A Detailed Study on the Range Fluctuations of 10¹¹eV to 10¹⁸eV Muons in Water and the Fluctuations of the Cherenkov Lights due to the Accompanied Cascade Showers initiated by Muons

The validity of our Monte Carlo simulation procedure ({\it the integral method}) had been verified by the corresponding analytical procedure of which is quite independent of our method methodologically. Also, the results obtained by our procedure are compared with those obtained by the different Monte Carlo simulation procedure ({\it the differential method}) which have been exclusively utilized by the different authors and the agreement between them are found to be well. By utilizing our Monte Carlo procedures, the validity of which is guaranteed in two different procedures, we investigate not only the fluctuation of high energy muons themselves but also fluctuation of the various quantities related to the energy losses by the muons, which are difficult to obtain by {\it the differential method}. Namely, we obtain fluctuation on energy losses of the muons, fluctuation on Cherenkov lights due to the accompanied cascade showers initiated by the muon and the correlations between them . Finally, we obtain the transition curves for Cherenkov lights in KM3 detector, taking into account of all possible fluctuations in the stochastic processes and point out the difficulty of the reliable estimation of the energy of the muons which are resultants of muon neutrino events in the KM3 detectors.