Collision rates estimated from exact N-body simulations of a one-dimensional plasma

Abstract

In a plasma, the charged particles interact via long-range forces, and this interaction causes the plasma to exhibit collective effects. If the graininess or coupling parameter g goes to zero (ideal collisionless plasma), two-body collisions are negligible, while collective effects dominate the dynamics. In contrast, when [Formula: see text] collisions play a significant role. To study the transition between a collisionless and a collisional regime, a N-body code was developed and used in this work. The code solves exactly, in one spatial dimension, the dynamics of N infinite parallel plane sheets for both ion and electron populations. We illustrate the transition between individual and collective effects by studying two basic plasma phenomena, the two-stream instability and Langmuir waves, for different values of g. The numerical collision rates given by the N-body code increase linearly with g for both phenomena, although with proportionality factors that differ by roughly a factor of two, a discrepancy that may be accounted for by the different initial conditions. All in all, the usual collision rates published in the literature (Spitzer collisionality) appear to compare rather well with the rates observed in our simulations.

Learning the dynamics of a one-dimensional plasma model with graph neural networks

“…Not only is it a gridless code (thus making fewer approximations than PIC codes) but also provides an exact description of one-dimensional non-relativistic electrostatic plasmas with arbitrary degrees of collisionality. Furthermore, direct extensions and/or similar models exist, which allow one to study multi-species scenarios [38,42], electron-neutral collisions [57], electron-ion 3D scattering events [58], and radiative effects in the presence of an external magnetic field [59] . Therefore, the demonstration that the GNS can accurately provide a fast surrogate for the sheet model can open new directions for the exploration of the fundamentals of statistical mechanics of plasmas, and further extensions of the framework can be developed to integrate multi-species and stochastic processes.…”

“…We represent this table as a standard priority queue implemented using a heap (Dawson uses a slightly different mechanism [38] but with similar end results). This is also the approach followed by Gravier et al [42] for the multi-species scenario (refer to this work for illustrations on the heap structure). In our case, each entry of the priority queue corresponds to a list with [tc ij , c, tag ij ] where tc ij is the estimated crossing time between sheets i and j (by definition j = i + 1), c a unique sequence count (used to break ties for crossings occurring at the exact same time), and tag ij a tag containing the array indices of the sheets that crossed (not their labels).…”

“…In order to provide stronger evidence of the GNS generalization capabilities, we showcase a broad range of known kinetic plasma processes that the simulator is able to recover. These examples, present in both the original sheet model benchmarks [37][38][39][40][41] and other kinetic codes benchmarks [2,3,9,42,50], aim to demonstrate the capability of the GNS to simulate collective behavior in accordance with known kinetic theory. An important point to stress is that the surrogate simulator was not explicitly trained to reproduce these effects.…”

“…We also provide details regarding the modified version of the synchronous algorithm which restricts the number of neighbors checked for crossings. Information regarding the multi-species model, its description and a possible implementation can be found in recent work by Gravier et al [42].…”

“…This is an ideal initial testbench since it provides a simpler scenario, in terms of the problem structure and possible computational gains, while capturing a wide range of kinetic plasma physics phenomena that go beyond 'collisionless' physics, including Coulomb collisions, and collisional thermalization [37][38][39][40][41]. Moreover, recent studies in the fundamental statistical physics processes in plasmas have been using the sheet model and/or direct extensions [42]. We show how to leverage previous work on GNN-based simulators by Sanchez-Gonzalez et al [22] for kinetic plasma simulations and for the one-dimensional sheet model by introducing domain knowledge into the graph construction and simulator update mechanisms which enforce the desired symmetries and invariances.…”