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The role of the equation of state (EOS) is reviewed as an analytical vehicle to express material pressure as a function of density and temperature (or internal energy in the case of adiabatic transition). Starting from the fundamental laws of thermodynamics, three criteria are developed to measure the stability of equations of state, and are applied specifically to the Mie–Grüneisen EOS, with Hugoniot reference, to investigate the stability characteristics thereof. This EOS is singled out for study because of its relative importance in the computational modeling of shock transition. Results indicate numerous possibilities of instability under various circumstances. Various remedies are proposed to correct the observed instabilities and compared with existing models.