Holographic Equilibration of Nonrelativistic Plasmas

We study far-from-equilibrium physics of strongly interacting plasmas at criticality and zero charge density for a wide range of dynamical scaling exponents $z$ in $d$ dimensions using holographic methods. In particular, we consider homogeneous isotropization of asymptotically Lifshitz black branes with full backreaction. We find stable evolution and equilibration times that exhibit small dependence of $z$ and are of the order of the inverse temperature. Performing a quasinormal mode analysis, we find a corresponding narrow range of relaxation times, fully characterized by the fraction $z/(d-1)$. For $z\ge d-1$, equilibration is overdamped, whereas for $z<d-1$, we find oscillatory behavior. Finally, and most interestingly, we observe that also the nonlinear evolution, although differing significantly from a quasinormal mode fit, is to a high degree of precision characterized by the fraction $z/(d-1)$.

Figure 1

(a) Nonlinear evolution of $B(u,t)$ for $d=4$, $z=2$. The colors indicate equal height. (b) Evolution of the pressure difference for $d=4$ and $z$ from 1 (rightmost) to 4 (leftmost). The inset shows a small oscillation below 0 for $z=2$. Dashed lines are the evolution resulting from a fit of the first 10 QNMs to the initial profile of $B$. (c) Evolution of pressure difference, comparing cases with the same $\alpha \equiv z/(d-1)$ but different $d$, showing close agreement. The initial profile is a factor of 10 bigger than in (b). Dashed lines denote evolution using the first 10 QNMs.

Figure 2

Relaxation times from QNMs (blue points) as a function of $\alpha \equiv z/(d-1)$. At $\alpha =1$ the modes bifurcate, with the second lowest shown in red. Also shown are relaxation times $4\pi T_0\tilde{\tau }$ obtained from the nonlinear evolution for $d=3$ (black squares), $d=4$ (green circles), and $d=5$ (orange diamonds). The hollow markers denote larger initial profiles.

Figure 3

We display the lowest modes found from Eq. (7), using numerics. Following the flow of the arrows corresponds to increasing $\alpha$. For $\alpha >1$, one mode moves up and asymptotes towards ${\omega }_{\mathrm{Im}}/(4\pi T)=-0.5$ as $\alpha \to \infty$. The other mode moves down, initially, but at some point reverses and for $\alpha \to \infty$ asymptotes towards the $\alpha =1$ location. Dots denote points with $\alpha$ corresponding to an AdS space.