We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

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具有自旋效应的 Vlasov-Maxwell 方程的几何粒子单元方法

我们提出了一种数值方案来求解具有自旋的电子的半经典 Vlasov-Maxwell 方程。电子气由分布函数描述 $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ 在扩展的 9 维相空间中演化 $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ ， 在哪里 $\粗体符号 s$ 表示自旋向量。使用合适的近似和对称，扩展的相空间可以减少到五个维度： $(x,{{p_x}}, {\boldsymbol s})$ . 可以证明，自旋 Vlasov-Maxwell 方程具有哈密顿结构，这促使使用最近开发的几何粒子在细胞 (PIC) 方法。在这里，几何 PIC 方法被推广到具有自旋的电子的情况。总能量守恒非常满意，相对误差如下 $0.05\,\%$ . 作为一个相关的例子，我们研究了电磁波与低密度等离子体相互作用的受激拉曼散射，其中电子被部分或完全自旋极化。结果表明，拉曼不稳定性在破坏电子极化方面非常有效。