Far from equilibrium attractors in phase space

TL;DR

This paper explores the nature of hydrodynamic attractors in relativistic fluid dynamics, clarifying their phase space structure, independence from conformal symmetry, and implications for understanding early-time hydrodynamisation.

Contribution

It demonstrates how to identify attractor sections in phase space and shows that conformal symmetry is not essential for the emergence of early-time attractors.

Findings

Attractors can be characterized as submanifolds in phase space.

Conformal symmetry is not crucial for early-time attractor emergence.

Hydrodynamisation can be understood without conformal symmetry constraints.

Abstract

We clarify the connection between attractor solutions known from studies of Bjorken flow in conformal models of relativistic fluid dynamics and the more general description of attractors as submanifolds in phase space. We show how to determine sections of the attractor on slices of constant proper time, and point out that in conformal models one may choose a parametrisation of the extended phase space such that the intersection of the attractor with a slice of constant proper time is the same on all slices. In consequence, it is possibleto project these intersections and one finds that the result reproduces the attractor solution. This projection allows for an expressive picture of hydrodynamisation, but the attractor can be equally well described without taking advantage of this convenience. An immediate implication of these observations is that conformal symmetry does not play a significant role in the emergence of early-time attraction. We illustrate this by describing the hydrodynamic attractor in a simple model where conformal symmetry is broken by the equation of state. Our findings can be applied to studies of hydrodynamic attractors in more general situations, also beyond the context of quark-gluon plasma dynamics.

Equations

$$u\cdot \nabla \mathcal{E}$$

$$(\mathcal{E}+p)u\cdot \nabla u_{\mu }$$

$${\Delta }_{\mu \alpha }{\Delta }_{\nu \beta }u\cdot \nabla {\pi }^{\alpha \beta }$$

$$\mathcal{E}=3p=a{T}^4,\eta =\frac{4}{3}a{C}_{\eta }{T}^3,{\tau }_{\pi }=C_{\pi }/T,$$

$$\tau {\partial }_{\tau }\ln T$$

$$\tau {\partial }_{\tau }\mathcal{A}$$

$$\mathcal{A}\sim {\mathcal{A}}_{\pm },T\sim (\mu \tau {)}^{\beta -1},$$

$${\mathcal{A}}_{\pm }\equiv \pm 6\sqrt{\frac{C_{\eta }}{C_{\pi }}}$$

$$\beta =\frac{\sqrt{C_{\eta }}+2\sqrt{C_{\pi }}}{3\sqrt{C_{\pi }}}.$$

$$T\sim \Lambda (\Lambda \tau {)}^{-\frac{1}{3}},\mathcal{A}\sim 8C_{\eta }(\Lambda \tau {)}^{-\frac{2}{3}},$$

$$w(\Lambda )\sim (\Lambda \tau {)}^{\frac{2}{3}},\mathcal{A}(\Lambda )\sim 8C_{\eta }(\Lambda \tau {)}^{-\frac{2}{3}}.$$

$$w(\Lambda )$$

$$\mathcal{A}(\Lambda )$$

$$C_{\pi }w\left(1+\frac{\mathcal{A}}{12}\right){\partial }_w\mathcal{A}=12C_{\eta }-\frac{3}{2}w\mathcal{A}-\frac{1}{3}{C}_{\pi }{\mathcal{A}}^2,$$

$$\tau {\partial }_{\tau }w=\frac{2}{3}w\left(1+\frac{1}{12}\mathcal{A}\right).$$

$$\eta /s=C_{\eta }T/m,{\tau }_{\pi }=C_{\pi }/m.$$

$$\tau {\partial }_{\tau }\ln T$$

$$\tau {\partial }_{\tau }\mathcal{A}$$

$$p=\frac{a}{3}\frac{T^4}{1+\frac{m^2}{T^2}},$$

$$s$$

$$\mathcal{E}$$

$$\Delta (T)\equiv \frac{\mathcal{E}-3p}{T^4}=\frac{2a}{3}\frac{m^2{T}^2}{{\left(m^2+T^2\right)}^2}.$$

$$c_s^2(T)=\frac{1}{3}\frac{\left(1+\frac{m^2}{T^2}\right)\left(1+\frac{3m^2}{2T^2}\right)}{\left(1+\frac{17m^2}{6T^2}+\frac{5m^4}{2T^4}\right)}.$$

$$c_s^2(T)$$

$$\tau {\partial }_{\tau }\ln T+c_s(T)=0.$$

$$T\sim \frac{\Lambda }{(\Lambda \tau {)}^{1/5}}+\frac{4{\Lambda }^2}{15m^2}\frac{\Lambda }{(\Lambda \tau {)}^{3/5}}+\dots ,$$

$$\tau {\partial }_{\tau }\log T$$

$$\tau {\partial }_{\tau }\mathcal{A}$$

$$R_1$$

$$R_2$$

$$S_1$$

$$S_2$$

$$S_3$$

$$T$$

$$\mathcal{A}$$

$$T$$

Full text

I Introduction

At the macroscopic level, the evolution of complex systems toward equilibrium involves an effective loss of memory of the initial conditions. While the approach to equilibrium is captured by the hydrodynamic gradient expansion at sufficiently late times, it can happen that universal behaviour sets in significantly earlier, well before it can be attributed to hydrodynamics. Such a situation may be interpreted as a prehydrodynamic attractor. An important example appears in the context of quark-gluon plasma (QGP) physics, where Mueller-Israel-Stewart theory [1, 2, 3] (MIS) is a key element of the phenomenological description of heavy ion collisions. Simulations of QGP dynamics involve flows such that at the early stages the energy-momentum tensor is very anisotropic [4]. The success of these simulations in describing experimental data suggests a measure of universality emerging while the system is still very far from equilibrium. Indeed, an idealised description of QGP dynamics by Bjorken flow within MIS theory features an early-time, prehydrodynamic attractor [5]. A similar situation arises for Bjorken flow within kinetic theory, where the early-time regime is characterised by free-streaming while the emergence of hydrodynamic behaviour at late times signals the dominance of interactions [6, 7].

A key feature of these models is that the equations of motion are singular at early time. This singularity can be traced to the dominance of the longitudinal expansion in the initial stages of a heavy-ion collision. Its origins lie in the set of approximations underlying Bjorken flow: longitudinal boost invariance and homogeneity in the transverse plane. Of these, the crucial dynamical assumption is boost invariance, which is a characteristic property of particle production in QCD at high energies [8]. Imposing boost invariance on MIS theory introduces dynamical information that is essential for the emergence of early-time attraction — this may explain why MIS theory is able to approximate the significantly more complex dynamics of QCD. Boost invariance is also a key element of the emergence of hydrodynamic behaviour in other studies of early-time physics [9, 10, 11]. The importance of boost invariance in connection with early-time attraction was highlighted in the recent paper [12] (see also Ref. [13]), where it was found that breaking boost-invariance in the initial conditions tends to wash out the early-time attraction, in contrast to relaxing the condition of transverse homogeneity [14, 15, 16, 17, 18, 19].

The appearance of the attractor solution in conformal MIS theory relies on a partial decoupling of the equations of motion, which can be achieved be a suitable change of variables; specifically, by introducing a dimensionless “clock variable” $w\equiv\tau T$, where $\tau$ is the proper time and $T$ is the effective temperature. The attractor is then described by a specific solution of a single ordinary differential equation which determines the pressure anisotropy ${\cal A}$ as a function of $w$. However, in less constrained scenarios such a decoupling is typically not possible, and this poses a technical obstacle to the exploration of attractors in more general settings.

A natural re-framing of the problem, which applies regardless of whether a decoupling of the equations of motion can be brought about, is to recognise attractors as submanifolds of phase space where generic solutions congregate. This perspective is general enough to accommodate many types of attractor phenomena. In this context, Ref. [20] introduced a local notion of attractor that avoids referencing any asymptotic limits — as dimensionality reduction of phase space regions. It was shown there that for conformal MIS theory generic solutions converge to a region of phase space coinciding with a hypersurface determined from the known attractor solution. This raised the question whether the attractor hypersurface could be found directly without appealing to an attractor solution in the sense of Ref. [5]. Recently, this question was resolved in Ref. [19], where it was shown that the attractor hypersurface of conformal MIS theory in phase space can be found directly using a regularity condition at asymptotically small proper times.

The remaining piece of the puzzle is to clarify the relationship between the attractor hypersurface determined in this way and the concept of an attractor solution in the sense of Ref. [5]. We will resolve this issue by showing that these notions are identical in cases when an attractor solution is available. In brief, an explicit description of the attractor hypersurface can be obtained by means of its intersections with constant proper time slices. Generally, these sections evolve from slice to slice; however, in the special cases where the equations can be partially decoupled, they can be projected and the resulting curve coincides with the attractor solution. This is illustrated below, in sections II and III, by considering two models of MIS theory where this projection comes about in two rather different ways.

These developments lead to a broader question: can the singularity at early times be leveraged to determine the attractor hypersurface also in cases when the equations of motion cannot be partially decoupled? We suggest that the answer is affirmative. We provide evidence in support of this assertion in section IV, where we analyse a model derived from conformal MIS theory via a conformal symmetry-breaking modification of the equation of state. Although this model cannot be effectively treated using the original approach of Ref. [5], the regularity condition at early proper time unambiguously determines the attractor which captures the collective behaviour of generic solutions in a manner qualitatively similar to what is seen in conformal MIS theory.

We also point out that in all the models discussed here, an analytic description of the attractor sections on constant proper time slices can be obtained in parametric form. This comes about by reinterpreting the transseries solutions in proper time as functions of the initial data. In particular, in examples where an attractor solution ${\cal A}(w)$ exists, one can formally recover its transseries form from the transseries expansion in proper time.

II Conformal MIS theory

In this section we describe how the attractor hypersurface of conformal MIS theory is determined, with an emphasis on the role of early time asymptotics. We will also show how this surface can be projected when the special coordinates ${\cal A},w$ are used to parameterise the phase space, thus resolving the question concerning the relationship of the attractor solution to the attractor hypersurface.

MIS theory is expressed in terms of the classical fields ${\cal E}$ (the energy density), $u^{\mu}$ (the flow velocity) and $\pi^{\mu\nu}$ (the shear-stress tensor). They satisfy the following set of partial differential equations

[TABLE]

where $\nabla_{\mu}$ is the covariant derivative, $\Delta_{\mu\nu}\equiv g_{\mu\nu}+u_{\mu}u_{\nu}$ is the transverse projector, $\sigma_{\mu\nu}=\frac{1}{2}\Delta_{\mu\alpha}\Delta_{\nu\beta}(\nabla^{\alpha}u^{\beta}+\nabla^{\beta}u^{\alpha}-\frac{2}{3}\Delta^{\alpha\beta}\nabla\cdot u)$ is the shear tensor, $\eta$ is the shear viscosity and $\tau_{\pi}$ is the relaxation time. The first two of the above equations express conservation of the energy-momentum tensor, while the last is the MIS equation which determines the evolution of the shear-stress tensor. Throughout this paper we set the bulk viscosity as well as the bulk pressure to zero. In this section we will also adopt equations of state and transport coefficients dictated by conformal invariance:

[TABLE]

where $C_{\eta}$, $C_{\pi}$ and $a$ are dimensionless constants and $T$ is the effective temperature.

With the above assumptions, denoting lab frame time by $t$ and the coordinate along the collision axis by $z$, the energy-momentum tensor for Bjorken flow can be parameterised in terms of two functions of the proper time $\tau\equiv\sqrt{t^{2}-z^{2}}$: the effective temperature $T$, and the pressure anisotropy ${\cal A}\equiv 9\pi^{k}_{k}/2{\cal E}$. The physical meaning of the pressure anisotropy is manifest if one expresses it in terms of eigenvalues of the energy-momentum tensor $(T_{\mu}^{\nu})\equiv\mathrm{diag}(-{\cal E},p_{L},p_{T},p_{T})$ as ${\cal A}=(p_{T}-p_{L})/p$, (for details see, e.g. [21, 22]). The equations of motion given in Eqs. 1a, 1b and 1c can then be written in a form describing evolution in proper time as [19]

[TABLE]

This system of equations is not autonomous: in addition to $T,{\cal A}$ one also needs the value of $\tau$ to fully describe the state at a given instant. Formally, such non-autonomous systems can be equivalently presented in an autonomous form at the price of introducing an additional variable (see e.g.  [23]). The resulting extended phase space is thus effectively $3$-dimensional, parameterised by $(T,{\cal A},\tau)$. In what follows, it will be most convenient to think of this extended phase space as a set slices at constant values of the proper time. On each such slice at some specified value of $\tau$, a solution is represented by the point $(T(\tau),{\cal A}(\tau))$.

A crucial fact is that the system of evolution equations given in Eqs. 3a and 3b is singular at $\tau=0$, so solutions are defined only on the domain $\tau>0$. There are however two special families of solutions (each labelled by a single integration constant denoted below by $\mu$) for which the pressure anisotropy is regular at $\tau=0$. Denoting asymptotic equality by $\sim$, one finds that as $\tau\to 0$

[TABLE]

where

[TABLE]

and

[TABLE]

The family of solutions with ${\cal A}\sim{\cal A}_{+}$ as $\tau\to 0$ defines the attractor hypersurface in phase space. These initial conditions can be used to calculate series expansions that provide an analytic representation of the attractor at early time [19]. Since the extended phase space is $3$-dimensional here, sections of the attractor are curves in the $(T,{\cal A})$ planes at constant values of $\tau$. The behaviour of generic solutions is shown in Fig. 1.

One can also describe the attractor analytically in the hydrodynamic regime, where approximate asymptotic solutions are provided by the late proper time expansion (related to the hydrodynamic gradient expansion, see, e.g., Refs. [24, 25, 22]): Eqs. 3a and 3b imply the well-known asymptotic behaviour of generic solutions

[TABLE]

where $\Lambda$ is an integration constant which depends on the initial conditions. One usually views these relations as expressing the proper time dependence of a specific solution $(T(\tau),{\cal A}(\tau))$ with some value of $\Lambda$. Here we propose a change of perspective: to interpret these equations as a parametric representation of a curve $(T(\Lambda),{\cal A}(\Lambda))$ on the slice of phase space at a fixed value of $\tau$. This implies that on a proper time slice at a sufficiently large value of the proper time, generic solutions lie asymptotically close to this curve, which is a section of the hydrodynamic attractor on that particular time slice. Obviously, the shape of this curve changes from slice to slice. The value of $\Lambda$ determines which point on the attractor section corresponds to a specific solution (or, equivalently, a specific initial condition).

By including subleading terms, along with the correct transseries contributions [19], one can extend the validity of Eq. 7 to early times. Thus, at least in principle, sections of the attractor can be parametrically determined on any time slice. However, this is not a practical method at early times, because it is not known a priori which value of the transseries parameter singles out the attractor [5, 26]. In practice, one can determine the attractor locus numerically by imposing the initial condition ${\cal A}(\tau_{0})={\cal A}_{+}$ along with initial conditions for $T(\tau_{0})$ spanning a range of temperatures at some very small proper time $\tau_{0}>0$. In this way, a set of solutions is obtained which threads the attractor surface in phase space. One can then determine the attractor section on any slice by evaluating these solutions on that slice and interpolating – this is how the red curves in Fig. 1 were determined. It is clear that the attractor sections evolve from slice to slice. Fig. 1 also demonstrates the three distinct stages of evolution noted in [20] (see also Ref. [19]). The first stage is a rapid approach to the nonequilibrium part of the attractor which can be attributed to the longitudinal expansion; the second stage describes descent onto the attractor and is characterised by the decay of exponentially damped nonhydrodynamic degrees of freedom; and then the third, hydrodynamic stage, encompassing evolution along the attractor which is captured by the gradient expansion. It is also clear from Fig. 1 that on a given proper time slice solutions with higher temperature are closer to the attractor than solutions with smaller temperatures.

We now turn to an important issue raised implicitly by Ref. [19]: what is the precise connection between the attractor hypersurface in phase space and the attractor solution in the sense of Ref. [5]? We will show that these two notions coincide, and they do so at all times. To make this connection, we will now use the dimensionless variable $w\equiv\tau T$ as one of the extended phase space coordinates instead of $T$. With this choice, the slice at each value of $\tau$ is parameterised by $(w,{\cal A})$. It is simplest to begin with the hydrodynamic regime, where Eq. 7 can be reinterpreted as representing the attractor section on a constant $\tau$ slice in the $(w,{\cal A})$ variables:

[TABLE]

These relations define a curve on this slice, expressed in parametric form, with $\Lambda$ playing the role of a parameter along it, labelling different solutions in this regime. By eliminating the parameter $\Lambda$, one recognises this curve as the well-known Navier-Stokes result ${\cal A}\sim 8C_{\eta}/w$ (see e.g. Ref. [24]). In conformal models $\Lambda$ always enters in the combination $\Lambda\tau$, so when it is eliminated the resulting curve does not depend on $\tau$: it is the same across all slices in the late proper time regime where Eq. 8 is valid.

The above observation can easily be extended to all times, because it remains true also when the full transseries representation of the solution is used instead of just the leading terms given in Eq. 8:

[TABLE]

where $\Phi_{k}(\Lambda\tau),\Psi_{k}(\Lambda\tau)$ are power series in $\Lambda\tau$, with $\Lambda$-independent coefficients determined by the equations of motion, $\sigma$ is the transseries parameter and $\tilde{\sigma}$ is related to it. Formally, eliminating the parameter $\Lambda$ order by order leads to a curve which is independent of $\tau$. This means that when the phase space is parameterised using $(w,{\cal A},\tau)$, the attractor section is identical across all proper time slices. This conclusion can be corroborated numerically, and is illustrated in Fig. 2, which plots exactly the same data as Fig. 1, but using the $(w,{\cal A},\tau)$ parameterisation in place of $(T,{\cal A},\tau)$.

Since sections of the attractor hypersurface are the same on each slice, they may be projected to the $(w,{\cal A})$ plane. A hypersurface with this property is sometimes called a cylindrical surface (or a generalised cylinder, see e.g. Ref. [27]). This projected curve can be interpreted as a functional dependence ${\cal A}(w)$ and coincides with the attractor solution in the sense of Ref.[5]. Thus, one can in principle recover the attractor solution ${\cal A}(w)$ analytically — as a transseries — by eliminating the parameter $\Lambda$ order by order in the transseries expansion given in Eqs. 9a and 9b.

It is clear that this projection comes about due to the fact that the equations of motion in the $(w,{\cal A},\tau)$ variables can be partially decoupled, in the sense that ${\cal A}(w)$ satisfies [5, 24]

[TABLE]

while $w(\tau)$ can subsequently be determined from

[TABLE]

This is convenient, but from the phase space perspective it does not offer a significant advantage, since the attractor hypersurface can also be identified in other parameterisations.

III The Denicol-Noronha model

In this section we briefly discuss another case when the evolution equations partially decouple: this is the model introduced by Denicol and Noronha, who found its attractor solution analytically [28] 111I am grateful to Gabriel Denicol for emphasising the significance of this example.. This model is defined by the MIS equations given in Eqs. 1a, 1b and 1c, with the same equations of state as in Eq. 2, but with the transport coefficients taken as

[TABLE]

Here $m$ is a mass scale and $C_{\eta}$ and $C_{\pi}$ are dimensionless constants.

In this model, the MIS equations of motion for Bjorken flow, Eqs. 3a and 3b, take the form

[TABLE]

The MIS equation, Eq. 13b, is decoupled from the conservation equation, Eq. 13a, without the need to introduce a new independent variable (such as $w$ in conformal MIS theory). This example is especially instructive, since the general solution of this equation is known. However, for the present discussion, the essential point is that the equations of motion, Eqs. 13a and 13b, are singular at $\tau=0$. While solutions exist only for $\tau>0$, there are two special solutions of Eq. 13b for which the pressure anisotropy is regular at $\tau=0$, with the early time behaviour ${\cal A}\sim{\cal A}_{\pm}$, where ${\cal A}_{\pm}$ is given in Eq. 5. The solution with the upper sign acts as an attractor.

Here we reinterpret this attractor from the phase space perspective. Since the equations of motion Eqs. 13a and 13b are non-autonomous, the phase space is extended and parameterised by $(T,{\cal A},\tau)$. The attractor can be described in full analogy with the discussion in section II: one can determine a set of solutions threading the attractor hypersurface by solving the equations of motion numerically, setting initial conditions consistent with the condition of regularity of the pressure anisotropy at $\tau=0$. The results of this are displayed in Fig. 3, which shows sections of the attractor along with a set of generic solutions on four slices at fixed proper times. The attractor is evolving from slice to slice, but since it is constant in $T$, it can be projected to the $(\tau,{\cal A})$ plane. The projection coincides with the attractor solution of Eq. 13b.

IV A model of nonconformal MIS theory

The previous sections have demonstrated that in two models where the equations of motion can be partially decoupled one can determine the full attractor hypersurface in phase space without appealing to the attractor solution. The only essential feature of the dynamics which had led to early time attraction was the singularity at early time. This suggests that this may be a more general feature: that as long as the evolution equations are singular at early times, the attractor hypersurface in phase space can be determined using regularity conditions at $\tau=0$. This section provides some evidence for this claim: we will consider a model of MIS theory where the equations of motion can no longer be partially decoupled and thus one cannot find an attractor solution in the sense of Ref. [5]. However, since the evolution equations are singular at $\tau=0$, the final outcome is essentially the same: there is an attractor hypersurface in phase space where generic solutions congregate in the course of evolution.

We consider MIS theory with an equation of state which is conformal at high temperatures but introduces a mass scale, denoted by $m$, that parameterises the departure from conformality. This approach takes inspiration from parameterisations of the QCD equation of state proposed to emulate the results of lattice computations (see e.g. Ref. [29]), but is simple enough to allow some analytic calculations. We will also neglect bulk viscosity, which is known to be very small at high temperature, rising sharply only close to the transition temperature. This simplification will be taken a step further by neglecting the bulk pressure entirely, which leads to the technical convenience that the extended phase space remains $3$-dimensional so it is very easy to compare with conformal MIS theory. At temperatures above the chiral transition, the resulting model captures some of the physics at a qualitative level.

The model equation of state we adopt reads:

[TABLE]

where $a,m$ are constant parameters. When $T\gg m$, at leading order one recovers the conformal result $p\sim aT^{4}/3$, with the subleading correction of a form appropriate for a gas of massive particles. Standard thermodynamic relations imply that the entropy density $s$ and energy density ${\cal E}$ are given by

[TABLE]

The energy momentum tensor in this case acquires a non-zero trace. The mass scale $m$ happens to coincide with the position of the maximum of the normalised trace anomaly

[TABLE]

In the following, when numerical values are needed, we set $m=200$ MeV and take $a=8\pi^{2}/15$, which is the value appropriate for pure $SU(3)$ Yang-Mills theory in the limit of high temperature. The speed of sound is

[TABLE]

Its asymptotic behaviour at $T\gg m$ and $T\ll m$ is

[TABLE]

To study boost invariant flow with the nonconformal equation of state given in Eq. 14 it is useful to start with the ideal fluid. Imposing the symmetries of Bjorken flow, conservation of the energy momentum tensor implies

[TABLE]

Unlike the ideal fluid evolution equation for the conformal case, this equation does not have an exact power law solution, but it can be directly integrated. At late time

[TABLE]

where $\Lambda$ is an integration constant. The leading asymptotic behaviour reflects the limiting value of the speed of sound at low temperature, Eq. 17. All powers of the proper time appear in the above series, but the entropy density falls off as $1/\tau$ exactly, which expresses the conservation of entropy in ideal fluid flow.

We now turn to the MIS evolution equations for the model defined by Eq. 14. For simplicity, we will keep the transport coefficients the same as in the conformal case, given in Eq. 2. Proceeding in this way one obtains

[TABLE]

where the $R_{k},S_{k}$ are rational functions of the effective temperature:

[TABLE]

They all equal unity when $m$ is set to zero – in that case Eqs. 21a and 21b reduce to Eqs. 3a and 3b. Importantly, this system of equations is singular at $\tau=0$.

As in the conformal case, Eqs. 21a and 21b imply a second order differential equation for $T$ alone (or ${\cal A}$ alone), which can be convenient but not essential for the derivation of asymptotic solutions. At late times, $\tau\to\infty$, one finds the asymptotic behaviour

[TABLE]

The dependence on the relaxation time enters at one order higher than in the conformal case due to the subleading ideal fluid contribution in Eq. 20. Only one integration constant, $\Lambda$, appears in this series. The other integration constant, denoted below by $\sigma$, enters as an exponentially-suppressed transseries correction to the above power-law asymptotics:

[TABLE]

where

[TABLE]

and $\Phi_{k}(\Lambda\tau,\Lambda)$, $\Psi_{k}(\Lambda\tau,\Lambda)$ are power series in $\Lambda\tau$ with coefficients that depend on $\Lambda$. This is in contrast to the case of conformal MIS theory, where the corresponding coefficients appearing in Eqs. 9a and 9b are universal for all solutions. The constant $\sigma$ is the transseries parameter and $\tilde{\sigma}$ can be expressed in terms of $\sigma$. The leading terms in $\Phi_{0}(\Lambda\tau,\Lambda)$ and $\Psi_{0}(\Lambda\tau,\Lambda)$ are given in Eqs. 23a and 23b; coefficients of the remaining series can be calculated using the equations of motion. While this transseries has some novel elements as compared to conformal MIS, the basic picture of hydrodynamisation does not differ from that discussed in section II. It follows the pattern familiar from earlier studies [5, 30, 31, 26, 19]: a part of the data contained in the initial conditions enters via exponentially-suppressed terms which are nonperturbative from the point of view of classical asymptotics, that is, which vanish faster than any power of the proper time.

At late times, Eq. 23 can be interpreted as a parametric representation of the section of the attractor on a slice of fixed $\tau$, with $\Lambda$ as the parameter along this curve. Clearly, the result of eliminating this parameter will depend on the proper time slice, so the attractor sections evolve in time. This argument can also be applied to earlier times by using the transseries given in Eq. 24. To assess whether the attractor extends into the far from equilibrium domain, we proceed numerically as in the previous sections. The asymptotic behaviour of solutions at $\tau=0$ leads to the conclusion that solutions for which the pressure anisotropy remains finite satisfy the same initial conditions as in the conformal case, ${\cal A}\sim{\cal A}_{+}$, where ${\cal A}_{+}$ is given in Eq. 5. This way one obtains the set of plots shown in Fig. 4. Both here and in Fig. 1 we have used the $(T,{\cal A},\tau)$ parameterisation of the extended phase space so one can easily compare the evolution of the attractor section to its behaviour in conformal MIS theory. The shape of the attractor here is different, but the role it plays in the dynamics of the system is the same: generic solutions first coalesce on the attractor and then evolve along it. In this model there is no parameterisation in which the attractor sections would be the same on each slice. Despite this, there is no doubt when comparing Fig. 4 and Fig. 1 that generic solutions follow the attractor in a very similar way. While these results cannot be taken as indication of what will be found in systems with phase spaces of higher dimension, such as those of Refs. [32, 33, 34, 35, 36, 37, 38], the same approach can be applied.

V Summary and outlook

The attractor of conformal MIS theory was originally defined as a specific solution to a first order ordinary differential equation satisfied by the pressure anisotropy ${\cal A}(w)$. The partial decoupling of evolution equations which is crucial for such an attractor solution to exist is usually not attainable, so in order to address more general situations it is useful to adopt a different perspective, recognising the attractor as a hypersurface in an extended phase space. Two different approaches to determining this manifold have been proposed: Ref. [20] introduced a local notion of attractor that is not connected to any asymptotic limits, while in Ref. [19] the attractor hypersurface was defined by imposing regularity of the pressure anisotropy at asymptotically small proper time. These two approaches should be compatible, and we have illustrated this in two cases where an attractor solution is known to exist: conformal MIS theory and the Denicol-Noronha model. We have demonstrated that in these cases, in the appropriate coordinates, the projection of the attractor hypersurface obtained using regularity coincides with the attractor solution in the sense of Ref. [5], and it does so at all times. This we showed formally by reinterpreting the transseries representation of the attractor, as well as numerically by observing that attractor sections in these cases are independent of the proper time and thus can be projected.

We took these developments a step further by proposing that in situations where the evolution equations are singular at early time, regularity conditions should uniquely determine the attractor hypersurface regardless of whether the evolution equations can be partially decoupled or not. To illustrate the efficacy of this prescription, we have considered a model of MIS theory where the evolution equations do not decouple, but regularity at early time still determines a unique attractor hypersurface and generic solutions evolve toward it already at high pressure anisotropy: the essential feature which leads to early-time attraction is the singularity at $\tau=0$. Importantly, this approach works directly with the equations describing evolution in proper time, without requiring any special coordinates. The basic picture of hydrodynamisation remains the same: it begins with a rapid approach to the attractor followed by exponential, nonhydrodynamic mode decay (captured by a transseries containing contributions carrying nonhydrodynamic data present in the initial conditions) and then finally a hydrodynamic approach to equilibrium [20].

There are several possible applications of the approach advocated here. Perhaps the simplest one would be to reconsider the fluid-dynamical description of the massive relativistic gas [32, 33, 34, 35] from this perspective. A little further afield is the treatment of nonconformal kinetic theory models [37, 36, 38]. It may also be interesting to consider the impact of a possible singularity at early time on the hydrodynamic attractors that were recently described in the context of a Fermi gas close to unitarity with a time-dependent scattering length [39, 40, 41]. In this setting, early-time attraction may appear because the system is externally driven in a specific manner. This drive could lead to consequences similar to the boost invariance of particle production in QCD and could result in a singularity of the equations of motion at early time.

From a broader perspective, universal features in the dynamics of nonequilibrium systems can arise in various ways. The appearance of attractors in boost-invariant flow is linked to the strong longitudinal expansion and the ensuing singularity at early proper time, resulting in the emergence of prehydrodynamic universality in the sense of many different initial states being dynamically driven to a specific region in phase space. There are also other examples of universal far from equilibrium behaviour, such as nonthermal fixed points [42, 43, 44, 45], which arise for special classes of initial conditions. It would be of great interest to understand how these phenomena relate to attractors of the type discussed here, particularly in the context of heavy ion collisions.

Acknowledgements.

I would like to thank Jean-Paul Blaizot, Gabriel Denicol, Tuomas Lappi and Derek Teaney for helpful conversations. This research was supported by the National Science Centre, Poland, under Grant No. 2021/41/B/ST2/02909.