypothesis classes for PSD shapes. We then present two inverse routes: (i) time-resolved parametric fitting of transport/spectral parameters, and (ii) a regularized steady-state inversion that reconstructs an effective potential from a measured steady PSD. To address practical validation, we include numerical benchmarks: forward simulation of the jump transport model (CTMC discretization) and its drift--diffusion reduction, quantitative discrepancy metrics, and inverse parameter recovery on an Airy half-line synthetic benchmark under controlled multiplicative noise.