We have implemented the Tsallis statistics in a Blast-Wave model and applied it to mid-rapidity transverse-momentum spectra of identified particles measured at RHIC. This new Tsallis Blast-Wave function fits the RHIC data very well for $p_T<$3 GeV/$c$. We observed that the collective flow velocity starts from zero in p+p and peripheral Au+Au collisions growing to 0.470 $pm$ 0.009($c$) in central Au+Au collisions. The $(q-1)$ parameter, which characterizes the degree of non-equilibrium in a system, changes from $0.100pm0.003$ in p+p to $0.015pm0.005$ in central Au+Au collisions, indicating an evolution from a highly non-equilibrated system in p+p collisions toward an almost thermalized system in central Au+Au collisions. The temperature and collective velocity are well described by a quadratic dependence on $(q-1)$. Two sets of parameters in our Tsallis Blast-Wave model are required to describe the meson and baryon groups separately in p+p collisions while one set of parameters appears to fit all spectra in central Au+Au collisions.
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