A Lindblad and Non-Hermitian Spectral Framework for Fragmentation Dynamics and Particle Size Distributions

Abstract: Population balance equations (PBEs) for pure fragmentation describe how particle size distributions (PSDs) evolve under breakage and fragment redistribution. We map a self-similar fragmentation class to: a conservative pure-jump master equation in log-size space; an exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dilation whose diagonal sector reproduces that master equation; and a controlled small-jump limit where the dynamics reduce to a Fokker-Planck operator that can be symmetrized into a Schrodinger-type spectral problem. Two points ensure correctness and applicability. First, the Lindblad embedding is exact when the daughter kernel is interpreted in mass-weighted form (equivalently, when z*kappa(z) is a probability measure). Second, for genuinely non-Hermitian dynamics the stationary PSD is naturally a biorthogonal product of left and right ground states, not a naive modulus square; the usual modulus squared of psi appears only in pseudo-Hermitian or effectively dephased regimes. We then give a spectral dictionary linking typical PSD shapes to low-dimensional potential families in log-size space and outline inverse routes to infer effective potentials from data: parametric fitting with time-resolved PSDs, a direct steady-state inversion from a smoothed PSD, and an outlook toward inverse spectral ideas. A synthetic example demonstrates forward simulation and inverse parameter recovery in an Airy-type half-line model.