Strong Coupling Isotropization of Non-Abelian Plasmas Simplified

We study the isotropization of a homogeneous, strongly coupled, non-Abelian plasma by means of its gravity dual. We compare the time evolution of a large number of initially anisotropic states as determined, on the one hand, by the full nonlinear Einstein’s equations and, on the other, by the Einstein’s equations linearized around the final equilibrium state. The linear approximation works remarkably well even for states that exhibit large anisotropies. For example, it predicts with a 20% accuracy the isotropization time, which is of the order of $t_{\mathrm{iso}}\lesssim 1/T$, with $T$ the final equilibrium temperature. We comment on possible extensions to less symmetric situations.

Figure 1

(a) Setup of Ref. 3. (b) Our setup. The initial state is specified on a $t=\mathrm{const}$ surface spanned by the radial coordinate $r$—see Eq. (3).

Figure 2

(a) Solution $B(t,z)$ (with $z\equiv 1/r$) obtained from the full Einstein’s equations. The initial profile $B(t=0,z)=\frac{4}{5}(z/z_h{)}^4\sin (8z/z_h)$ is shown as a thick red curve. The thick blue curve shows $\Delta \mathcal{P}(t)/\mathcal{E}$ as obtained from the full Einstein’s equations. The thin magenta curve shows the value of $\Delta \mathcal{P}(t)/\mathcal{E}$ as obtained from the linear approximation. (b) Difference between the full solution and the linear approximation.

Figure 3

Time evolution of the areas of the event (top, blue curve) and apparent (bottom, red curve) horizons for the initial state of Fig. 2a. The red dot at the origin signifies that there is no apparent horizon for this state at the initial time. From that time until the start of the red curve, there is no apparent horizon within the range of the radial coordinate covered by our grid, but there could be one at a deeper position.

Figure 4

Results for the isotropization times obtained from the full evolution of 1000 initial states and for the differences between the full and the linearized evolution (normalized by the full isotropization time). The height of each bar indicates the number of states in each bin.