A conservative master equation in logarithmic size for fragmentation population-balance equations: jump transport, drift–diffusion asymptotics, and particle-size distribution inference

Fragmentation population-balance equations describe how particle-size distributions evolve under breakage and daughter fragment redistribution. By log-size we mean the logarithmic internal coordinate ξ = ln( x / x 0 ), where x is a size-like variable taken proportional to a conserved mass. From a standard self-similar fragmentation class we derive an exact conservative transport equation in ξ for the normalized mass fraction : a state-dependent 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 optional structure-preserving operator representation and constrained parameterization for inverse modeling. 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 Schrödinger-type spectral problem, motivating compact parametric hypothesis classes for distribution shapes. We then present two inverse routes: (i) time-resolved parametric fitting of transport/spectral parameters, and (ii) a regularized steady inversion that reconstructs an effective potential from a measured steady PSD. We include numerical benchmarks: forward simulation of the jump transport model (continuous-time Markov chain discretization) and its drift–diffusion reduction, quantitative discrepancy metrics, and inverse parameter recovery on an Airy half-line synthetic benchmark under multiplicative noise.