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Validity of relativistic hydrodynamics beyond local equilibrium

Reghukrishnan Gangadharan

Abstract

We examine the applicability of relativistic hydrodynamics far from equilibrium by constructing formal solutions of the Boltzmann moment equations in the relaxation time approximation. These solutions naturally decompose into a divergent gradient series and exponentially decaying non-perturbative modes that encode initial conditions. The non-perturbative contributions are essential for understanding causality, the divergence of the gradient series, and the unexpected effectiveness of relativistic hydrodynamics far from equilibrium. In the 0+1D Bjorken scenario, we demonstrate that the exact evolution of non-equilibrium terms shares the same structural form as the gradient expansion, differing only through modified transport coefficients that reflect both initial data and free-streaming dynamics. Extending to 3+1D, we find that hydrodynamics remains effective not because the system is close to equilibrium, but because it interpolates smoothly between free streaming and collective behavior. This perspective offers a natural explanation for the remarkable success of hydrodynamics in modeling quark-gluon plasma evolution.

Validity of relativistic hydrodynamics beyond local equilibrium

Abstract

We examine the applicability of relativistic hydrodynamics far from equilibrium by constructing formal solutions of the Boltzmann moment equations in the relaxation time approximation. These solutions naturally decompose into a divergent gradient series and exponentially decaying non-perturbative modes that encode initial conditions. The non-perturbative contributions are essential for understanding causality, the divergence of the gradient series, and the unexpected effectiveness of relativistic hydrodynamics far from equilibrium. In the 0+1D Bjorken scenario, we demonstrate that the exact evolution of non-equilibrium terms shares the same structural form as the gradient expansion, differing only through modified transport coefficients that reflect both initial data and free-streaming dynamics. Extending to 3+1D, we find that hydrodynamics remains effective not because the system is close to equilibrium, but because it interpolates smoothly between free streaming and collective behavior. This perspective offers a natural explanation for the remarkable success of hydrodynamics in modeling quark-gluon plasma evolution.

Paper Structure

This paper contains 32 sections, 210 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relativistic Hydrodynamics
  3. Kinetic theory to hydrodynamics
  4. The Chapman-Enskog expansion
  5. Divergence of the Gradient Expansion
  6. The Boltzmann Kernel
  7. Validity of the Gradient Expansion
  8. Hydrodynamics in ${\mathbf0+1}$ D
  9. Moment Equations
  10. Solution of the moment equations
  11. Gradient expansion of the moments
  12. Dynamics of anisotropy
  13. Hydrodynamic Equations
  14. Generalisation to $3+1$D
  15. The linearised moment equations
  16. ...and 17 more sections

Figures (2)

  • Figure 1: The $\log-\log$ plot of the free streaming of the moment $\rho_{1,1}$ as a function of time for the analytical solution (dotted) obtained in \ref{['Eq:FSExSol']} vs the numerical solution (solid) for the moment equation.
  • Figure 2: Relativistic hydrodynamic evolution $\sim$ gradient expansion with modified transport coefficients