Natural time analysis together with non-extensive statistical mechanics shorten the time window of the impending 2011 Tohoku M9 earthquake in Japan

Applying natural time analysis (NTA) to the Japanese seismic data, we have found that the system enters the critical stage upon the occurrence of M=4.2-5.0 earthquakes from 08:36 to 13:14 local time (LT) on 10 March 2011, i.e., almost one day before the M9 Tohoku earthquake on 11 March 2011. In addition, here, we find that just after this period the entropy change ΔS under time reversal of NTA along with the Tsallis entropy of non-extensive statistical mechanics (NESM) show distinct simultaneous changes. This simultaneous appearance enables the shortening of the time window of the impending mainshock to several hours. Furthermore, upon the occurrence of the M7.3 foreshock at 11:45 LT on 9 March 2011, the following fact emerged: The Tsallis entropy of NESM exhibited a scaling behavior with a characteristic exponent 1/3 that conforms to Lifshitz–Slyozov–Wagner theory for phase transitions as had been also observed in NTA for the fluctuations of the entropy change ΔS under time reversal upon a M7.8 earthquake occurrence in Japan on 22 December 2010. The latter obeyed the form A(t−t0)c where c is approximately equal to 1/3 and the pre-factors A are proportional to the scale i (number of events) while t0 is almost 0.2 days after this M7.8 earthquake.