  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 $0.100\pm0.003$ in p+p to
$0.015\pm0.005$ 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.
