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

TL;DR

This paper develops a quantum field-theoretic framework for particle fragmentation processes, capturing fluctuations and correlations beyond mean-field models, and identifies universal behaviors in specific kernel cases.

Contribution

It introduces a non-Hermitian quantum field theory approach to model stochastic fragmentation, deriving exact generators and connecting kernels to universality classes.

Findings

Exact Markov jump generator for log-size distribution

Lindblad embedding and second-quantized extension

Explicit correlations in the Airy kernel case

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.