Passive two-plateau relaxation from Tricomi confluent hypergeometric kernels

TL;DR

This paper introduces a novel non-fractional passive modeling framework using Tricomi hypergeometric functions to accurately represent broad-memory relaxation phenomena in various materials.

Contribution

It develops a new two-plateau dispersive law that extends classical models, with proven properties like complete monotonicity and passivity, and provides practical rational approximations.

Findings

The model captures asymmetric two-plateau responses with tunable exponents.

Finite-dimensional approximations converge well across different regimes.

Application to experimental data shows improved fitting and interpretability.

Abstract

Anomalous relaxation with memory spectra arises in disordered solids, soft matter, biological tissues and electrochemical interfaces. Fractional-order models capture broad power-law behaviour efficiently, but they can obscure spectral structure and are not always convenient for passive realisation or finite-dimensional simulation. We introduce a non-fractional passive framework based on the Tricomi confluent hypergeometric function, combined with a bounded Moebius normalisation that enforces prescribed low-frequency and high-frequency plateaux while preserving a broad dispersive transition. The resulting family contains the Debye and Cole-Cole responses as exact subcases, while extending them to asymmetric two-plateau dispersive laws with independently tunable low- and high-frequency exponents. For an admissible parameter range, we prove that the bounded block admits a Stieltjes representation with nonnegative spectral density, implying complete monotonicity, passivity, causality and compatibility with standard circuit and state-space descriptions. Building on this structure, we derive a passive Gauss-Stieltjes discretisation leading to Foster-type rational approximations and first-order state-space realisations with positive poles and residues. Numerical experiments show convergence of these finite-dimensional approximations across moderate-memory and long-tail regimes, enabling passive reduced-order representations of broad-memory responses. The framework is then validated on broadband dielectric data and battery electrochemical impedance spectra. In tissues, multi-block Tricomi mixtures improve complex-domain fitting accuracy relative to classical Cole-Cole baselines while preserving interpretable modal structure.