Electrodynamic particle-in-cell algorithm with exact momentum conservation

From the Sun to the interior of the ITER tokamak, plasma is a fundamental component of systems of scientific and engineering interest. Particle-in-cell (PIC) is a class of computational methods where individual particles, whose positions and momenta occupy a continuous range, interact with fields defined on a mesh. PIC has been successfully applied to study laser-plasma interactions, magnetohydrodynamics, magnetic reconnection, and more. One of the main limiting factors in PIC applications is the rapid scaling of computational complexity with system size. It is essential to achieve the greatest possible accuracy given finite computational resources. The recent discovery of an energy-conserving electrodynamic PIC algorithm has enabled the simulation of relativistic plasmas without numerical heating. But while momentum conservation would eliminate unphysical self-forces and improve stability for long simulations, this conservation law is yet to be realized in electrodynamic PIC. In this work, by formulating Yee's discretization of Maxwell's equations in the least-action formalism, we derive an exactly conserved stress-energy tensor for the electromagnetic field. Using an iterative algorithm, we ensure that each particle's momentum change per time step exactly cancels the momentum change it induces in the electromagnetic field. This leads to exact conservation of the combined field and particles momentum. Our results contribute to the understanding of conservation laws in PIC simulations.