Time irreversibility, which characterizes nonequilibrium processes, can be measured based on the probabilistic differences between symmetric vectors. To simplify the quantification of time irreversibility, symmetric permutations instead of symmetric vectors have been employed in some studies. However, although effective in practical applications, this approach is conceptually incorrect. Time irreversibility should be measured based on the permutations of symmetric vectors rather than symmetric permutations, whereas symmetric permutations can instead be employed to determine the quantitative amplitude irreversibility – a novel parameter proposed in this paper for nonequilibrium calculated by means of the probabilistic difference in amplitude fluctuations. Through theoretical and experimental analyses, we highlight the strong similarities and close associations between the time irreversibility and amplitude irreversibility measures. Our paper clarifies the connections of and the differences between the two types of permutation-based parameters for quantitative nonequilibrium, and by doing so, we bridge the concepts of amplitude irreversibility and time irreversibility and broaden the selection of quantitative tools for studying nonequilibrium processes in complex systems.

可以基于对称向量之间的概率差异来度量时间不可逆性，该时间可逆性是非平衡过程的特征。为了简化时间不可逆性的量化，在某些研究中采用了对称置换而不是对称向量。但是，尽管在实际应用中有效，但是此方法在概念上是错误的。时间不可逆性应该基于对称矢量的排列而不是对称性排列来测量，而对称排列可以用来确定定量的幅度不可逆性–本文提出的一种新参数，用于通过振幅波动的概率差异来计算非平衡。通过理论和实验分析，我们强调了时间不可逆性和幅度不可逆性度量之间的强相似性和紧密联系。我们的论文阐明了定量非平衡的两种基于置换的参数之间的联系和差异，通过这样做，我们架起了振幅不可逆和时间不可逆的概念，并拓宽了研究复杂环境中非平衡过程的定量工具的选择系统。