Temporal Theil scaling in diffusive trajectory time series

Temporal fluctuation scaling (TFS) is a power-law relation between the variance (Ξ) and the mean (Υ) which is present in cumulative time series. Taking into account that Theil index (T) can be assumed as a measure of dispersion and considering diffusive trajectory time series, we find a power-law relation between T and Υ of the form T∼(1-cΥ)^{β}, which we call temporal Theil scaling (TTS). Specifically, by analyzing data of volatility and absolute log-return for 24 nonstationary time series of financial markets, meteorology, and COVID-19 spread, we find that TTS is present in diffusive trajectory time series, while TFS is not present. Furthermore, we show that the power-law relation of TTS has a form that is similar to the relation between order parameter and temperature, which is found in the Ginzburg-Landau theory when the nontrivial critical points of an energy functional F_{η,δ} containing arbitrary powers η and δ of the order parameter are calculated.