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.

W. Yao, J. Wang, M. Perc, W. Yao, J. Dai, D. Guo, D. Yao, Time irreversibility and amplitude irreversibility measures for nonequilibrium processes, Communication in Nonlinear Science and Numerical Simulation 96 (2021) 105688.