In this article, we investigate some issues related to the quantification of uncertainties associated with the electrical properties of graphene nanoribbons. The approach is suited to understand the effects of missing information linked to the difficulty of fixing some material parameters, such as the band gap, and the strength of the applied electric field. In particular, we focus on the extension of particle Galerkin methods for kinetic equations in the case of the semiclassical Boltzmann equation for charge transport in graphene nanoribbons with uncertainties. To this end, we develop an efficient particle scheme which allows us to parallelize the computation and then, after a suitable generalization of the scheme to the case of random inputs, we present a Galerkin reformulation of the particle dynamics, obtained by means of a generalized polynomial chaos approach, which allows the reconstruction of the kinetic distribution. As a consequence, the proposed particle-based scheme preserves the physical properties and the positivity of the distribution function also in the presence of a complex scattering in the transport equation of electrons. The impact of the uncertainty of the band gap and applied field on the electrical current is analyzed.

翻译：本文研究了与石墨烯纳米带电学特性相关的不确定性量化问题。该方法适用于理解因某些材料参数（如带隙）及外加电场强度难以确定所导致的信息缺失效应。我们着重将粒子伽辽金方法拓展至含不确定性的石墨烯纳米带电荷输运半经典玻尔兹曼方程情形。为此，我们开发了一种高效的粒子格式以实现计算并行化，经随机输入情形下的适当推广后，基于广义多项式混沌方法构建了粒子动力学的伽辽金重构，从而重建动力学分布函数。该粒子格式即使在电子输运方程存在复杂散射项时，仍能保持分布函数的物理特性与正性。最后分析了带隙与外加电场不确定性对电流的影响。