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

TL;DR

This paper explains the instability of first-order relativistic dissipation theories by showing their entropy has no maximum, and discusses how stability can be achieved through conditions on the entropy's extremum.

Contribution

It reveals that the instability stems from the entropy not having an upper bound and connects stability conditions to the entropy's maximum, offering a new perspective on relativistic fluid theories.

Findings

First-order theories lack an entropy maximum, leading to instability.

Stability conditions correspond to the entropy having an absolute maximum.

Recent stable theories do not fully resolve instability but rely on small violations of the second law.

Abstract

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 behaviour 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.