  The Butler-Volmer (BV) equation links the current flux crossing an
electrochemical interface to the electric potential drop across it with the
assumption of Arrhenius kinetics and the Boltzmann factor. Applying the
semilogarithmic Tafel analysis in which the logarithm of current is plotted vs.
the overpotential one expects straight lines from which the fundamental
reaction rate of the kinetic process can be computed. However, some Li-ion
battery data, which is the focus here, show nonlinear convex profiles that
cannot be adequately fitted with the standard BV model. We propose instead two
deformed BV models for the analysis of such types of behaviors constructed from
the superposition of cells exhibiting only local equilibrium and thus giving
rise to the power-law $q$-exponential and $\kappa$-exponential functions.
Non-Boltzmann distributions have been successfully employed for the modeling of
a wide spectrum of physical systems in nonequilibrium situations, but not yet
for batteries. We verify the validity of the deformed BV models on experimental
data obtained from \ce{LiFePO4} and \ce{Li}-\ce{O2} batteries.
