[Paper Review] Relaxation-time approximation and relativistic viscous hydrodynamics from kinetic theory

Amaresh Jaiswal|arXiv (Cornell University)|Jul 3, 2014

High-Energy Particle Collisions Research1 references2 citations

TL;DR

This paper derives a third-order viscous hydrodynamics equation using an iterative relaxation-time approximation to the Boltzmann equation, enabling accurate phase-space distribution functions that preserve the experimentally observed 1/pT scaling of femtoscopic radii. The method yields results in excellent agreement with exact Boltzmann solutions and transport simulations in one-dimensional scaling flows.

ABSTRACT

Using the iterative solution of Boltzmann equation in the relaxation-time approximation, the derivation of a third-order evolution equation for shear stress tensor is presented. To this end we first derive the expression for viscous corrections to the phase-space distribution function, f (x; p), up to second-order in derivative expansion. The expression for f (x; p) obtained in this method does not lead to violation of the experimentally observed 1= p mT scaling of the femtoscopic radii, as opposed to the widely used Grad’s 14-moment approximation. Subsequently, we present the derivation of a third-order viscous evolution equation and demonstrate the significance of this derivation within one-dimensional scaling expansion. We show that results obtained using third-order evolution equations are in excellent accordance with the exact solution of Boltzmann equation as well as with transport results.

Motivation & Objective

• To develop a viscous hydrodynamics framework that avoids violating the 1/pT scaling of femtoscopic radii observed in heavy-ion collisions.
• To derive a third-order evolution equation for the shear stress tensor using an iterative solution of the Boltzmann equation in the relaxation-time approximation.
• To improve the accuracy of viscous corrections to the phase-space distribution function f(x; p) beyond the standard Grad’s 14-moment approximation.
• To validate the third-order viscous evolution equation against exact solutions of the Boltzmann equation and transport simulations in one-dimensional scaling expansion.

Proposed method

• Apply the relaxation-time approximation to the Boltzmann equation to iteratively solve for viscous corrections to the phase-space distribution function f(x; p) up to second order in derivative expansion.
• Derive the shear stress tensor evolution equation up to third order in the derivative expansion using the iterative solution method.
• Ensure the derived distribution function maintains the 1/pT scaling of femtoscopic radii, avoiding the violation seen in Grad’s 14-moment method.
• Perform numerical analysis in one-dimensional scaling expansion to compare the third-order viscous hydrodynamics with exact Boltzmann solutions and transport results.
• Use the iterative relaxation-time approach to systematically include higher-order viscous corrections without introducing unphysical behavior.
• Validate the method by demonstrating consistency with exact solutions and transport simulations under standard hydrodynamic conditions.

Experimental results

Research questions

• RQ1 Does the iterative relaxation-time approximation yield a viscous hydrodynamics framework that preserves the 1/pT scaling of femtoscopic radii, unlike the Grad’s 14-moment method?
• RQ2 Can a third-order evolution equation for the shear stress tensor be consistently derived from kinetic theory using the relaxation-time approximation?
• RQ3 How accurately does the third-order viscous hydrodynamics reproduce the exact solution of the Boltzmann equation in one-dimensional scaling flows?
• RQ4 What is the quantitative agreement between the derived viscous hydrodynamics and transport simulations in the same kinematic regime?

Key findings

• The iterative relaxation-time approximation produces viscous corrections to f(x; p) that do not violate the experimentally observed 1/pT scaling of femtoscopic radii.
• The derived third-order viscous evolution equation accurately reproduces the exact solution of the Boltzmann equation in one-dimensional scaling expansion.
• The results from the third-order viscous hydrodynamics show excellent agreement with transport simulations, validating its reliability.
• The method successfully avoids the unphysical behavior associated with the Grad’s 14-moment approximation in the context of femtoscopic radius scaling.
• The third-order formalism provides a consistent and accurate framework for relativistic viscous hydrodynamics derived directly from kinetic theory.
• The iterative solution method enables systematic inclusion of higher-order viscous corrections while maintaining physical consistency with experimental observables.