[Paper Review] Universality Classes of Relativistic Fluid Dynamics: Foundations

Lorenzo Gavassino, Marcelo M. Disconzi+1 more|arXiv (Cornell University)|Feb 4, 2023

Advanced Thermodynamics and Statistical Mechanics89 references4 citations

TL;DR

This paper introduces a universal organizing principle for relativistic fluid dynamics near equilibrium, showing that causal and thermodynamically stable systems fall into universality classes defined by their degrees of freedom, conservation laws, and information current. It proves that all such theories can be recast to resemble Israel-Stewart theory near equilibrium, revealing deep equivalences across seemingly distinct hydrodynamic models, including fluids, solids, and superfluids.

ABSTRACT

A general organizing principle is proposed that can be used to derive the equations of motion describing the near-equilibrium dynamics of causal and thermodynamically stable relativistic systems. The latter are found to display some new type of universal behavior near equilibrium that allows them to be grouped into universality classes defined by their degrees of freedom, information content, and conservation laws. The universality classes expose a number of surprising equivalences between different theories, shedding new light on the near-equilibrium behavior of relativistic systems.

Motivation & Objective

• To identify a general organizing principle for deriving equations of motion in near-equilibrium relativistic fluid dynamics.
• To resolve the proliferation of ad hoc relativistic hydrodynamic theories by revealing underlying universal structures.
• To demonstrate that diverse theories become equivalent near equilibrium through a common mathematical form.
• To classify relativistic systems into universality classes based on degrees of freedom, conservation laws, and information current.
• To show that even solids and superfluids can be understood as special cases of fluid-like hydrodynamic theories with additional conserved charges.

Proposed method

• The authors introduce the information current $ E^{ u} $, a quadratic functional of perturbations, which acts as a Lyapunov functional and ensures thermodynamic stability.
• They prove that the linearized equations of motion for any causal and thermodynamically stable system can be derived from a 4-vector field $ E^{ u} $, which encodes the system's information content.
• The method systematically constructs $ E^{ u} $ from perturbations grouped by their $ SO(3) $ transformation properties, enabling classification by symmetry and conservation laws.
• The approach reveals that all such systems can be recast into a form resembling Israel-Stewart theory near equilibrium, regardless of their original formulation.
• The universality classes are defined by the structure of $ E^{ u} $, the number of independent perturbation fields, and their transformation behavior under rotations.
• The framework uses a derivative expansion and thermodynamic constraints to ensure hyperbolicity and positivity of entropy production.

Experimental results

Research questions

• RQ1 Can a unified framework be developed to describe the near-equilibrium dynamics of all causal and thermodynamically stable relativistic systems?
• RQ2 Why do different relativistic hydrodynamic theories (e.g., Israel-Stewart, kinetic theory, effective field theory) become mathematically equivalent near equilibrium?
• RQ3 What underlying structure explains the observed equivalences between theories with different degrees of freedom and dynamical equations?
• RQ4 How can the information current $ E^{ u} $ be systematically constructed for arbitrary hydrodynamic theories based on symmetry and conservation laws?
• RQ5 What determines the universality class of a given relativistic fluid or solid, and how does this classify their near-equilibrium behavior?

Key findings

• All causal and thermodynamically stable relativistic fluid dynamics theories can be recast into a form resembling Israel-Stewart theory near equilibrium, revealing a universal structure.
• The information current $ E^{ u} $, a quadratic functional of perturbations, universally governs the linearized dynamics and ensures entropy production via $ abla_{ u}E^{ u} \leq 0 $.
• Universality classes are defined by the number of independent perturbation fields, their $ SO(3) $ transformation properties, and conservation laws, leading to equivalence between seemingly distinct theories.
• An isotropic solid is shown to be equivalent to a fluid with an additional conserved symmetric $(0,2)$-tensor charge in the linear regime.
• The framework classifies multiple physical systems—such as Maxwell, Burgers, elastic, and supersolid materials—by their $ (n_{\text{fields}}, n_{\text{conserved}}, n_{\text{symmetry}}) $ structure, with explicit expressions for $ TE^0 $, $ TE^j $, and $ T\sigma $ provided.
• For example, in the elastic heat-conducting material, $ T\sigma = \delta q^k \delta q_k / (\kappa T) $, and $ TE^0 $ includes terms for temperature, velocity, heat flux, and stress perturbations, confirming its classification.