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Late-time attractors in relativistic spin hydrodynamics in Gubser flow

Gen-Hui Li, Xiang Ren, Dong-Lin Wang, Shi Pu

Abstract

We investigate the late-time asymptotic solutions and attractor structure of the spin density in minimal causal spin hydrodynamics in Gubser flow. After deriving the differential equation governing the spin density, we obtain its late-time asymptotic solutions and identify both attractors and repellers in the corresponding numerical solutions. We then map these solutions back to flat Minkowski space and find parameter regions where the spin density exhibits a power-law decay. We further show that, when the characteristic length scale of the system is much larger than the proper time, several components of the spin density can decay as slowly as conventional thermodynamic variables in relativistic hydrodynamics. In this regime, the spin density behaves as a hydrodynamic mode governed by the late-time scaling laws of the flow.

Late-time attractors in relativistic spin hydrodynamics in Gubser flow

Abstract

We investigate the late-time asymptotic solutions and attractor structure of the spin density in minimal causal spin hydrodynamics in Gubser flow. After deriving the differential equation governing the spin density, we obtain its late-time asymptotic solutions and identify both attractors and repellers in the corresponding numerical solutions. We then map these solutions back to flat Minkowski space and find parameter regions where the spin density exhibits a power-law decay. We further show that, when the characteristic length scale of the system is much larger than the proper time, several components of the spin density can decay as slowly as conventional thermodynamic variables in relativistic hydrodynamics. In this regime, the spin density behaves as a hydrodynamic mode governed by the late-time scaling laws of the flow.

Paper Structure

This paper contains 9 sections, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Relativistic spin hydrodynamics with Gubser expansion
  3. Brief review on relativistic spin hydrodynamics in canonical form
  4. Coordinates for Gubser flow
  5. Spin hydrodynamics in $dS_3\times \mathbb{R}$ spacetime
  6. Asymptotic solutions at late time
  7. Late time attractor
  8. Late-time power-law decay of spin density
  9. Conclusion

Figures (2)

  • Figure 1: Asymptotic behavior of $f(w)$ at large $w$ for different parameter choices. The red solid and blue dashed curves denote the attractors and repellers summarized in Table \ref{['tab:Solutions_f']}, respectively. The gray solid curves show numerical solutions for different initial conditions.
  • Figure 2: Late-time asymptotic behavior of $\hat{S}$ for different choices of $\Delta_{1,2}$. Here, we take $\tau/\tau_\phi$ as an effective Knudsen number, denoted by $\mathrm{Kn}$.