  We develop a random batch Ewald (RBE) method for molecular dynamics
simulations of particle systems with long-range Coulomb interactions, which
achieves an $O(N)$ complexity in each step of simulating the $N$-body systems.
The RBE method is based on the Ewald splitting for the Coulomb kernel with a
random "mini-batch" type technique introduced to speed up the summation of the
Fourier series for the long-range part of the splitting. Importance sampling is
employed to reduce the induced force variance by taking advantage of the fast
decay property of the Fourier coefficients. The stochastic approximation is
unbiased with controlled variance. Analysis for bounded force fields gives some
theoretic support of the method. Simulations of two typical problems of charged
systems are presented to illustrate the accuracy and efficiency of the RBE
method in comparison to the results from the Debye-H\"uckel theory and the
classical Ewald summation, demonstrating that the proposed method has the
attractiveness of being easy to implement with the linear scaling and is
promising for many practical applications.
