Internal energy and hydrostatic pressure of a quantum relativistic ideal gas

In the classical case, the hydrostatic pressure of an ideal gas is defined as being two-thirds of the specific kinetic energy of the gas: p = 2/3 n < Uk >, where < Uk > is the average kinetic energy of the particles. However, this is no longer the case when relativistic cases are considered. Further concerns may be raised by quantum considerations. In the present paper, both pressure and kinetic energy are calculated taking the appropriate averages weighted with the equilibrium distribution function of the gas. For the purpose of this work the quantum relativistic Fermi–Dirac and Bose–Einstein distribution functions, will be considered for the cases of bosons and weakly degenerate fermions. Integration yields results in a closed form containing modified Bessel functions. A large-argument approximation is then taken, leading to an equation of state composed of the classic part plus correction terms.