  Understanding the properties of charge dynamics is crucial to many practical
applications, such as electrochemical energy devices and transmembrane ion
channels. This work proposes a Maxwell-Amp\`{e}re Nernst-Planck (MANP)
framework for the description of charge dynamics. The MANP model with a
curl-free condition on the electric displacement is shown to be energy
dissipative with respect to a convex free-energy functional, and demonstrated
to be equivalent to the Poisson-Nernst-Planck model. By the energy dissipation
law, the steady state of the MANP model reproduces the charge conserving
Poisson--Boltzmann (PB) theory, providing an alternative energy stable approach
to study the PB theory. In order to achieve the curl-free condition, a
companion local curl-free relaxation algorithm, which is shown to naturally
preserve the discrete Gauss's law and converge robustly with linear
computational complexity, is developed for the MANP model. One of the main
advantages of our development is that it can efficiently deal with
space-dependent permittivity instead of solving the variable-coefficient
Poisson's equation. Many-body effects such as ionic steric effects and Coulomb
correlations can be incorporated within the MANP framework to derive modified
MANP models for problems in which the mean-field approximation fails. Numerical
results on the charge dynamics with such beyond mean-field effects in
inhomogeneous dielectric environments are presented to demonstrate the
performance of the MANP models in the description of charge dynamics,
illustrating that the proposed MANP model provides a general framework for
modeling charge dynamics.
