Comminution as a Non-Hermitian Quantum Field Theory: Log-Size Jump Generators, Branching Embeddings, and the Airy Solvable Sector

TL;DR

This work develops a foundational field-theoretic framework for comminution PBEs by deriving an exact log-size jump generator from homogeneous kernels and embedding it in a completely positive Lindblad form, then extending to a bosonic NHQFT with two stochastic embeddings: a linear tagged-mass theory and an interacting branching theory. It demonstrates how the deterministic PBE emerges as the one-body sector, while higher-order correlators capture finite-N fluctuations, and shows that coarse-graining around a linear Airy-type kernel yields a solvable quadratic sector with explicit Gaussian two-point functions. The Airy-type analysis provides explicit mode-sum expressions for equal-time correlations, linking PSD shapes and fluctuations to a universal quadratic sector. The framework clearly separates kernel-determined fixed structure from stochastic modelling, and connects comminution kernels to universality classes in non-Hermitian and growth–fragmentation physics, while outlining precise open problems in coarse-graining, RG, and parameter inference.

Abstract

Pure-breakage population balance equations (PBEs) give the standard deterministic description of fragmentation and comminution. They predict mean particle size distributions, but they do not determine fluctuations, size-size correlations, or universality under coarse-graining. We develop a field-theoretic framework anchored in the PBE kernel inputs (selection rate and daughter distribution) and compatible with the stochastic Doi-Peliti approach. From homogeneous kernels we derive an exact Markov jump generator in log-size for a mass-weighted (tagged-mass) distribution, with a jump law that is a probability density fixed by the daughter distribution. The generator is generically non-self-adjoint, admits a Lindblad embedding, and has a second-quantized extension. The deterministic PBE appears as the one-body sector, while multi-point correlators encode finite-population fluctuations. We also give a binary-fragmentation embedding whose mean-field limit reproduces the PBE but whose higher correlators capture multiplicative cascade noise. For a linear Airy-type kernel, long-wavelength coarse-graining yields an effective Airy operator as a solvable quadratic sector about a stationary profile, producing explicit mode-sum formulas for equal-time connected two-point correlations. Overall, the framework separates what is fixed by kernel data from what requires additional stochastic modeling and links comminution kernels to universality classes.

Figures (2)

• Figure 1: Schematic mapping developed in this article. Starting from a homogeneous pure-breakage PBE, one may: (i) form a log-size jump generator for a mass-weighted (tagged-mass) density, which admits a Lindblad embedding and a free (Gaussian) NHQFT; and/or (ii) specify a microscopic branching rule (e.g. binary splitting) whose mean-field equation reproduces the PBE, yielding an interacting branching NHQFT. In both cases, coarse-graining can produce solvable quadratic sectors (e.g. Airy) that organise PSD profiles and fluctuation spectra.
• Figure 2: Illustration of the coarse-grained Airy operator: a linear effective potential in log-size around a reference scale $\xi_\star$ yields Airy eigenmodes controlling the long-wavelength profile and (in the NHQFT) the leading fluctuation spectrum.