The study of uncertainty propagation is of fundamental importance in plasma physics simulations. To this end, in the present work we propose a novel stochastic Galerkin (sG) particle {method} for collisional kinetic models of plasmas under the effect of uncertainties. This class of methods is based on a generalized polynomial chaos (gPC) expansion of the particles' position and velocity. In details, we introduce a stochastic particle approximation for the Vlasov-Poisson system with a BGK term describing plasma collisions. A careful reformulation of such dynamics is needed to perform the sG projection and to obtain the corresponding system for the gPC coefficients. We show that the sG particle method preserves the main physical properties of the problem, such as conservations and positivity of the solution, while achieving spectral accuracy for smooth solutions in the random space. Furthermore, in the fluid limit the sG particle solver is designed to possess the asymptotic-preserving property necessary to obtain a sG particle scheme for the limiting Euler-Poisson system, thus avoiding the loss of hyperbolicity typical of conventional sG methods based on finite differences or finite volumes. We tested the schemes considering the classical Landau damping problem in the presence of both small and large initial uncertain perturbations, the two stream instability and the Sod shock tube problems under uncertainties. The results show that the proposed method is able to capture the correct behavior of the system in all test cases, even when the relaxation time scale is very small.

翻译：不确定性的传播研究在等离子体物理模拟中具有根本重要性。为此，本工作提出了一种新颖的随机伽辽金粒子方法，用于描述不确定性影响下的碰撞等离子体动力学模型。该方法基于粒子的位置和速度的广义多项式混沌展开。具体而言，我们针对包含描述等离子体碰撞的BGK项的Vlasov-Poisson系统，引入了随机粒子近似。为执行随机伽辽金投影并获得相应的广义多项式混沌系数方程组，需要对上述动力学过程进行精心重构。研究表明，该随机伽辽金粒子方法在随机空间中对于光滑解具有谱精度，同时能保持问题的主要物理特性（如守恒性和解的正性）。此外，在流体极限下，该方法被设计为具有渐近保持特性，可得到极限Euler-Poisson系统的随机伽辽金粒子格式，从而避免了基于有限差分或有限体积的传统随机伽辽金方法中常见的双曲性丧失问题。我们分别针对存在小初始不确定扰动和大初始不确定扰动的经典朗道阻尼问题、双流不稳定性以及Sod激波管问题进行了数值验证。结果表明，即使在弛豫时间尺度极小的情况下，该方法在所有测试案例中均能准确捕捉系统的正确行为。