  In this paper we review several approaches to mathematical modeling of simple
battery cells and develop these ideas further with emphasis on charge recovery
and the response behavior of batteries to given external load. We focus on
models which use few parameters and basic battery data, rather than detailed
reaction and material characteristics of a specific battery cell chemistry,
starting with the coupled ODE linear dynamics of the kinetic battery model. We
show that a related system of PDE with Robin type boundary conditions arises in
the limiting regime of a spatial kinetic battery model, and provide a new
probabilistic representation of the solution in terms of Brownian motion with
drift reflected at the boundaries on both sides of a finite interval. To
compare linear and nonlinear dynamics in kinetic and stochastic battery models
we study Markov chains with states representing available and remaining
capacities of the battery. A natural scaling limit leads to a class of
nonlinear ODE, which can be solved explicitly and compared with the capacities
obtained for the linear models. To indicate the potential use of the modeling
we discuss briefly comparison of discharge profiles and effects on battery
performance.
