Chromatographic Peak Shape beyond the High-Péclet Approximation: Analytic Time-Domain Expression and Derivatives
Abstract
Analytic time-domain expressions for the elution-time density and its parameter derivatives were recently derived from a stochastic model of linear isocratic chromatography incorporating advective-diffusive transport, injection dispersion, and fast and slow retention mechanisms. A Gaussian approximation, justified asymptotically at high column-scale Péclet numbers, was adopted for the distribution of the combined mobile-phase and fast-retention time, while slow retention was incorporated through a series of Gaussian-Gamma convolutions. Here, the underlying normal-inverse-Gaussian density is approximated by a finite Gaussian mixture using an existing quadrature rule. The elution-time PDF is a weighted sum of series of the previously derived Gaussian-Gamma form. Its parameter derivatives are also obtained. Newly derived recurrences reduce the cost of evaluating the coefficients of that series and their derivatives from quadratic to linear in truncation order for a fixed number of mechanisms. Through a probabilistic analysis of the series coefficients, the required number of terms is found to be small enough for rapid evaluation under usual chromatographic conditions. A relative roundoff-error estimator is derived to quantify finite-precision loss in the inherited forward recurrence for Gaussian-Gamma convolutions. For the cancellation-prone regime, a backward reconstruction is developed, initialized from an exact ratio of consecutive Gaussian-Gamma terms expressed through parabolic-cylinder functions. A highly accurate approximation is derived for rapid evaluation of this ratio. Independent high-precision calculations confirm the high accuracy of the resulting double-precision evaluation in the tested numerical case.
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BibTeX
@misc{sanchez2026chromatographic,
title = {{Chromatographic Peak Shape beyond the High-Péclet Approximation: Analytic Time-Domain Expression and Derivatives}},
author = {Hernán R. Sánchez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17297},
eprint = {2610.17297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17297},
}
RIS
TY - UNPB
TI - Chromatographic Peak Shape beyond the High-Péclet Approximation: Analytic Time-Domain Expression and Derivatives
AU - Hernán R. Sánchez
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17297
UR - https://arxiv.org/abs/2610.17297
AB - Analytic time-domain expressions for the elution-time density and its parameter derivatives were recently derived from a stochastic model of linear isocratic chromatography incorporating advective-diffusive transport, injection dispersion, and fast and slow retention mechanisms. A Gaussian approximation, justified asymptotically at high column-scale Péclet numbers, was adopted for the distribution of the combined mobile-phase and fast-retention time, while slow retention was incorporated through a series of Gaussian-Gamma convolutions. Here, the underlying normal-inverse-Gaussian density is approximated by a finite Gaussian mixture using an existing quadrature rule. The elution-time PDF is a weighted sum of series of the previously derived Gaussian-Gamma form. Its parameter derivatives are also obtained. Newly derived recurrences reduce the cost of evaluating the coefficients of that series and their derivatives from quadratic to linear in truncation order for a fixed number of mechanisms. Through a probabilistic analysis of the series coefficients, the required number of terms is found to be small enough for rapid evaluation under usual chromatographic conditions. A relative roundoff-error estimator is derived to quantify finite-precision loss in the inherited forward recurrence for Gaussian-Gamma convolutions. For the cancellation-prone regime, a backward reconstruction is developed, initialized from an exact ratio of consecutive Gaussian-Gamma terms expressed through parabolic-cylinder functions. A highly accurate approximation is derived for rapid evaluation of this ratio. Independent high-precision calculations confirm the high accuracy of the resulting double-precision evaluation in the tested numerical case.
ER -