  In the realm of statistical mechanics, it has been established that there is
no distinction between the micro-canonical and canonical ensembles in the
thermodynamic limit. However, this paradigm may alter when addressing
statistical mechanics in the context of a moving sample with a velocity $ v $.
Our investigation reveals significant disparities between the two ensembles
when considering relativistic effects up to the order of $ (v/c)^2 $. While the
temperature remains the same in the former, it experiences an increase in the
latter. If the system undergoes a finite-temperature phase transition, the
critical temperature decreases in the co-moving frame of the latter ensemble.
The implications of these findings on the thermodynamic zeroth to the third
laws and the eigenstate thermalization hypothesis are analysed. The potential
for the experimental detection of these novel effects in condensed matter
systems are discussed.
