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We studied the escort averages in microcanonical and canonical ensembles inthe Tsallis statistics of entropic parameter $q>1$. The quantity $(q-1)$ is themeasure of the deviation from the Boltzmann-Gibbs statistics. We derived therelation between the escort average in the microcanonical ensemble and theescort average in the canonical ensemble. Conditions arise by requiring thatthe integrals appeared in the canonical ensemble do not diverge. A condition isthe relation between the heat capacity $C_V^{\mathrm{CE}}$ at constant volumein 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 derivedrelation, we calculated the energy, the energy fluctuation, and the differencebetween the canonical ensemble and the microcanonical ensemble in theexpectation value of the square of Hamiltonian. The difference between themicrocanonical ensemble and the canonical ensemble in energy is small becauseof 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 shownthat 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 microcanonicalensemble and $S^{\mathrm{CE}}_{\mathrm{R}q}$ is the R\'enyi entropy in thecanonical ensemble. The similar result was also obtained for the Tsallisentropy.

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