Remarks on Statistical mechanics of a moving system

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.