Hardly any theoretically formulated realistic problem can be solved exactly. Therefore, as a standard, we resort to approximations. In this context, expansions play a major role. We are used to relying on lowest-order expansions and confining our point of view accordingly. However, one should always bear in mind that such considerations may fail at some point. Here, we address a very common example situation, namely, the motion of a Brownian particle. We know that the associated mean-squared displacement in the long term increases linearly in time. Yet, when we take the Fokker–Planck approach in combination with a low-order expansion, the direct route towards this result fails. That is, in the expansion the term linear in time vanishes. Instead, the treatment requires consideration of all higher-order contributions. Together, they restore the linear increase in time. In this way, we stress that care is always mandatory when resorting to low-order expansions, and we present in a traceable way a route to solving the considered problem.

几乎没有任何理论上表述的现实问题可以准确地解决。因此，作为标准，我们采用近似值。在这种情况下，扩张发挥了重要作用。我们习惯于依赖最低阶的扩展并相应地限制我们的观点。但是，应该始终牢记，这种考虑可能会在某些时候失败。在这里，我们解决一个非常常见的示例情况，即布朗粒子的运动。我们知道，长期相关的均方位移随时间线性增加。然而，当我们将 Fokker-Planck 方法与低阶展开相结合时，通往该结果的直接路径就失败了。也就是说，在扩展中，时间线性项消失了。相反，治疗需要考虑所有高阶贡献。它们一起恢复了时间的线性增长。通过这种方式，我们强调在诉诸低阶扩展时始终需要注意，并且我们以可追溯的方式提出了解决所考虑问题的途径。