Attractor and fixed points in Bjorken flows

Jean-Paul Blaizot and Li Yan
Phys. Rev. C 104, 055201 – Published 10 November 2021

Abstract

We consider a plasma of massless particles undergoing Bjorken expansion, mimicking the matter created in ultrarelativistic heavy ion collisions. We study the transition to hydrodynamics using kinetic theory in the relaxation time approximation. By allowing the relaxation time to depend on time, we can monitor the speed of the transition from the collisionless regime to hydrodynamics. By using a special set of moments of the momentum distribution, we reduce the kinetic equation to a coupled mode problem which encompasses all versions of second-order viscous hydrodynamics for Bjorken flows. This coupled mode problem is analyzed first using techniques of linear algebra. Then we transform this two-mode problem into a single nonlinear differential equation and proceed to a fixed point analysis. We identify an attractor solution as the particular solution of this nonlinear equation that joins two fixed points: one corresponding to the collisionless, early time regime, the other corresponding to late time hydrodynamics. We exploit the analytic solution of this equation in order to test several approximations and to identify generic features of the transition to hydrodynamics. We argue that extending the accuracy of hydrodynamics to early time, i.e., to the region of large gradients, amounts essentially to improve the accuracy of the location of the collisionless fixed point. This is demonstrated by showing that a simple renormalization of a second-order transport coefficient puts the free streaming fixed point at the right location, and allows us to reproduce accurately the full solution of the kinetic equation within second-order viscous hydrodynamics, even in regimes far from local equilibrium.

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  • Received 3 August 2021
  • Accepted 8 September 2021

DOI:https://doi.org/10.1103/PhysRevC.104.055201

©2021 American Physical Society

Physics Subject Headings (PhySH)

Fluid DynamicsNonlinear DynamicsInterdisciplinary Physics

Authors & Affiliations

Jean-Paul Blaizot

  • Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France

Li Yan

  • Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE) & Institute of Modern Physics Fudan University, 220 Handan Road, 200433 Yangpu District, Shanghai, China

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Issue

Vol. 104, Iss. 5 — November 2021

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