  The necessary and sufficient condition for a conservative perfect fluid
energy tensor to be the energetic evolution of a classical ideal gas is
obtained. This condition forces the square of the speed of sound to have the
form $c_s^2 = \frac{\gamma p}{\rho+p}$ in terms of the hydrodynamic quantities,
energy density $\rho$ and pressure $p$, $\gamma$ being the (constant) adiabatic
index. The {\em inverse problem} for this case is also solved, that is, the
determination of all the fluids whose evolutions are represented by a
conservative energy tensor endowed with the above expression of $c^2_s$, and it
shows that these fluids are, and only are, those fulfilling a Poisson law. The
relativistic compressibility conditions for the classical ideal gases and the
Poisson gases are analyzed in depth and the values for the adiabatic index
$\gamma$ for which the compressibility conditions hold in physically relevant
ranges of the hydrodynamic quantities $\rho, p$ are obtained. Some scenarios
that model isothermal or isentropic evolutions of a classical ideal gas are
revisited, and preliminary results are presented in applying our hydrodynamic
approach to looking for perfect fluid solutions that model the evolution of a
classical ideal gas or of a Poisson gas.
