Background So if I think of a gas of say having only $2$ point particles $A$ and $B$. Then, the Hamiltonian in the relativistic case is given by: $$ H_1 = \gamma_A m_Ac^2 + \gamma_B m_Bc^2 $$ where $\gamma= \frac{1}{\sqrt{1 - v^2/c^2}}$ and $m$ is mass. However, this doesn't really make sense...