Distribution of Relaxation Times through the Lens of Distribution Theory

Abstract The distribution of relaxation times (DRT) model is widely used to deconvolve electrochemical impedance spectra into timescale distributions. However, standard DRT formulations assume integrable impedance. This condition is violated by electrochemical systems with blocking boundaries where diffusion or charge accumulation causes the low-frequency response to diverge. We formulate DRT inversion in the space of tempered distributions and derive a distributional Fuoss–Kirkwood boundary-value relation in S′(R), under explicit analyticity, growth, and boundary-value assumptions on the strip continuation of the impedance. In this setting, the boundary values of the impedance on the strip determine the DRT, with simple poles yielding Dirac distributions, higher-order poles giving their derivatives, and non-integrable low-frequency behavior yielding power-law tails. Each contribution is a tempered distribution, with the power-law tails defined as (Abel) limits of tempered functions. This formulation clarifies reported discrepancies for capacitive and diffusive systems by making the boundary-value convention explicit and separating distributional identities from pointwise evaluations. We derive closed-form distributional DRTs for generalized resistor-capacitor circuits, constant-phase elements, and finite-length diffusion transmission-line models in multiple geometries, and establish relationships between the DRT and the distributions of diffusion and capacitive times.