  This paper discusses a novel fully implicit formulation for a 1D
electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier
implicit electrostatic PIC approaches (which are based on a linearized
Vlasov-Poisson formulation), ours is based on a nonlinearly converged
Vlasov-Amp\`ere (VA) model. By iterating particles and fields to a tight
nonlinear convergence tolerance, the approach features superior stability and
accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit
PIC implementations. In particular, the formulation is stable against temporal
(CFL) and spatial (aliasing) instabilities. It is charge- and energy-conserving
to numerical roundoff for arbitrary implicit time steps. While momentum is not
exactly conserved, errors are kept small by an adaptive particle sub-stepping
orbit integrator, which is instrumental to prevent particle tunneling. The VA
model is orbit-averaged along particle orbits to enforce an energy conservation
theorem with particle sub-stepping. As a result, very large time steps,
constrained only by the dynamical time scale of interest, are possible without
accuracy loss. Algorithmically, the approach features a Jacobian-free
Newton-Krylov solver. A main development in this study is the nonlinear
elimination of the new-time particle variables (positions and velocities). Such
nonlinear elimination, which we term particle enslavement, results in a
nonlinear formulation with memory requirements comparable to those of a fluid
computation, and affords us substantial freedom in regards to the particle
orbit integrator. Numerical examples are presented that demonstrate the
advertised properties of the scheme. In particular, long-time ion acoustic wave
simulations show that numerical accuracy does not degrade even with very large
implicit time steps, and that significant CPU gains are possible.
