Hasegawa Yoshihiko, Van Vu Tan
Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan.
Phys Rev E. 2019 Jun;99(6-1):062126. doi: 10.1103/PhysRevE.99.062126.
The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound. Inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities, we apply the latter to systems described by Langevin equations, and we derive the bound for the fluctuation of thermodynamic quantities. When applying the Cramér-Rao inequality, the obtained inequality reduces to the fluctuation-response inequality. We find that the thermodynamic uncertainty relation is a particular case of the Cramér-Rao inequality, in which the Fisher information is the total entropy production. Using the equality condition of the Cramér-Rao inequality, we find that the stochastic total entropy production is the only quantity that can attain equality in the thermodynamic uncertainty relation. Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations.
热力学不确定性关系是一个不等式，它表明不可能达到比由熵产生所定义的界限更高的精度。在统计推断理论中，信息不等式断言任何估计器都不可能实现比规定界限更小的误差。受热力学不确定性关系与信息不等式之间相似性的启发，我们将后者应用于由朗之万方程描述的系统，并推导了热力学量涨落的界限。当应用克拉美 - 罗不等式时，所得到的不等式简化为涨落 - 响应不等式。我们发现热力学不确定性关系是克拉美 - 罗不等式的一个特殊情况，其中费希尔信息是总熵产生。利用克拉美 - 罗不等式的等式条件，我们发现随机总熵产生是唯一能在热力学不确定性关系中达到等式的量。此外，我们应用查普曼 - 罗宾斯不等式，并得到了系统方差与对任意扰动响应灵敏度之比下限的一个关系。
