Effects of the Tsallis distribution in the linear sigma model
The effects of the Tsallis distribution which has two parameters, $q$ and $T$,on physical quantities are studied using the linear sigma model in chiral phase transitions. The Tsallis distribution approaches the Boltzmann-Gibbs distribution as $q$ approaches $1$. The parameter $T$ dependences of the condensate and mass for various $q$ are shown, where $T$ is called temperature. The critical temperature and energy density are described with digamma function, and the $q$ dependences of these quantities and the extension of Stefan-Boltzmann limit of the energy density are shown. The following facts are clarified. The chiral symmetry restoration at $q>1$ occurs at low temperature, compared with the restoration at $q=1$.The sigma mass and pion mass reflect the restoration. The critical temperature decreases monotonically as $q$ increases. The small deviation from the Boltzmann-Gibbs distribution results in the large deviations of physical quantities, especially the energy density. It is displayed from the energetic point of view that the small deviation from the Boltzmann-Gibbs distribution is realized for $q>1$. The physical quantities are affected by the Tsallis distribution even when $|q-1|$ is small.