Applying a Weyl–Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.

中文翻译：

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线性电磁场中的维格纳输运

将 Weyl-Stratonovich 变换应用于电磁场中维格纳函数的演化方程，会产生一个多维规范不变方程，该方程在数值上求解起来非常困难。在这项工作中，我们对线性电磁场和平面中电子的演化（二维输运）应用了简化假设，这降低了复杂性，并能够获得规范不变维格纳方程的初步经验。我们提出了方程分析，并表明求解高阶导数的有限差分方法允许重新表述为 Fredholm 积分方程。后者的求解展开包含连续积分，这有利于蒙特卡洛求解方法。为此，我们提出了两种随机（蒙特卡罗）算法，用于评估通用物理量的平均值或直接评估维格纳函数。这些算法产生了量子粒子模型，该模型以启发式术语解释量子传输。