We propose and analyse a spatial discretization of the non-local Quantum Drift Diffusion (nlQDD) model by Degond, M\`{e}hats and Ringhofer in one space dimension. With our approach, that uses consistently matrices on ${\mathbb C}^N$ instead of operators on $L^2$, we circumvent a variety of analytical subtleties in the analysis of the original nlQDD equation, e.g. related to positivity of densities or to the quantum exponential function. Our starting point is spatially discretized quantum Boltzmann equation with a BGK-type collision kernel, from which we derive the discretized nlQDD model in the diffusive limit. Then we verify that solutions dissipate the von-Neumann entropy, which is a known key property of the original nlQDD, and prove global existence of positive solutions, which seems to be a particular feature of the discretization. Our main result concerns convergence of the scheme: discrete solutions converge -- locally uniformly with respect to space and time -- to classical solutions of the the original nlQDD model on any time interval $[0,T)$ on which the latter remain positive. In particular, this extends the existence theory for nlQDD, that has been established only for initial data close to equilibrium so far.

翻译：我们提出并分析了一维空间中Degond、Méhats和Ringhofer提出的非局域量子漂移扩散（nlQDD）模型的空间离散化方法。通过采用在${\mathbb C}^N$上使用矩阵而非$L^2$上算子的自洽方法，我们规避了原始nlQDD方程分析中的诸多解析难点，例如密度正定性或量子指数函数相关问题。我们的出发点是对具有BGK型碰撞核的空间离散化量子玻尔兹曼方程进行推导，在扩散极限下得到离散化的nlQDD模型。随后验证了该模型解具有冯·诺依曼熵耗散特性——这是原始nlQDD模型的已知关键性质，并证明了正解的整体存在性（这似乎是离散化特有的性质）。我们的主要结果关注格式的收敛性：在原始nlQDD模型的经典解保持正性的任意时间区间$[0,T)$上，离散解在空间和时间上局部一致地收敛于该经典解。特别地，这将nlQDD的存在性理论从原先仅适用于平衡态附近初始数据的情形进行了扩展。