The geometric theory of charge conservation in particle-in-cell simulations

Abstract

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

Machine learning and serving of discrete field theories

“…Compared with continuous field theories, discrete field theories can be served more easily and with improved long-term accuracy and fidelity. The serving algorithm of discrete field theories belongs to the family of structure-preserving geometric algorithms 46 – 75 , which have been proven to be superior to the conventional algorithms based on discretization of differential equations. The demonstrated advantages of discrete field theories relative to continuous theories in terms of machine learning compatibility are consistent with Bostrom’s simulation hypothesis 78 .…”

“…The DEL equation is employed to solve for the discrete field ψ on the spacetime lattice when a discrete Lagrangian density L d is prescribed. This has been the only usage of the DEL equation in the literature that I am aware of so far 48,49,[51][52][53]55,57,58,61,64,68,[70][71][72][73][74][75] . I will come back to this shortly.…”

“…The principle of variational integrators is to discretize the action and Lagrangian density instead of the associated EL equations. Methods and techniques of variational integrators have been systematically developed in the past decade 46 – 75 . The advantages of variational integrators over standard integration schemes based on discretization of differential equations have been demonstrated.…”

“…For example, variational integrators in general are symplectic or multi-symplectic 48 , 49 , 51 – 53 , 55 , 57 , 58 , 61 , 64 , 68 , 70 – 75 , and as such are able to bound globally errors on energy and other invariants of the system for all simulation time-steps. More sophisticated discrete field theories have been designed to preserve other geometric structures of physical systems, such as the gauge symmetry 52 , 75 and Poincaré symmetry 72 , 80 , 81 , 83 . What proposed in this paper is to learn the discrete field theory directly from observational data and then serve the learned discrete field theory to predict new observations.…”

“…The training of is straightforward. Learning serves the purpose of serving, and the most effective way to serve a field theory with long term accuracy and fidelity is by offering the discrete version of the theory, as has been proven by the recent advances in structure-preserving geometric algorithms 46 – 75 . Therefore, learning a discrete field theory directly from the training data and then serving it constitute an attractive approach for discovering physical models by artificial intelligence.…”