How much data does a memory kernel need? Measurement requirements for fractional models of battery and corrosion degradation

Abstract

How much data does a memory kernel need before a fit is meaningful? For a distributed-order model, the answer is about 3.1 decades of observation, more than standard battery-degradation tests provide. Battery capacity fade and corrosion-film growth depend on their history, so they are often modelled using general fractional calculus (GFC), where memory is described by an arbitrary Sonine kernel rather than a single power law. We tested this using two mechanistic models without fractional operators and de Levie's porous-electrode model, whose high-frequency order is exactly 0.5; our estimator recovered 0.500. A single aging curve can recover an effective order, alpha, to within 5% noise using a weak formulation, while separating the tempered parameters (alpha, lambda) requires several Arrhenius conditions, with errors improving from (0.25, 0.43) with one condition to (0.04, 0.07) with four. A distributed-order memory needs 3.1 decades of data to be resolved to 90%; below that, a two-order memory can collapse into one intermediate order that fits the data better than the true spectrum. For saturating passivation, however, a general kernel improves the fit substantially, raising R2 from 0.862 to 0.998. Across 20 cells from two public datasets, observation windows span only 1.23 to 2.24 decades, making the 3.1-decade result a practical design rule: it defines what must be measured before fitting a memory kernel, not only what cannot be fitted.