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 analysed.

翻译：本文研究了石墨烯纳米带电学特性相关不确定性的量化问题。该方法适用于理解因材料参数（如带隙）和外加电场强度难以精确测定所导致信息缺失的影响。具体而言，我们重点扩展了动理学方程中的粒子Galerkin方法，以处理存在不确定性的石墨烯纳米带电荷传输半经典玻尔兹曼方程。为此，我们开发了一种高效的粒子格式，可实现计算并行化；随后通过将格式推广至随机输入情形，采用广义多项式混沌方法对粒子动力学进行Galerkin重构，从而恢复动理学分布函数。该粒子格式在电子输运方程包含复杂散射作用时，仍能保持分布函数的物理特性与正定性。最后，我们分析了带隙与外加电场不确定性对电流特性的影响。