How does relativistic kinetic theory remember about initial conditions?

Understanding hydrodynamization in microscopic models of heavy-ion collisions has been an important topic in current research. Many lessons obtained within the strongly coupled (holographic) models originate from the properties of transient excitations of equilibrium encapsulated by short-lived quasinormal modes of black holes. This paper aims to develop similar intuition for expanding plasma systems described by a simple model from the weakly coupled domain: the Boltzmann equation in the relaxation time approximation. We show that in this kinetic theory setup there are infinitely many transient modes carrying information about the initial distribution function. They all have the same exponential damping set by the relaxation time but are distinguished by different power-law suppressions and different frequencies of oscillations, logarithmic in proper time. Finally, we also analyze the resurgent interplay between the hydrodynamics and transients in this setup.

Figure 1

Singularities of the Borel transform of the hydrodynamic gradient expansion of $\mathcal{A}$ for sample values of allowed $\mathrm{\Delta }$; see Eq. (29). As a method of analytic continuation we use Padé approximants. The cases of $\mathrm{\Delta }=0$ and $\mathrm{\Delta }=1$ were studied before in, respectively, Refs. [17, 18]. In the plots sequences of poles represent branch cuts, a known feature of the Padé approximation; see, e.g., Ref. [54]. The singularities on the real axis are physical and give rise to transients of the form dictated by Eq. (36). The arguments in Sec. 4 make it clear that this singularity is an infinite set of branch cuts with the same branch point, but of a different order. The singularities off the real axis are unphysical and follow from contour deformations in the integral in Eq. (21), as explained for $\mathrm{\Delta }=1$ in Ref. [18]. Surprisingly, the unphysical singularities, whose location is at $1+(-1{)}^{\pm \mathrm{\Delta }/3}$, start controlling the radius of convergence of the hydrodynamic series for $\mathrm{\Delta }>2$.

Figure 2

Points in the figure shows roots of the function $M(\beta )$. For $\mathrm{\Delta }=0$, each root gives rise to a transient mode of the form $e^{-\tau }{\tau }^{\beta }$. For other $\mathrm{\Delta }$, the modes behave as in Eq. (41). The roots with the largest real part will be the dominant ones. The first three are ${\beta }_1\approx -3.4313$, ${\beta }_{\pm 2}\approx -5.4584\pm 0.5614i$, ${\beta }_{\pm 3}\approx -7.4746\pm 0.6648i$.

Figure 3

Shown here is ${\mathcal{A}}_1(w)$ defined in Eq. (47). The plots provide overwhelming evidence that Eq. (50) accurately describes the first transient mode. Note that $\mathrm{\Delta }=0$. (Top) All curves approach the exponential decay rate of the transient modes $-1$. (Bottom) All curves approach the power-law decay rate of the first transient mode ${\beta }_1$.

Figure 4

This figure compares the numerically evaluated ${\mathcal{A}}_2$, i.e., the second transient, with theoretical predictions for $\mathrm{\Delta }=0$. Dashed red lines are of the form of Eq. (40), where $\theta$ is fitted using data in the continuous red colored region at late times. (Top) Equation (40) describes the curves very well. Fitting also the value of ${\beta }_2$, it differs from the analytical value by less than 1%. The vertical segments represent a singularity of the tangent function appearing in Eq. (51). (Bottom) Likely due to interference from subleading transients with large amplitudes, the fit does not work well.