Phase transition for the system of small volume in the $\phi^4$ theory in the Tsallis nonextensive statistics

Abstract

We studied the effects of the nonextensivity on the phase transition for the system of small volume $V$ in the $\phi^4$ theory in the Tsallis nonextensive statistics of entropic parameter $q$ and temperature $T$, when the deviation from the Boltzmann-Gibbs statistics, $|q-1|$, is small. We calculated the condensate and the mass to the order $q-1$ with the normalized $q$-expectation value under the massless free particle approximation. The following facts were found. The condensate $\Phi$ divided by $v$, $\Phi/v$, at $q$ is smaller than that at $q'$ for $q>q'$ as a function of $T_{\mathrm{ph}}/v$ which is the physical temperature $T_{\mathrm{ph}}$ divided by $v$, where $T_{\mathrm{ph}}$ at $q=1$ coincides with $T$ and $v$ is the value of the condensate at $T=0$. The mass decreases, reaches minimum, and increases after that, as $T_{\mathrm{ph}}$ increases. The mass at $q>1$ is lighter than the mass at $q=1$ at low physical temperature and heavier than the mass at $q=1$ at high physical temperature. The effects of the nonentensivity on the physical quantity as a function of $T_{\mathrm{ph}}$ become strong as $|q-1|$ increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.