Abstract
In a relativistic context, the main purpose of Extended Irreversible Thermodynamics (EIT) is to generalize the principles of non-equilibrium thermodynamics to the domain of fluid dynamics. In particular, the theory aims at modeling any diffusion-type process (like heat as diffusion of energy, viscosity as diffusion of momentum, charge-conductivity as diffusion of particles) directly from thermodynamic laws. Although in Newtonian physics this task can be achieved with a first-order approach to dissipation (i.e. Navier–Stokes–Fourier like equations), in a relativistic framework the relativity of simultaneity poses serious challenges to the first-order methodology, originating instabilities which are, instead, naturally eliminated within EIT. The first part of this work is dedicated to reviewing the most recent progress made in understanding the mathematical origin of this instability problem. In the second part, we present the formalism that arises by promoting non-equilibrium thermodynamics to a classical effective field theory. We call this approach Unified Extended Irreversible Thermodynamics (UEIT), because it contains, as particular cases, EIT itself, in particular the Israel-Stewart theory and the divergence-type theories, plus Carter’s approach and most branches of non-equilibrium thermodynamics, such as relativistic chemistry and radiation hydrodynamics. We use this formalism to explain why all these theories are stable by construction (provided that the microscopic input is correct), showing that their (Lyapunov) stability is a direct consequence of the second law of thermodynamics.
Highlights
In Newtonian physics, the Navier–Stokes equations emerge as an almost (Jain and Kovtun, 2020) universal behavior of dissipative simple fluids when the following double limit is taken (Huang, 1987): i- Small deviations from equilibrium. ii- Slow macroscopic evolution in an assigned reference frame A.Only when both these conditions are met, more fundamental descriptions can be consistently replaced by the Navier–Stokes equations at the macroscopic scale (Kadanoff and Martin, 1963)
To set terminology and notation, we provide a minimal introduction to relativistic hydrodynamics, interpreted as a classical effective field theory (Dubovsky et al, 2012; Kovtun, 2012)
It has been recently noted that a particular class of theories resembling Carter’s original formalism can be derived directly from an action principle (Andersson and Comer, 2015; Celora et al, 2021). These theories are constructed within a Lagrangian specification of the flow field, see Section 2.2, so that are not presented in a natural Unified Extended Irreversible Thermodynamics (UEIT) form: the hydrodynamic equations for the fields Φi are of the second order and the constitutive relations involve first derivatives
Summary
In Newtonian physics, the Navier–Stokes equations emerge as an almost (Jain and Kovtun, 2020) universal behavior of dissipative simple fluids when the following double limit is taken (Huang, 1987): i- Small deviations from equilibrium. ii- Slow macroscopic evolution in an assigned reference frame A. The solution proposed by Israel and Stewart (1979) consists of dropping the slow limit assumption (ii) and working only under the assumption of small deviations from equilibrium Their theory includes a consistent description of the (possibly fast) relaxation processes of the fluid elements toward local thermodynamic equilibrium. Even in the simple case of pure bulk viscosity, there is an infinitely large class of systems (consistently modeled using non-equilibrium thermodynamics) that cannot be described within the Israel-Stewart theory (Gavassino et al, 2021a) The indices i, h are always abstract labels, counting the fields of a specific hydrodynamic theory
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