# A hydrodynamic approach to the classical ideal gas

## TL;DR

This paper derives conditions under which a perfect fluid energy tensor models a classical ideal gas, linking the speed of sound to hydrodynamic quantities, and explores the inverse problem and physical constraints for such gases.

## Contribution

It provides a hydrodynamic framework for characterizing classical ideal gases and solves the inverse problem, identifying fluids with a specific Poisson law as the only solutions.

## Key findings

- Speed of sound squared has a specific form in terms of pressure and energy density.
- The inverse problem is solved, showing fluids with a Poisson law fulfill the conditions.
- Relativistic compressibility conditions are analyzed for different adiabatic indices.

## Abstract

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.