Thermodynamic relations and fluctuations in the Tsallis statistics

The thermodynamic relations in the Tsallis statistics were studied with physical quantities. An additive entropic variable related to the Tsallis entropy was introduced by assuming the form of the first law of the thermodynamics. The fluctuations in the Tsallis statistics were derived with physical quantities with the help of the introduced entropic variable. It was shown that the mean squares of the fluctuations of the physical quantities in the Tsallis statistics are the same as those in the conventional statistics. The mean square of the fluctuation of the Tsallis entropy and the mean square of the fluctuation of the Tsallis temperature were also derived. The mean square of the relative fluctuation of the Tsallis entropy and the mean square of the relative fluctuation of the Tsallis temperature are represented with heat capacities. It was shown that these fluctuations of the Tsallis quantities have the $q$-dependent terms in the Tsallis statistics of the entropic parameter $q$.