When the entropy has no maximum: A new perspective on the instability of the first-order theories of dissipation

The first-order relativistic fluid theories of dissipation proposed by Eckart and Landau-Lifshitz have been proved to be unstable. They admit solutions which start in proximity of equilibrium and depart exponentially from it. We show that this behavior is due to the fact that the total entropy of these fluids, restricted to the dynamically accessible states, has no upper bound. As a result, these systems have the tendency to constantly change according to the second law of thermodynamics and the unstable modes represent the directions of growth of the entropy in state space. We, then, verify that the conditions of stability of Israel and Stewart's theory are exactly the requirements for the entropy to have an absolute maximum. Hence, we explain how the instability of the first-order theories is a direct consequence of the truncation of the entropy current at the first order, which turns the maximum into a saddle point of the total entropy. Finally, we show that recently proposed first-order stable theories, constructed using more general frames, do not solve the instability problem by providing a maximum for the entropy, but, rather, are made stable by allowing for small violations of the second law.