We propose a fractional diffusion equation for anomalous transport at short spatial scales, where the anomalous diffusion can be observed due to non-Markovian process of scattering. This equation describes the anomalous diffusion under conditions of non-zero total flux of particles. The fractional operator (the Marchaud fractional derivative) of variable order provides transition to a normal diffusion for a bulk media.

The Nernst–Einstein correlation for anomalous diffusion at short scales was obtained. It is evident from this equation that kinetic coefficients depend on the scale. Possibly, this correlation explains the observed dependence of conductivity of thin films on their thickness.

我们提出了一个分数扩散方程，用于在短空间尺度上的异常传输，其中由于非马尔可夫散射过程可以观察到异常扩散。该方程描述了粒子总通量非零的情况下的异常扩散。可变阶数的分数算子（马尔科夫分数导数）为大容量介质提供了向正态扩散的过渡。

获得了小尺度异常扩散的能斯特-爱因斯坦相关性。从该方程式可以明显看出，动力学系数取决于比例。可能的是，这种相关性解释了所观察到的薄膜电导率与其厚度的相关性。