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Relaxation time approximation revisited and non-analytical structure in retarded correlators

Jin Hu

TL;DR

This work interrogates the validity of the relaxation time approximation (RTA) in relativistic kinetic theory, showing that a energy-independent Anderson–Witting RTA is mathematically justifiable only when collisions are hard enough to yield a spectral gap near the origin of the linearized collision operator. It analyzes energy-dependent RTAs with $\tau_R=(\beta E_p)^\alpha t_R$, demonstrating that naive truncations fail to define a consistent RTA for both $\alpha>0$ and $\alpha<0$, and linking the viability of RTA to the spectrum structure and particle mass. To resolve collision invariance, the authors introduce a novel RTA (a mutilated operator) that preserves the five collision invariants by projecting out the zero modes, providing a controlled interpolation between full $\mathcal{L}_0$ and standard RTA with tunable accuracy. They further show that the non-analytic structure of retarded correlators reflects the interaction softness: hard interactions yield hydrodynamic poles separated by a gap from non-hydrodynamic modes, while soft interactions produce gapless branch cuts; for massive or inhomogeneous perturbations, the structures become more intricate. The paper also outlines practical tools, such as finite-element analyses, to map field theories to their eigenspectrum and discusses extensions to nonlinear kinetics and hydrodynamic frame choices.

Abstract

In this paper, we give a rigorous mathematical justification for the relaxation time approximation (RTA) model. We find that only the RTA with an energy-independent relaxation time can be justified in the case of hard interactions. Accordingly, we propose an alternative approach to restore the collision invariance lacking in traditional RTA. Besides, we provide a general statement on the non-analytical structures in the retarded correlators within the kinetic description. For hard interactions, hydrodynamic poles are the long-lived modes. Whereas for soft interactions, commonly encountered in relativistic kinetic theory, the gapless eigenvalue spectrum of linearized collision operator leads to gapless branch-cuts. We note that particle mass and inhomogeneous perturbations would complicate the above-mentioned non-analytical structures.

Relaxation time approximation revisited and non-analytical structure in retarded correlators

TL;DR

This work interrogates the validity of the relaxation time approximation (RTA) in relativistic kinetic theory, showing that a energy-independent Anderson–Witting RTA is mathematically justifiable only when collisions are hard enough to yield a spectral gap near the origin of the linearized collision operator. It analyzes energy-dependent RTAs with , demonstrating that naive truncations fail to define a consistent RTA for both and , and linking the viability of RTA to the spectrum structure and particle mass. To resolve collision invariance, the authors introduce a novel RTA (a mutilated operator) that preserves the five collision invariants by projecting out the zero modes, providing a controlled interpolation between full and standard RTA with tunable accuracy. They further show that the non-analytic structure of retarded correlators reflects the interaction softness: hard interactions yield hydrodynamic poles separated by a gap from non-hydrodynamic modes, while soft interactions produce gapless branch cuts; for massive or inhomogeneous perturbations, the structures become more intricate. The paper also outlines practical tools, such as finite-element analyses, to map field theories to their eigenspectrum and discusses extensions to nonlinear kinetics and hydrodynamic frame choices.

Abstract

In this paper, we give a rigorous mathematical justification for the relaxation time approximation (RTA) model. We find that only the RTA with an energy-independent relaxation time can be justified in the case of hard interactions. Accordingly, we propose an alternative approach to restore the collision invariance lacking in traditional RTA. Besides, we provide a general statement on the non-analytical structures in the retarded correlators within the kinetic description. For hard interactions, hydrodynamic poles are the long-lived modes. Whereas for soft interactions, commonly encountered in relativistic kinetic theory, the gapless eigenvalue spectrum of linearized collision operator leads to gapless branch-cuts. We note that particle mass and inhomogeneous perturbations would complicate the above-mentioned non-analytical structures.
Paper Structure (10 sections, 2 theorems, 34 equations)

This paper contains 10 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Linearized Boltzmann equation
  3. Revisiting the relaxation time approximation
  4. Anderson and Witting model
  5. Energy-dependent relaxation time approximation ($\alpha >0$)
  6. Energy-dependent relaxation time approximation ($\alpha < 0$)
  7. Hard interaction VS soft interaction
  8. Pole or cut --- the non-analytical structures in retarded correlators
  9. The novel relaxation time approximation
  10. Summary and outlook

Key Result

Theorem 1

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2