Lévy Formulation of the Stochastic Theory of Chromatography and Extension to Phase-Type Markov Renewal Process

The stochastic theory of chromatography describes solute migration as the cumulative result of random retention events superimposed on convective transport and axial dispersion. Classical Poisson-based formulations offer analytical transparency but are limited in their ability to represent heterogeneous, multistep, or multipathway adsorption kinetics increasingly revealed by single-molecule measurements. Here, we reformulate stochastic chromatography within a Lévy-Khintchine characteristic function framework and extend the underlying event structure to phase-type Markov renewal processes, including Erlang and hyper-Erlang waiting-time distributions. This representation preserves analytical tractability through matrix-exponential evaluation while enabling flexible descriptions of heterogeneous adsorption-desorption pathways. We further develop an extended classical characteristic function Fourier inversion method that operates directly in the first-passage domain and incorporates multisite log-normal sojourn heterogeneity, γ distributed stationary residence times, and inverse-Gaussian treatment of mobile-phase dispersion. Application to experimental DNA chromatograms demonstrates accurate reconstruction of peak position, width, asymmetry, and tailing behavior. A hybrid Markov renewal Monte Carlo simulator was also introduced as a mechanistic first-passage benchmark. Identifiability analysis indicated that effective sojourn times and transport parameters are robustly constrained, whereas several microscopic kinetic constants remain structurally nonidentifiable from single chromatograms alone. Overall, the proposed Lévy and Fourier-inversion framework links ensemble chromatographic peak shapes with microscopic adsorption statistics and provides a practical analytical route for modeling heterogeneous stationary phases, biomolecular separations, and single-molecule-informed chromatographic method development.