The putative generalization of the thermodynamic uncertainty relation (TUR) to underdamped dynamics is still an open problem. So far, bounds that have been derived for such a dynamics are not particularly transparent and they do not converge to the known TUR in the overdamped limit. Furthermore, it was found that there are restrictions for a TUR to hold such as the absence of a magnetic field. In this article we first analyze the properties of driven free diffusion in the underdamped regime and show that it inherently violates the overdamped TUR for finite times. Based on numerical evidence, we then conjecture a bound for one-dimensional driven diffusion in a potential which is based on the result for free diffusion. This bound converges to the known overdamped TUR in the corresponding limit. Moreover, the conjectured bound holds for observables that involve higher powers of the velocity as long as the observable is odd under time reversal. Finally, we address the applicability of this bound to underdamped dynamics in higher dimensions.

中文翻译：

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自由扩散限制了欠阻尼动力学中电流的精度。

将热力学不确定性关系（TUR）推定为阻尼不足的动力学的普遍性仍然是一个未解决的问题。到目前为止，为这种动力学得出的边界不是特别透明，并且在超限极限内它们还没有收敛到已知的TUR。此外，发现存在诸如不存在磁场等限制TUR的限制。在本文中，我们首先分析了欠阻尼状态下驱动自由扩散的特性，并表明它固有地在有限时间内违反了过阻尼TUR。基于数字证据，我们然后根据自由扩散的结果推测势中一维驱动扩散的界。该界限在相应的极限处收敛到已知的超阻尼TUR。此外，只要在时间倒转下可观察到的是奇数，猜想界就适用于涉及较高速度幂的可观察到的事物。最后，我们解决了此约束在更高维度上阻尼不足的动力学的适用性。