Route from discreteness to the continuum for the Tsallis q-entropy

The existence and exact form of the continuum expression of the discrete nonlogarithmic q-entropy is an important open problem in generalized thermostatistics, since its possible lack implies that nonlogarithmic q-entropy is irrelevant for the continuous classical systems. In this work, we show how the discrete nonlogarithmic q-entropy in fact converges in the continuous limit and the negative of the q-entropy with continuous variables is demonstrated to lead to the (Csiszár type) q-relative entropy just as the relation between the continuous Boltzmann-Gibbs expression and the Kullback-Leibler relative entropy. As a result, we conclude that there is no obstacle for the applicability of the q-entropy to the continuous classical physical systems.