  The classical treatment of the electrical double-layer (EDL) structure at a
planar metal/electrolyte junction via the Gouy-Chapman-Stern (GCS) approach is
based on the Poisson equation relating the electrostatic potential to the net
mean charge density. The ions concentration in the diffuse layer are assumed to
follow the Boltzmann distribution law, i.e. $\propto \exp(-\tilde{\psi})$ where
$\tilde{\psi}$ is the dimensionless electrostatic potential. However, even in
stationary equilibrium in which variables are averaged over a large number of
elementary stochastic events, deviations from the mean-value are expected. In
this study we evaluate the behavior of the EDL by assuming some small
perturbations superposed on top of its Boltzmann distribution of ion
concentrations using the Tsallis nonextensive statistics framework. With this
we assume the ion concentrations to be proportional to $ [ 1- (1-q)
\tilde{\psi}]^{{1}/{(1-q)}} = \exp_q({- \tilde{\psi}})$ with $q$ being a real
parameter that characterizes the system's statistics. We derive analytical
expression and provide computational results for the overall differential
capacitance of the EDL structure, which, depending on the values of the
parameter $q$ can show both the traditional inverse bell-shaped curves for
aqueous solutions and bell curves observed with ionic liquids.
