Bose-Einstein Condensation in the Framework of $\kappa$-Statistics

In the present work we study the main physical properties of a gas of $\kappa$-deformed bosons described through the statistical distribution function $f_\kappa=Z^{-1}[\exp_\kappa (\beta({1/2}m v^2-\mu))-1]^{-1}$. The deformed $\kappa$-exponential $\exp_\kappa(x)$, recently proposed in Ref. [G.Kaniadakis, Physica A {\bf 296}, 405, (2001)], reduces to the standard exponential as the deformation parameter $\kappa \to 0$, so that $f_0$ reproduces the Bose-Einstein distribution. The condensation temperature $T_c^\kappa$ of this gas decreases with increasing $\kappa$ value, and approaches the $^{4}He(I)-^{4}He(II)$ transition temperature $T_{\lambda}=2.17K$, improving the result obtained in the standard case ($\kappa=0$). The heat capacity $C_V^\kappa(T)$ is a continuous function and behaves as $B_\kappa T^{3/2}$ for $T<T_c^\kappa$, while for $T>T_c^\kappa$, in contrast with the standard case $\kappa=0$, it is always increasing. Pacs: 05.30.Jp, 05.70.-a Keywords: Generalized entropy; Boson gas; Phase transition.