Relation between the escort average in microcanonical ensemble and the escort average in canonical ensemble in the Tsallis statistics

Abstract: We studied the escort averages in microcanonical and canonical ensembles in the Tsallis statistics of entropic parameter $q>1$. The quantity $(q-1)$ is the measure of the deviation from the Boltzmann-Gibbs statistics. We derived the relation between the escort average in the microcanonical ensemble and the escort average in the canonical ensemble. Conditions arise by requiring that the integrals appeared in the canonical ensemble do not diverge. A condition is the relation between the heat capacity $C_V^{\mathrm{CE}}$ at constant volume in the canonical ensemble and the entropic parameter $q$: $0 < (q-1) C_V^{\mathrm{CE}} < 1$. This condition gives the known condition when $C_V^{\mathrm{CE}}$ equals the number of ingredients $N$. With the derived relation, we calculated the energy, the energy fluctuation, and the difference between the canonical ensemble and the microcanonical ensemble in the expectation value of the square of Hamiltonian. The difference between the microcanonical ensemble and the canonical ensemble in energy is small because of the condition. The heat capacity $C_V^{\mathrm{CE}}$ and the quantity $(q-1)$ are related to the energy fluctuation and the difference. It was shown that the magnitude of the relative difference $|(S^{\mathrm{CE}}{\mathrm{R}q}-S^{\mathrm{ME}}{\mathrm{R}q})/S^{\mathrm{ME}}_{\mathrm{R}q}|$ is small when the number of free particles is large, where $S^{\mathrm{ME}}_{\mathrm{R}q}$ is the R\'enyi entropy in the microcanonical ensemble and $S^{\mathrm{CE}}_{\mathrm{R}q}$ is the R\'enyi entropy in the canonical ensemble. The similar result was also obtained for the Tsallis entropy.