  Porous electrodes{made of hierarchically nanostructured materials{are
omnipresent in various electrochemical energy technologies from batteries and
supercapacitors to sensors and electrocatalysis. Modeling the system-level
macroscopic transport and relaxation in such electrodes given their complex
microscopic geometric structure is important to better understand the
performance of the devices in which they are used. The discharge response of
capacitive porous electrodes in particular do not necessarily follow the
traditional exponential decay observed with at electrodes, which is good enough
for describing the general dynamics of processes in which the rate of a dynamic
quantity (such as charge) is proportional to the quantity itself. Electric
double-layer capacitors (EDLCs) and other similar systems exhibit instead power
law-like discharge profiles that are best described with differential equations
involving non-integer derivatives. Using the fractional-order integral in the
Riemann-Liouville sense and superstatistics we present a treatment of the
macroscopic response of such type of electrode systems starting from the
mesoscopic behavior of sub-parts of it. The solutions can be in terms of the
Mittag-Leffer (ML) function or a power law-like function depending on the
underlying assumptions made on the physical parameters of initial charge and
characteristic time response. The generalized three-parameter ML function is
found to be the best suited to describe experimental results of a commercial
EDLC at different time scales of discharge.
