Relativistic Hydrodynamics 2026, 05: Equations of State

A fluid system can be fully defined by an equilibrium function, which can be phenomenological between the energy-momentum tensor and pressure and other hydrodynamic quantities, if the Boltzmann equation is too difficult to compute. These expressions are the equations of state. It can be as simple as p = 0 in cases of "dust". One can identify (non-)degenerate fluids by a fugacity much (smaller) greater than 1, and a (non-)relativistic one by a coldness much (larger) smaller than 1. A quantitative variable can be found in the specific heat, which can also be helpful at some constant volume or pressure, from which derives the adiabatic index

One can assume that for perfect fluids, the adiabatic index comes out to 5/3, for ultrarelativistic fluids of massless particles, to 4/3. The relativistic sound speed is given by

Basic thermodynamic quantities are the particle number n, the pressure p, the entropy e, and