The large-time asymptotics of the density matrix solving a drift-diffusion-Poisson model for the spin-polarized electron transport in semiconductors is proved. The equations are analyzed in a bounded domain with initial and Dirichlet boundary conditions. If the relaxation time is sufficiently small and the boundary data is close to the equilibrium state, the density matrix converges exponentially fast to the spinless near-equilibrium steady state. The proof is based on a reformulation of the matrix-valued cross-diffusion equations using spin-up and spin-down densities as well as the perpendicular component of the spin-vector density, which removes the cross-diffusion terms. Key elements of the proof are time-uniform positive lower and upper bounds for the spin-up and spin-down densities, derived from the De Giorgi-Moser iteration method, and estimates of the relative free energy for the spin-up and spin-down densities.

中文翻译：

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矩阵自旋漂移扩散模型的大时间渐近

证明了求解半导体中自旋极化电子传输的漂移扩散泊松模型的密度矩阵的大时间渐近性。在具有初始和狄利克雷边界条件的有界域中分析方程。如果弛豫时间足够小并且边界数据接近平衡状态，则密度矩阵以指数方式快速收敛到无自旋近平衡稳态。证明基于使用自旋向上和自旋向下密度以及自旋矢量密度的垂直分量对矩阵值交叉扩散方程的重新表述，这去除了交叉扩散项。证明的关键要素是自旋向上和自旋向下密度的时间均匀正下限和上限，源自 De Giorgi-Moser 迭代方法，