Hydrodynamic Attractors in Phase Space

Hydrodynamic attractors have recently gained prominence in the context of early stages of ultrarelativistic heavy-ion collisions at the RHIC and LHC. We critically examine the existing ideas on this subject from a phase space point of view. In this picture the hydrodynamic attractor can be seen as a special case of the more general phenomenon of dynamical dimensionality reduction of phase space regions. We quantify this using principal component analysis. Furthermore, we adapt the well known slow-roll approximation to this setting. These techniques generalize easily to higher dimensional phase spaces, which we illustrate by a preliminary analysis of a dataset describing the evolution of a five-dimensional manifold of initial conditions immersed in a 16-dimensional representation of the phase space of the Boltzmann kinetic equation in the relaxation time approximation.

Figure 1

Three snapshots of evolution in MIS-BRSSS phase space of a cloud of about 10 000 random states in the region determined by the ranges of the top plot. The red curve denotes the family of solutions corresponding to the attractor ${\mathcal{A}}_{\star }(\tau T)$. The background color represents the speed at which the points move in phase space, with magenta being faster than blue according to the velocity defined in the text. The dark blue denotes the slow region beyond the color coding scale. For the purposes of visualizing local dimensionality reduction, see also Fig. 2, we track three initially spherical regions. The plots were made for $C_{\eta }=0.75$ and $C_{{\tau }_{\pi }}=1$, see Eqs. (3) and (4), and ${\tau }_0$ denotes initialization time.

Figure 2

Top: evolution of explained variance ratio of each principal component in MIS-BRSSS models for circles (radius: ${10}^{-4}$) of initial conditions with centres lying in the middle of initial dots of corresponding color in Fig. 1. Bottom: for large enough values of $w$ there is an exponential decay with a decay rate consistent with twice the transient mode contribution to $\mathcal{A}(w)$. The initial dimensional reduction of the brown region we view as triggered by the expansion rather than by nonhydrodynamic mode decay [22].

Figure 3

In HJSW, the evolution of a cloud in phase space can be split into three stages, corresponding to the dimensionality of the cloud. The reduction from three to two dimensions corresponds to a collapse onto the slow region (blue region in plots). The plots were made with $C_{\eta }=0.75$, $C_{{\tau }_{\pi }}=1.16$, and $C_{\omega }=9.8$.

Figure 4

The six largest principal components and their explained variance ratios as a function of $\tau$. Due to the bias in initial conditions, one component dominates already at the initial time. This direction is not dynamically preferred, as the other components initially grow in importance and only at $\tau \approx 15{\tau }_0$ we see their expected exponential decay. At $\tau \approx 22{\tau }_0$, the second principal component stabilizes. This feature may reflect the limitations of PCA to capture effects of curved regions in phase space.